CHAPTER IV DEFINITIONS OF HIGHER K-THEORY

The higher algebraic K-groups of a ring R are defined to be the homotopy groups K_{n}(R) = π_{n}K(R) of a certain topological space K(R), which we shall construct in this chapter. Of course, the space K(R) is rigged so that if n = 0, 1, 2 then π_{n}K(R) agrees with the groups K_{n}(R) constructed in chapters II and III.

We shall also define the higher K-theory of a category A in each of the three settings where K_{0}(A) was defined in chapter II: when A is a symmetric monoidal category (§3), an exact category (§4) and a Waldhausen category (§6). In each case we build a “K-theory space” KA and define the group K_{n}A to be its homotopy groups: K_{n}A = π_{n}KA. Of course the group π_{0}KA will agree with the corresponding group K_{0}A defined in chapter II.

We will show these definitions of K_{n}A coincide whenever they coincide for K_{0}. For example, the group K_{0}(R) of a ring R was defined in §II.2 as K_{0} of the category P(R) of f.g. projective R-modules, but to define K_{0}P(R) we could also regard the category P(R) as being either a symmetric monoidal category (II.5.2), an exact category (II.7.1) or a Waldhausen category (II.9.1.3). We will show that the different constructions give homotopy equivalent spaces KP(R), and hence the same homotopy groups. Thus the groups K_{n}(R) = π_{n}KP(R) will be independent of the construction used.

Many readers will not be interested in the topological details, so we have designed this chapter to allow “surfing.” Since the most non-technical way to construct K(R) is to use the “+”-construction, we will do this in §1 below.

In §2, we summarize the basic facts about the geometric realization BC of a category C, and the basic connection between category theory and homotopy theory needed for the rest of the constructions. Indeed, the K-theory space KA is constructed in each setting using the geometric realization BC of some category C, concocted out of A. For this, we assume only that the reader has a slight familiarity with cell complexes, or CW complexes, which are spaces obtained by successive attachment of cells, with the weak topology.

Sections 3–6 give the construction of the K-theory spaces. Thus in §3 we have group completion constructions for symmetric monoidal categories (and the connection to the +-construction). Quillen’s Q-construction for abelian and exact categories is given in §4; in §5 we prove the “+ = Q” theorem, that the Q-construction and group completion constructions agree for split exact categories (II.7.1.2). The wS_{·} construction for Waldhausen categories is in §6, along with its connection to the Q-construction.

We conclude with a short section (§7) on homotopy groups with finite coefficients.

Typeset by AMS-TEX

2 IV. DEFINITIONS OF HIGHER K-THEORY

These have shown to be remarkably useful in describing the structure of the groups K_{n}(A), especially as related to étale cohomology.

§1. The BGL^{+} definition for Rings

Let R be an associative ring with unit. Recall from chapter III that the infinite general linear group GL(R) is the union of the finite groups GL_{n}(R), and that its commutator subgroup is the perfect group E(R) generated by the elementary matrices e_{ij}(r). Moreover the group K_{1}(R) is defined to be the quotient GL(R)/E(R).

In 1969, Quillen proposed defining the higher K-theory of a ring R to be the homotopy groups of a certain topological space, which he called “BGL(R)^{+}.” Before describing the elementary properties of Quillen’s construction, and the related subject of acyclic maps, we present Quillen’s description of BGL(R)^{+} and define the groups K_{n}(R) for n ≥ 1.

For any group G, we can naturally construct a connected topological space BG whose fundamental group is G, but whose higher homotopy groups are zero. Moreover, the homology of the topological space BG coincides with the algebraic homology of the group G. Details of this construction are in §2 below. For G = GL(R) we obtain the space BGL(R), which is central to the following definition.

Definition 1.1. The notation BGL(R)^{+} will denote any CW complex X which has a distinguished map BGL(R) → BGL(R)^{+} such that

(1) π_{1}BGL(R)^{+} = K_{1}(R), and the natural map from GL(R) = π_{1}BGL(R) to π_{1}BGL(R)^{+} is onto with kernel E(R);

(2) H_{∗}(BGL(R); M) ⟶^{≅} H_{∗}(BGL(R)^{+}; M) for every GL(R)-module M. We will sometimes say that X is a model for BGL(R)^{+}.

For n ≥ 1, K_{n}(R) is defined to be the homotopy group π_{n}BGL(R)^{+}.

By Theorem 1.4 below, any two models are homotopy equivalent, i.e., the space BGL(R)^{+} is uniquely defined up to homotopy. Hence the homotopy groups K_{n}(R) of BGL(R)^{+} are well-defined.

By construction, K_{1}(R) agrees with the group K_{1}(R) = GL(R)/E(R) defined in chapter III. We will see in 1.6.1 below that K_{2}(R) = π_{2}BGL^{+}(R) agrees with the group K_{2}(R) defined in chapter III.

Several different models for BGL(R)^{+} are described in 1.8 below. We will construct even more models for BGL(R)^{+} in the rest of this chapter: the space P^{−1}P(R) of §3, the space ΩBQP(R) of §4 and the space Ω(iso S_{·}S) arising from the Waldhausen construction in §6.

Definition 1.1.1. Write K(R) for the product K_{0}(R) × BGL(R)^{+}. That is, K(R) is the disjoint union of copies of the connected space BGL(R)^{+}, one for each element of K_{0}(R). By construction, K_{0}(R) = π_{0}K(R). Moreover, it is clear that π_{n}K(R) = π_{n}BGL(R)^{+} = K_{n}(R) for n ≥ 1.

Functoriality 1.1.2. Each K_{n} is a functor from rings to abelian groups, while the topological spaces BGL(R)^{+} and K(R) are functors from rings to the homotopy category of topological spaces. However, without more information about the models used, the topological maps BGL(R)^{+} → BGL(R′)^{+} are only well-defined up to homotopy.

To see this, note that any ring map R → R′ induces a natural group map GL(R) → GL(R′), and hence a natural map BGL(R) → BGL(R′). This induces a map BGL(R)^{+} → BGL(R′)^{+}, unique up to homotopy, by Theorem 1.4 below. Thus the group maps K_{n}(R) → K_{n}(R′) are well defined. Since the identity of R induces the identity on BGL(R)^{+}, only composition remains to be considered. Given a second map R′ → R′′, the composition BGL(R) → BGL(R′) → BGL(R′′) is induced by R → R′′ because BGL is natural. By uniqueness in Theorem 1.4, the composition BGL(R)^{+} → BGL(R′)^{+} → BGL(R′′)^{+} must be homotopy equivalent to any a priori map BGL(R)^{+} → BGL(R′′)^{+}.

It is possible to make K(R) = K_{0}(R)×BGL(R)^{+} into a homotopy-commutative H-space in a functorial way, using other constructions (see 3.8.1). As an H-space, K(R) is not the product of the H-space BGL(R)^{+} and the discrete group K_{0}(R). (See 1.1.4.)

Homotopy Fiber 1.1.3. The maps π_{∗}X → π_{∗}Y induced by a continuous map X ⟶^{f} Y can always be made to fit into a long exact sequence, in a natural way. The homotopy fiber F (f) of a f, relative to a basepoint ∗_{Y} of Y, is the space of pairs (x, γ), where x ∈ X and γ: [0, 1] → Y is a path in Y starting at the baspoint γ(0) = ∗_{Y}, and ending at γ(1) = f(x). The key property of the homotopy fiber is that (given a basepoint ∗_{X} with f(∗_{X}) = ∗_{Y}) there is a long exact sequence of homotopy groups/pointed sets

⋯ π_{n+1}Y ⟶^{∂}π_{n}F (f) → π_{n}X → π_{n}Y ⟶^{∂} π_{n−1}F (f) → ⋯
⋯ ⟶^{∂}π_{1}F (f) → π_{1}X → π_{1}Y ⟶^{∂} π_{0}F (f) → π_{0}X → π_{0}Y.

When X → Y is an H-map of H-spaces, F (f) is also an H-space, and the maps ending the sequence are product-preserving.

Relative groups 1.1.4. Given a ring homomorphism f: R → R′, let K(f) be the homotopy fiber of K(R) → K(R′), and set K_{n}(f) = π_{n}K(f). This construction is designed so that these relative groups fit into a long exact sequence:

⋯ K_{n+1}(R′) ⟶^{∂}K_{n}(f) → K_{n}(R) → K_{n}(R′) ⟶^{∂} ⋯
⋯ K_{1}(R, I) → K_{1}(R) → K_{1}(R′) ⟶^{∂}K_{0}(f) → K_{0}(R) → K_{0}(R′).

Using the functorial homotopy-commutative H-space structure on K(R), it follows that each K_{n}(f), including K_{0}(f), is an abelian group.

When R′ = R/I for some ideal I, we write K(R, I) for K(R → R/I). It is easy to see (Ex. 1.10) that K_{0}(R, I) and K_{1}(R, I) agree with the relative groups defined in Ex. II.2.3 and III.2.2, and that the ending of this sequence is the exact sequence of III, 2.3 and 5.7. Keune and Loday have shown that K_{2}(R, I) agrees with the defined in III.5.7.

Acyclic Spaces and Acyclic Maps

The definition of BGL(R)^{+} fits into the general framework of acyclic maps, which we now discuss. Our discussion of acyclicity is taken from [HH] and [Berrick].

Definition 1.2 (Acyclic spaces). We call a topological space F acyclic if it has the homology of a point, that is, if H̃_{∗}(F; ℤ) = 0.

Lemma 1.2.1. Let F be an acyclic space. Then F is connected, its fundamental group G = π_{1}(F) must be perfect, and H_{2}(G; ℤ) = 0 as well.

Proof. The acyclic space F must be connected, as H_{0}(F) = ℤ. Because G/[G, G] = H_{1}(F; ℤ) = 0, we have G = [G, G], i.e., G is a perfect group. To calculate H_{2}(G), observe that the universal covering space F̃ has H_{1}(F̃; ℤ) = 0. Moreover, the homotopy fiber of the canonical map F → BG is homotopy equivalent to F̃ (consider the long exact sequence of homotopy groups 1.1.3 to see this). The Serre Spectral Sequence for this homotopy fibration is E_{pq}^{2} = H_{p}(G; H_{q}(F̃; ℤ)) ⇒ H_{p+q}(F; ℤ) and the conclusion that H_{2}(G; ℤ) = 0 follows from the associated exact sequence of low degree terms:

H_{2}(F; ℤ) → H_{2}(G; ℤ) ⟶^{d^{2}} H_{1}(F̃; ℤ)^{G} → H_{1}(F; ℤ) → H_{1}(G; ℤ).

Example 1.2.2 (Volodin Spaces). The Volodin space X(R) is an acyclic subspace of BGL(R), constructed as follows. For each n, let T_{n}(R) denote the subgroup of GL_{n}(R) consisting of upper triangular matrices. As n varies, the union of these groups forms a subgroup T (R) of GL(R). Similarly we may regard the permutation groups Σ_{n} as subgroups of GL_{n}(R) by their representation as permutation matrices, and their union is the infinite permutation group Σ ⊂ GL(R). For each σ ∈ Σ_{n}, let T_{n}^{σ}(R) denote the subgroup of GL_{n}(R) obtained by conjugating T_{n}(R) by σ. For example, if σ = (n … 1) then T_{n}^{σ}(R) is the subgroup of lower triangular matrices.

Since the classifying spaces BT_{n}(R) and BT_{n}(R)^{σ} are subspaces of BGL_{n}(R), and hence of BGL(R), we may form their union over all n and σ: X(R) = ⋃_{n,σ} BT_{n}(R)^{σ}. The space X(R) is acyclic (see [Suslin]). Since X(R) was first described by Volodin in 1971, it is usually called the Volodin space of R.

The image of the map π_{1}X(R) → GL(R) = π_{1}BGL(R) is the group E(R). To see this, note that the maps BT_{n}^{σ}(R) → BGL(R) factor through X(R); applying π_{1} shows that the image of π_{1}(X) contains all the subgroups T_{n}^{σ}(R) of GL(R). Hence these subgroups contain the generators e_{ij}(r) of E(R). Since the image of the perfect group π_{1}X(R) is contained in the commutator subgroup E(R) of GL(R), the image must be E(R).

Definition 1.3 (Acyclic maps). Let X and Y be based connected CW complexes. A cellular map f: X → Y is called acyclic if the homotopy fiber F (f) of f is acyclic (has the homology of a point).

From the long exact sequence of homotopy groups/pointed sets, we see that if X → Y is acyclic, then the map π_{1}(X) → π_{1}(Y) is onto, and its kernel P is a perfect normal subgroup of π_{1}(X).

Definition 1.3.1. Let P be a perfect normal subgroup of π_{1}(X), where X is a based connected CW complex. An acyclic map f: X → Y is called a +–construction on X (relative to P) if P is the kernel of π_{1}(X) → π_{1}(Y).

When Quillen introduced this definition in 1969, he observed that both Y and the map f are determined up to homotopy by the subgroup P. This is the content of the following theorem; its proof uses topological obstruction theory. Part (a) is proven in Ex. 1.3; an explicit proof may be found in §5 of [Berrick].

Theorem 1.4 (Quillen). Let P be a perfect normal subgroup of π_{1}(X). Then (1) There is a +–construction f: X → Y relative to P

(2) Let f: X → Y be a +–construction relative to P, and g: X → Z a map such that P vanishes in π_{1}(Z). Then there is a map h: Y → Z, unique up to homotopy, such that g = hf.

(3) In particular, if g is another +–construction relative to P, then the map h in (2) is a homotopy equivalence: h: Y ⟶^{∼} Z.

Remark. Every group G has a unique largest perfect subgroup P, called the perfect radical of G; see Ex. 1.4. If no mention is made to the contrary, the notation X^{+} will always denote the +–construction relative to the perfect radical of π_{1}(X).

The first construction along these lines was announced by Quillen in 1969, so we have adopted Quillen’s term “+–construction” as well as his notation. A good description of his approach may be found in [HH] or [Berrick].

Lemma 1.5. A map f: X → Y is acyclic iff H_{∗}(X, M) ≅ H_{∗}(Y, M) for every π_{1}(Y)-module M.

Proof. Suppose first that f is acyclic, with homotopy fiber F (f). Since the map π_{1}F (f) → π_{1}Y is trivial, π_{1}F (f) acts trivially upon M. By the Universal Coefficient Theorem, H_{q}(F (q); M) = 0 for q ≠ 0 and H_{0}(F; M) = M. Therefore E_{pq}^{2} = 0 for q ≠ 0 in the Serre Spectral Sequence for f:

E_{pq}^{2} = H_{p}(Y; H_{q}(F (f); M)) ⇒ H_{p+q}(X; M).

Hence the spectral sequence collapses to yield H_{p}(X; M) ⟶^{≅} H_{p}(Y; M) for all p.

Conversely, we suppose first that π_{1}Y = 0 and H_{∗}(X; ℤ) ≅ H_{∗}(Y; ℤ). By the Comparison Theorem for the Serre Spectral Sequences for F (f) → X ⟶^{f} Y and ∗ → Y ⟶^{=} Y, we have H_{∗}(F (f); ℤ) = 0. Hence F (f) and f are acyclic.

The general case reduces to this by the following trick. Let Ỹ denote the universal covering space of Y, and X̃ = X ×_{Y} Ỹ the corresponding covering space of X. Then there are natural isomorphisms H_{∗}(Ỹ; ℤ) ≅ H_{∗}(Y; M) and H_{∗}(X̃; ℤ) ≅ H_{∗}(X; M), where M = ℤ[π_{1}(Y)]. The assumption that H_{∗}(X; M) ≅ H_{∗}(Y; M) implies that the map f̃: X̃ → Ỹ induces isomorphisms on integral homology. But π_{1}(Ỹ) = 0, so by the special case above the homotopy fiber F (f̃) of f̃ is an acyclic space. But by path lifting we have F (f̃) ≅ F (f), so F (f) is acyclic. Thus f is an acyclic map.

Recall from III.3.3.3 that every perfect group P has a universal central extension E → P, and that the kernel of this extension is the abelian group H_{2}(P; ℤ).

Proposition 1.6. Let P be a perfect normal subgroup of a group G, with corresponding +–construction f: BG → BG^{+}. If F (f) is the homotopy fiber of f then π_{1}F (f) is the universal central extension of P, and π_{2}(BG^{+}) ≅ H_{2}(P; ℤ).

Proof. We have an exact sequence π_{2}(BG) → π_{2}(BG^{+}) → π_{1}F (f) → G → G/P → 1. But π_{2}(BG) = 0, and π_{2}(BG^{+}) is in the center of π_{1}F (f) by [Wh, IV.3.5]. Thus π_{1}F (f) is a central extension of P with kernel π_{2}(BG^{+}). But F (f) is acyclic, so π_{1}F (f) is perfect and H_{2}(F; ℤ) = 0 by 1.2. By the Recognition Theorem III.3.4, π_{1}F (f) is the universal central extension of P.

Recall from Theorem III.3.5 that the Steinberg group St(R) is the universal central extension of the perfect group E(R). Thus we have:

Corollary 1.6.1. The group K_{2}(R) = π_{2}BGL(R)^{+} is isomorphic to the group K_{2}(R) ≅ H_{2}(E(R); ℤ) of chapter III.

In fact, we will see in Ex. 1.6 and 1.7 that K_{n}(R) ≅ π_{n}(BE(R)^{+} for all n ≥ 2, and K_{n}(R) ≅ π_{n}(BSt(R)^{+} for all n ≥ 3, with K_{3}(R) ≅ H_{3}(St(R); ℤ).

Corollary 1.6.2. The fundamental group π_{1}X(R) of the Volodin space (1.2.2) is the Steinberg group St(R).

Construction Techniques

One problem with the +construction approach is the fact that BGL(R)^{+} is not a uniquely defined space. It is not hard to see that BGL(R)^{+} is an H-space (see Ex. 1.9). Quillen proved that that it is also an infinite loop space. We omit the proof here, because it will follow from the + = Q theorem in section 5.

Here is one of the most useful recognition criteria, due to Quillen. The proof is is an application of obstruction theory, which we omit (but see [Ger72, 1.5].)

Theorem 1.7. The map i: BGL(R) → BGL(R)^{+} is universal for maps into H-spaces. That is, for each map f: BGL(R) → H, where H is an H-space, there is a map g: BGL(R)^{+} → H so that f = gi, and such that the induced map π_{i}(BGL(R)^{+}) → π_{i}(H) is independent of g.

Constructions 1.8. Here are some ways that BGL(R)^{+} may be constructed:

(i) Using point-set topology, e.g., by attaching 2-cells and 3-cells to BGL(R). This method is described Ex. 1.3, and in the books [Berrick] and [Rosenberg].

(ii) By the Bousfield-Kan integral completion functor ℤ_{∞}: we set BGL(R)^{+} = ℤ_{∞}BGL(R). This approach has the advantage of being absolutely functorial in R, and is used in [Dror] and [Gersten].

(iii) “Group completing” the H-space ∐_{n=0}^{∞} BGL_{n}(R) yields an infinite loop space whose basepoint component is BGL(R)^{+}. This method will be discussed more in section 3, and is due to G. Segal [Segal].

(iv) By taking BGL of a free simplicial ring F_{∗}(R), as in [Swan].

(v) Volodin’s construction. Let X(R) denote the acyclic Volodin space of Example 1.2.2. By Ex. 1.5, the quotient group BGL(R)/X(R) is a model for BGL(R)^{+}.

An excellent survey of these constructions (excluding Volodin’s) may be found in [Ger72].

We conclude this section with a description of Quillen’s construction for the K-theory of finite fields, arising from his work on the Adams Conjecture [Q70]. Adams had shown that the Adams operations ψ^{k} on topological K-theory (II.4.4) were represented by maps ψ^{k}: BU → BU in the sense that for each X the induced map

K̃U(X) = [X, BU] ⟶^{ψ^{k}} [X, BU] = K̃U(X)

is the Adams operation.

Fix a finite field 𝔽_{q} with q elements. For each n, the Brauer lifting of the trivial and standard n-dimensional representations of GL_{n}(𝔽_{q}) are n-dimensional complex representations, given by homomorphisms 1_{n}, ρ_{n}: GL_{n}(𝔽_{q}) → U. Since BU is an H-space, we can form the difference b^{n} = B(ρ_{n}) − B(1_{n}) as a map BGL_{n}(𝔽_{q}) → BU. Quillen observed that b^{n} and b^{n+1} are compatible up to homotopy with the inclusion of BGL_{n}(𝔽_{q}) in BGL_{n+1}(𝔽_{q}). Hence there is a map b: BGL(𝔽_{q}) → BU, well defined up to homotopy. By Theorem 1.7, b induces a map from BGL(𝔽_{q})^{+} to BU.

Theorem 1.9. (Quillen) The map BGL(𝔽_{q})^{+} → BU identifies BGL(𝔽_{q})^{+} with the homotopy fiber of ψ^{q} − 1. That is, the following is a homotopy fibration.

BGL(𝔽_{q})^{+} ⟶^{b} BU ⟶^{ψ^{q}−1} BU

On homotopy groups, II.4.4.1 shows that ψ^{q} is multiplication by q^{i} on π_{2i}BU = K̃U(S^{2i}). Using the homotopy sequence 1.1.3 we immediately deduce:

Corollary 1.9.1. For every finite field 𝔽_{q}, and n ≥ 1, we have

K_{n}(𝔽_{q}) = π_{n}BGL(𝔽_{q})^{+} ≅ \{ \begin{matrix}ℤ/(q^{i} − 1) & n = 2i − 1, \\ 0 & n even. \end{matrix}

Moreover, if 𝔽_{q} ⊂ 𝔽_{q′} then K_{n}(𝔽_{q}) ⊂ K_{n}(𝔽_{q′}) for all n.

EXERCISES

1.1. (Kervaire) Let X be a homology n-sphere, i.e., a space with H_{∗}(X) = H_{∗}(S^{n}). Show that there is a homotopy equivalence S^{n} → X^{+}. Hint: Show that π_{1}(X) is perfect if n ≠ 1, so X^{+} is simply connected, and use the Hurewicz theorem. 1.2. a) If F is acyclic space, show that F^{+} is contractible.

b) If X ⟶^{f} Y is acyclic and π_{1}(X) = π_{1}(Y), show that f is a homotopy equivalence. 1.3 Here is a point-set construction of X^{+} relative to a perfect normal subgroup P. Form Y by attaching one 2-cell e_{p} for each element of P, so that π_{1}(Y) = π_{1}(X)/P. Show that H_{2}(Y; ℤ) is the direct sum of H_{2}(X; ℤ) and the free abelian group ℤ[P] on the [e_{p}]. Next, prove that the homology classes [e_{p}] are represented by maps h_{p}: S^{2} → Y, and form Z by attaching 3-cells to Y (one for each p ∈ P) using the h_{p}. Finally, prove that Z is a model for X^{+}. 1.4 Perfect Radicals. Show that the union of two perfect subgroups of any group G is itself a perfect subgroup. Conclude that G has a largest perfect subgroup P, called the perfect radical of G, and that it is normal subgroup of G.

1.5 Let cone(i) denote the mapping cone of a map F ⟶^{i} X. If F is an acyclic space, show that the map X → cone(i) is acyclic. If F is a subcomplex of X then cone(i) is homotopy equivalent to the quotient space X/F, so it too is acyclic. Conclude that if X(R) is the Volodin space of Example 1.2.2 then BGL(R)/X(R) is a model for BGL(R)^{+}. Hint: Consider long exact sequences in homology.

1.6 Let P be a perfect normal subgroup of G. Show that BP^{+} is homotopy equivalent to the universal covering space of BG^{+}. Hence π_{n}(BP^{+}) ≅ π_{n}(BG^{+}) for all n ≥ 2. Hint: BE is homotopy equivalent to a covering space of BG.

For G = GL(R) and P = E(R), This shows that BE(R)^{+} is homotopy equivalent to the universal covering space of BGL(R)^{+}. Thus K_{n}(R) ≅ π_{n}BE(R)^{+} for n ≥ 2.

If R is a commutative ring, show that the group map SL(R) → GL(R) induces isomorphisms π_{n}BSL(R)^{+} ≅ K_{n}(R) for n ≥ 2, and that π_{1}BSL(R)^{+} ≅ SK_{1}(R). 1.7 Suppose that A → S → P is a universal central extension (III.5.3). In particular, S and P are perfect groups. Show that there is a homotopy fibration BA → BS^{+} → BP^{+}. Conclude that π_{n}(BS^{+}) = 0 for n ≤ 2, and that π_{n}(BS^{+}) ≅ π_{n}(BP^{+}) ≅ π_{n}(BG^{+}) for all n ≥ 3. In particular, π_{3}(BP^{+}) ≅ H_{3}(S; ℤ).

Since the Steinberg group St(R) is the universal central extension of E(R), this shows that K_{n}(R) ≅ π_{n}St(R)^{+} for all n ≥ 3, and that K_{3}(R) ≅ H_{3}(St(R); ℤ). 1.8 For n ≥ 3, let P_{n} denote the normal closure of the perfect group E_{n}(R) in GL_{n}(R), and let BGL_{n}(R)^{+} denote the +–construction on BGL_{n}(R) relative to P_{n}. Corresponding to the inclusions GL_{n} ⊂ GL_{n+1} we can choose a sequence of maps BGL_{n}(R)^{+} → BGL_{n+1}(R)^{+}. Show that \underrightarrow{lim} BGL_{n}(R)^{+} is BGL(R)^{+}.

1.9 For each m and n, the group map □: GL_{m}(R) × GL_{n}(R) → GL_{m+n}(R) ⊂ GL(R) induces a map BGL_{m}(R) × BGL_{n}(R) → BGL(R) → BGL(R)^{+}. Show that these maps induce an H-space structure on BGL(R)^{+}. 1.10 Let I be an ideal in a ring R. Show that π_{0}K(R → R/I) is isomorphic to the group K_{0}(I) defined in Ex. II.2.3, and that the maps K_{1}(R/I) → K_{0}(I) → K_{0}(R) in loc. cit. agree with those of 1.1.4. Hint: π_{0}K(R ⊕ I → R) must be K_{0}(I).

Use Ex. III.2.8 to show that π_{1}K(R → R/I) is isomorphic to the group K_{1}(R, I) of III.2.2, and that the maps K_{2}(R/I) → K_{1}(R, I) → K_{1}(R) in III.2.3 agree with those of 1.1.4.

§2. Geometric realization of a small category

Recall (II.6.1.3) that a “small” category is a category whose objects form a set. If C is a small category, its geometric realization BC is a CW complex constructed naturally out of C. By definition, BC is the geometric realization |NC| of the nerve NC of C; see 2.1.3 below. However, it is characterized in a simple way.

Characterization 2.1. The realization BC of a small category C is the CW complex uniquely characterized up to homeomorphism by the following properties:

(1) (Naturality) A functor F: C → D induces a cellular map BF: BC → BD, and BF ∘ BG = B(F G);

(2) Bn is the standard (n − 1)–simplex, where n denotes the category with n objects {0, 1, ⋯, n − 1}, with exactly one morphism i → j for each i ≤ j.

(3) The n-skeleton of BC is the colimit colim_{Φ} Bi, where Φ = Φ(n) is the par-

tially ordered set of all functors i → C, i ≤ n + 1 (ordered by factorization). The following useful properties are consequences of this characterization:

(1) If C is a subcategory of D, BC is a subcomplex of BD;

(2) If C is the coproduct of categories C_{α}, BC = ∐ BC_{α};

(3) B(C × D) is homeomorphic to (BC) × (BD);

Here are some useful special cases of (2) for small n:

B0 = ∅ is the empty set, because 0 is the empty category.

B1 = {0} is a one-point space, since 1 is the one object-one morphism category. B2 = [0, 1] is the unit interval, whose picture is: 0 · ⟶ · 1. B3 is the 2-simplex, whose picture is:

1_{·}

g ↗ ↘^{f}

0 · ⟶_{f∘g} · 2

The small categories form the objects of a category CAT, whose morphisms are functors. By (1), we see that geometric realization is a functor from CAT to the category of CW complexes and cellular maps.

Recipe 2.1.1. The above characterization of the CW complex BC gives it the following explicit cellular decomposition. The 0-cells (vertices) are the objects of C. The 1-cells (edges) are the morphisms in C, excluding all identity morphisms, and they are attached to their source and target. For each pair (f, g) of composable maps in C, attach a 2-simplex, using the above picture of B3 as the model. (Ignore pairs (f, g) where either f or g is an identity.) Inductively, given an n-tuple of composable maps in C (none an identity map), attach an n-simplex, using B(n + 1) as the model. By (2), BC is the union of these spaces, equipped with the weak topology.

Notice that this recipe implies a canonical cellular homeomorphism between BC and the realization BC^{op} of the opposite category C^{op}. In effect, the recipe doesn’t notice which way the arrows run.

Example 2.1.2. Let C_{2} be the category with one object and one nontrivial morphism σ satisfying σ^{2} = 1. The recipe tells us that BC_{2} has exactly one n-cell for each n, attached to the (n − 1)-cell by a map of degree 2 (corresponding to the first and last faces of the n-simplex). Therefore the n-skeleton of BC_{2} is the projective n-space ℝℙ^{n}, and their union BC_{2} is the infinite projective space ℝℙ^{∞}.

Although the above recipe gives an explicit description of the cell decomposition of BC, we were a bit vague about the attaching maps. To be more precise, we shall assume that the reader has a slight familiarity with the basic notions in the theory of simplicial sets, as found for example in [WHomo] or [May]. For example, a simplicial set X is a sequence of sets X_{0}, X_{1}, …, together with “face” maps ∂_{i}: X_{n} → X_{n−1} and “degeneracy maps” σ_{i}: X_{n} → X_{n+1} (0 ≤ i ≤ n), subject to certain identities for the compositions of these maps.

The above recipe for BC is broken down into two steps. First we constructs a simplicial set NC, called the nerve of the category C, and then we set BC = |NC|.

Definition 2.1.3 (The nerve of C). The nerve NC of a small category C is the simplicial set defined by the following data. Its n-simplices are functors c: n + 1 → C, i.e., diagrams in C of the form

c_{0} → c_{1} → ⋯ → c_{n}.

The i^{th} face ∂_{i}(c) of this simplex is obtained by deleting c_{i} in the evident way; to get the i^{th} degeneracy σ_{i}(c), one replaces c_{i} by c_{i} ⟶^{=} c_{i}.

The geometric realization |X_{·}| of a simplicial set X_{·} is defined to be the CW complex obtained by following the recipe 2.1.1 above, attaching an n-cell for each nondegenerate n-simplex x, identifying the boundary faces of the simplex with the (n − 1)-simplices indexed by the ∂_{i}x. See [WHomo, 8.1.6] or [May, §14] for more details.

BC is defined as the geometric realization |NC| of the nerve of C. From this prescription, it is clear that BC is given by recipe 2.1.1 above.

By abuse of notation, we will say that a category is contractible, or connected, or has any other topological property if its geometric realization has that property. Similarly, we will say that a functor F: C → D is a homotopy equivalence if BF is a homotopy equivalence BC ≃ BD.

Homotopy-theoretic properties 2.2. A natural transformation η: F_{0} ⇒ F_{1} between two functors F_{i}: C → D gives a homotopy BC × [0, 1] → BD between the maps BF_{0} and BF_{1}. This follows from (1) and (3), because η may be viewed as a functor from C × 2 to D whose restriction to C × {i} is F_{i}.

As a consequence, any adjoint pair of functors L: C → D, R: D → C induces a homotopy equivalence between BC and BD, because there are natural transformations LR ⇒ id_{D} and id_{C} ⇒ RL.

Example 2.2.1 (Skeleta). Any equivalence C_{0} ⟶^{F} C between small categories induces a homotopy equivalence BC_{0} ⟶^{∼} BC, because F has an adjoint.

In practice, we will often work with a category C, such as P(R) or M(R), which is not actually a small category, but which is skeletally small (II.6.1.3). This means that C is equivalent to a small category, say to C_{0}. In this case, we can use BC_{0} instead of the mythical BC, because any other choice for C_{0} will have a homotopy equivalent geometric realization. We shall usually overlook this fine set-theoretic point in practice, just as we did in defining K_{0} in chapter II.

Example 2.2.2 (Initial objects). Any category with an initial object is contractible, because then the natural functor C → 1 has a left adjoint. Similarly, any category with a terminal object is contractible.

For example, suppose given an object d of a category C. The comma category C/d of objects over d has as its objects the morphisms f: c → d in C with target d. A morphism in the comma category from f to f′: c′ → d is a morphism h: c → c′ so that f = f′h. The comma category C/d is contractible because it has a terminal object, namely the identity map id_{d}: d ⟶^{=} d. The dual comma category d\C with objects d → c is similar, and left to the reader.

Example 2.2.3 (Comma categories). Suppose given a functor F: C → D and an object d of D. The comma category F/d has as its objects all pairs (c, f) with c an object in C and f a morphism in D from F (c) to d. By abuse of notation, we shall write such objects as F (c) ⟶^{f} d. A morphism in F/d from this object to

F (c′) ⟶^{f′} d is a morphism h: c → c′ in C so that the following diagram commutes in D.

F (c) ⟶^{F (h)} F (c′)
f ↘  ↙_{f′}
d

There is a canonical forgetful functor j: F/d → C, j(c, f) = c, and there is a natural transformation η_{(c,f)} = f from the composite F ∘j: F/d → D to the constant functor with image d. So B(F ∘ j) is a contractible map. It follows that there is a natural continuous map from B(F/d) to the homotopy fiber of BC → BD.

There is a dual comma category d\F, whose objects are written as d ⟶ F (c), and morphisms are morphisms h: c → c′ in C. It also has a forgetful functor to C, and a map from B(d\F) to the homotopy fiber of BC → BD. In fact, d\F = (d/F^{op})^{op}.

The set π_{0} of components of a category

The set π_{0}(X) of connected components of any CW complex X can be described as the set of vertices modulo the incidence relation of edges. For BC this takes the following form. Let obj(C) denote the set of objects of C.

Lemma 2.3. Let ∼ be the equivalence relation on obj(C) which is generated by the relation that c ∼ c′ if there is a morphism in C between c and c′. Then

π_{0}(BC) = obj(C)/ ∼ .

Translation categories 2.3.1. Suppose that G is a group, or even a monoid, acting on a set X. The translation category G∫ X is defined as the category whose objects are the elements of X, with Hom(x, x′) = {g ∈ G|g·x = x′}. By Lemma 2.3, π_{0}(G∫ X) is the orbit space X/G. The components of G∫ X are described in Ex. 2.2.

Thinking of a G-set X as a functor G → CAT, the translation category becomes a special case of the following construction, due to Grothendieck.

Example 2.3.2. Let I be a small category. Given a functor X: I → Sets, let I∫ X denote the category of pairs (i, x) with i an object of I and x ∈ X(i), in which a morphism (i, x) → (i′, x′) is a morphism f: i → i′ in I with X(f)(x) = x′. By Lemma 2.3 we have π_{0}(I∫ X) = colim_{i∈I} X(i).

More generally, given a functor X: I → CAT, let I∫ X denote the category of pairs (i, x) with i an object of I and x an object of X(i), in which a morphism (f, ϕ): (i, x) → (i′, x′) is given by a morphism f: i → i′ in I and a morphism ϕ: X(f)(x) → x′ in X(i′). Using Lemma 2.3, it is not hard to show that π_{0}(I∫ X) = colim_{i∈I} π_{0}X(i).

The fundamental group π_{1} of a category

Suppose that T is a set of morphisms in a category C. The graph of T is the 1-dimensional subcomplex of BC consisting of the edges corresponding to T and their incident vertices. We say that T is a tree in C if its graph is contractible (i.e., a tree in the sense of graph theory). If C is connected then (by Zorn’s Lemma) a tree T is maximal (a maximal tree) just in case every object of C is either the source or target of a morphism in T. The following well-known formula for the fundamental group of BC is a straight-forward application of Van Kampen’s Theorem.

Lemma 2.4. Suppose that T is a maximal tree in a small connected category C. Then the group π_{1}(BC) has the following presentation: it is generated by symbols [f], one for every morphism in C, modulo the relations that

(1) [t] = 1 for every t ∈ T, and [id_{c}] = 1 for the identity morphism id_{c} of each object c.

(2) [f] · [g] = [f ∘ g] for every pair (f, g) of composable morphisms in C.

This presentation does not depend upon the choice of the object c_{0} of C chosen as the basepoint. Geometrically, the class of f: c_{1} → c_{2} is represented by the unique path in T from c_{0} to c_{1}, followed by the edge f, followed by the unique path in T from c_{2} back to c_{0}.

Application 2.4.1 (Groups). Let G be a group, considered as a category with one object. Since BG has only one vertex, BG is connected. By Lemma 2.4 (with T empty) we see that π_{1}(BG) = G. In fact, π_{i}(BG) = 0 for all i ≥ 2. (See Ex. 2.2.) BG is often called the classifying space of the group G, for reasons discussed in Examples 2.8.2 and 2.8.3 below.

Application 2.4.2 (Monoids). If M is a monoid then BM has only one vertex. This time, Lemma 2.4 shows that the group π = π_{1}(BM) is the group completion (Ex. II.1.1) of the monoid M.

For our purposes, one important thing about BG is that its homology is the same as the ordinary Eilenberg-MacLane homology of the group G (see [WHomo, 6.10.5 or 8.2.3]). In fact, if M is any G-module then we may consider M as a local coefficient system on BG (see 2.5.1). The chain complex used to form the simplicial homology with coefficients in M is the same as the canonical chain complex used to compute the homology of G, so we have H_{∗}(BG; M) = H_{∗}(G; M). As a special case, we have H_{1}(BG; ℤ) = H_{1}(G; ℤ) = G/[G, G], where [G, G] denotes the commutator subgroup of G, i.e., the subgroup of G generated by all commutators [g, h] = ghg^{−1}h^{−1} (g, h ∈ G).

(2.5) The homology of C and BC. The ith homology of a CW complex X such as BC is given by the homology of the cellular chain complex C_{∗}(X). By definition, C_{n}(X) is the free abelian group on the n-cells of X. If e is an n + 1-cell and f is an n-cell, then the coefficient of [f] in the boundary of [e] is the degree of the attaching map of e, followed by projection onto f: S^{n} → X^{(n)} ⟶^{f} S^{n}. attached

For example, H_{∗}(BC; ℤ) is the homology of the cellular chain complex C_{∗}(BC), which in degree n is the free abelian group on the set of all n-tuples (f_{1}, ..., f_{n}) of composable morphisms in C, composable in the order c_{n} ⟶^{f_{n}} ⋯ → c_{1} ⟶^{f_{1}} c_{0}. The boundary map in this complex sends the generator (f_{1}, ..., f_{n}) to the alternating sum obtained by succesively deleting the c_{i} in the evident way:

(f_{2}, ..., f_{n}) − (f_{1}f_{2}, f_{3}, ..., f_{n}) + ⋯ ± (..., f_{i}f_{i+1}, ...) ∓ ⋯ ± (..., f_{n−1}f_{n}) ∓ (..., f_{n−1}).

More generally, for each functor M: C → Ab we let H_{i}(C; M) denote the i^{th} homology of the chain complex

⋯ → ∐_{c_{n}→···→c_{0}} M(c_{n}) → ⋯ → ∐_{c_{1}→c_{0}} M(c_{1}) → ∐_{c_{0}} M(c_{0}).

The final boundary map sends the copy of M(c_{1}) indexed by c_{1} ⟶^{f} c_{0} to M(c_{0}) ⊕ M(c_{1}) by x ↦ (fx, −x). The cokernel of this map is the usual description for the colimit of the functor M, so H_{0}(C; M) = colim_{c∈C} M(c).

Local coefficients 2.5.1. A functor C → Sets is said to be morphisminverting if it carries all morphisms of C into isomorphisms. By Ex. 2.1, morphisminverting functors are in 1–1 correspondence with covering spaces of BC. Therefore the morphism-inverting functors M: C → Ab are in 1–1 correspondence with local coefficient systems on the topological space BC. In this case, the groups H_{i}(C; M) are canonically isomorphic to H_{i}(BC; M), the topologist’s homology groups of BC with local coefficients M. The isomorphism is given in [Wh, VI.4.8].

Homotopy Fibers of Functors

If F: C → D is a functor, it is useful to study the realization map BF: BC → BD in terms of homotopy groups, and for this we want a category-theoretic interpretation of the homotopy fiber (1.1.3). The naïve approximations to the homotopy fiber are the realization of the comma categories F/d and its dual d\F. Indeed, we saw in 2.2.3 that there are continuous maps from both B(F/d) and B(d\F) to the homotopy fiber.

Here is the fundamental theorem used to prove that two categories are homotopy equivalent. We cite it without proof from [Q341].

2.6 Quillen’s Theorem A. Let F: C → D be a functor such that F/d is contractible for every d in D. Then BF: BC ⟶^{≃} BD is a homotopy equivalence.

Example 2.6.1. If F: C → D has a right adjoint G, then F/d is isomorphic to the comma category C/G(d), which is contractible by Example 2.2.2. In this case, Quillen’s Theorem A recovers the observation in 2.2 that C and D are homotopy equivalent.

Example 2.6.2. Consider the inclusion of monoids i: ℕ ↪ ℤ as a functor between categories with one object ∗. Then ∗\i isomorphic to the translation category ℕ∫ ℤ, which is contractible (why?). Quillen’s Theorem A shows that Bℕ ≃ Bℤ ≃ S^{1}.

The inverse image F^{−1}(d) of an object d is the subcategory of C consisting of all objects c with F (c) = d, and all morphisms h in C mapping to the identity of d. It is isomorphic to the full subcategory of F/d consisting of pairs (c, F (c) ⟶^{=} d), and also to the full subcategory of pairs (d ⟶^{=} F (c), c) of d\F. It will usually not be homotopy equivalent to either entire comma category.

One way to ensure that F^{−1}(d) is homotopic to a comma category is to assume that F is either pre-fibered or pre-cofibered in the following sense.

Fibered and Cofibered functors 2.6.2. (Cf. [SGA 1, Exp. VI]) We say that a functor F: C → D is pre-fibered if for every d in D the inclusion F^{−1}(d) ↪ d\F has a right adjoint. This implies that BF^{−1}(d) ≃ B(d\F), and the base-change functor f^{∗}: F^{−1}(d′) → F^{−1}(d) associated to a morphism f: d → d′ in D is defined as the composite F^{−1}(d′) ↪ (d\F) → F^{−1}(d). F is called fibered if it is pre-fibered and g^{∗}f^{∗} = (fg)^{∗} for every pair of composable maps f, g, so that F^{−1} gives a contravariant functor from D to CAT.

Dually, we say that F is pre-cofibered if for every d the inclusion F^{−1}(d) ↪ F/d has a left adjoint. In this case we have BF^{−1}(d) ≃ B(F/d). The cobase-change functor f_{∗}: F^{−1}(d) → F^{−1}(d′) associated to a morphism f: d → d′ in D is defined as the composite F^{−1}(d) ↪ (F/d′) → F^{−1}(d′). F is called cofibered if it is precofibered and (fg)_{∗} = f_{∗}g_{∗} for every pair of composable maps f, g, so that F^{−1} gives a covariant functor from D to CAT.

These notions allow us to state a variation on Quillen’s Theorem A.

Corollary 2.6.3. Suppose that F : C → D is either pre-fibered or pre-cofibered, and that F^{−1}(d) is contractible for each d in D. Then BF is a homotopy equivalence BC ≃ BD.

Example 2.6.4. Cofibered functors over D are in 1–1 correspondence with functors D → CAT. We have already mentioned one direction: if F: C → D is cofibered, F^{−1} is a functor from D to CAT. Conversely, for each functor X: D → CAT, the category D∫ X of Example 2.3.2 is cofibered over D by the forgetful functor (d, x) ↦ d. It is easy to check that these are inverses: C is equivalent to D∫ F^{−1}.

Here is the fundamental theorem use to construct homotopy fibration sequences of categories. We cite it without proof from [Q341], noting that it has a dual formulation in which F/d is replaced by d\F; see Ex. 2.6.

2.7 Quillen’s Theorem B. Let F: C → D be a functor such that for every morphism d → d′ in D the induced functor F/d → F/d′ is a homotopy equivalence. Then for each d in D the geometric realization of the sequence

F/d ⟶^{j} C ⟶^{F} D

is a homotopy fibration sequence. Thus there is a long exact sequence

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}B(F/d) ⟶^{j} π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

Corollary 2.7.1. Suppose that F is pre-fibered, and for every f: d → d′ in D the base-change f^{∗} is a homotopy equivalence. Then for each d in D the geometric realization of the sequence

F^{−1}(d) ⟶^{j} C ⟶^{F} D

is a homotopy fibration sequence. Thus there is a long exact sequence

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}BF^{−1}(d) ⟶^{j} π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

Topological Categories 2.8. If C = C^{top} is a topological category (i.e., the object and morphism sets form topological spaces), then the nerve of C^{top} is a simplicial topological space. Using the appropriate geometric realization of simplicial spaces, we can form the topological space BC^{top} = |NC^{top}|. It has the same underlying set as our previous realization BC^{δ} (the δ standing for “discrete,” i.e., no topology), but the topology of BC^{top} is more intricate. Since the identity may be viewed as a continuous functor C^{δ} → C^{top} between topological categories, it induces a continuous map BC^{δ} → BC^{top}.

For example, any topological group G = G^{top} is a topological category, so we need to distinguish between the two connected spaces BG^{δ} and BG^{top}. It is traditional to write BG for BG^{top}, reserving the notation BG^{δ} for the less structured space. As noted above, BG^{δ} has only one nonzero homotopy group: π_{1}(BG^{δ}) = G^{δ}. In contrast, the loop space Ω(BG^{top}) is G^{top}, so π_{i}BG^{top} = π_{i−1}G^{top} for i > 0.

Example 2.8.1. Let G = ℝ be the topological group of real numbers under addition. Then Bℝ^{top} is contractible because ℝ^{top} is, but Bℝ^{δ} is not contractible because π_{1}(Bℝ^{δ}) = ℝ.

Example 2.8.2 (BU). The unitary groups U_{n} are topological groups, and we see from I.4.10.1 that BU_{n} is homotopy equivalent to the infinite complex Grassmannian manifold G_{n}, which classifies n-dimensional complex vector bundles by Theorem I.4.10. The unitary group U_{n} is a deformation retract of the complex general linear group GL_{n}(ℂ)^{top}. Thus BU_{n} and BGL_{n}(ℂ)^{top} are homotopy equivalent spaces. Taking the limit as n → ∞, we have a homotopy equivalence BU ≃ BGL(ℂ)^{top}.

By Theorem II.3.2, KU(X) ≅ [X, ℤ × BU] and K̃U(X) ≅ [X, BU] for every compact space X. By Ex. II.3.11 we also have KU^{−n}(X) ≅ [X, Ω^{n}(ℤ×BU)] for all n ≥ 0. In particular, for the one-point space ∗ the groups KU^{−n}(∗) = π_{n}(ℤ × BU) are periodic of order 2: ℤ if n is even, 0 if not. This follow from the observation in II.3.2 that the homotopy groups of BU are periodic — except for π_{0}(BU), which is zero as BU is connected.

A refinement of Bott periodicity states that ΩU ≃ ℤ × BU. Since Ω(BU) ≃ U, we have Ω^{2}(ℤ×BU) ≃ Ω^{2}BU ≃ ℤ×BU and Ω^{2}U ≃ U. This yields the periodicity formula: KU^{−n}(X) = KU^{−n−2}(X).

Example 2.8.3 (BO). The orthogonal group O_{n} is a deformation retract of the real general linear group GL_{n}(ℝ)^{top}. Thus the spaces BO_{n} and BGL_{n}(ℝ)^{top} are homotopy equivalent, and we see from I.4.10.1 that they are also homotopy equivalent to the infinite real Grassmannian manifold G_{n}. In particular, they classify n-dimensional real vector bundles by Theorem I.4.10. Taking the limit as n → ∞, we have a homotopy equivalence BO ≃ BGL(ℝ)^{top}.

Bott periodicity states that the homotopy groups of BO are periodic of order 8 — except for π_{0}(BO) = 0, and that the homotopy groups of ℤ × BO are actually periodic of order 8. These homotopy groups are tabulated in I.3.1.1. A refinement of Bott periodicity states that Ω^{7}O ≃ ℤ × BO. Since Ω(BO) ≃ O, we have Ω^{8}(ℤ × BO) ≃ Ω^{8}(BO) ≃ ℤ × BO and Ω^{8}O ≃ O.

By Definition II.3.5 and Ex. II.3.11, the (real) topological K-theory of a compact space X is given by the formula KO^{−n}(X) = [X, Ω^{n}(ℤ×BO)], n ≥ 0. This yields the periodicity formula: KO^{−n}(X) = KO^{−n−8}(X).

EXERCISES

2.1 Covering spaces. If X: I → Sets is a morphism-inverting functor, use the recipe 2.1.1 to show that the forgetful functor I∫ X → I of Example 2.3.1 makes B(I∫ X) into a covering space of BI with fiber X(i) over each vertex i of BI.

Conversely, if E ⟶^{π} BI is a covering space, show that X(i) = π^{−1}(i) defines a morphism-inverting functor on I, where i is considered as a 0-cell of BI. Conclude that these constructions give a 1–1 correspondence between covering spaces of BI and morphism-inverting functors. 2.2 Translation categories. Suppose that a group G acts on a set X, and form the translation category G∫ X. Show that B(G∫ X) is homotopy equivalent to the disjoint union of the classifying spaces BG_{x} of the stabilizer subgroups G_{x}, one space for each orbit in X. For example, if X is the coset space G/H then B(G∫ X) ≃ BH.

In particular, if X = G is given the G-set structure g · g′ = gg′, this shows that B(G∫ G) is contractible, i.e., the universal covering space of BG. Use this to calculate the homotopy groups of BG, as described in Example 2.4.1.

2.3 Let H be a subgroup of G, and ι: H ↪ G the inclusion as a subcategory.

(a) Show that ι/∗ is the category H∫ G of Ex. 2.1. Conclude that the homotopy fiber of BH → BG is the discrete set G/H, while Bι^{−1}(∗) is a point.

(b) Use Ex. 2.2 to give another proof of (a).

2.4 If C is a filtering category [WHomo, 2.6.13], show that BC is contractible. Hint: By [Wh, V.3.5] it suffices to show that all homotopy groups are trivial. But any map from a sphere into a CW complex lands in a finite subcomplex, and every finite subcomplex of BC lands in the realization BD of a finite subcategory D of C. Show that D lies in another subcategory D′ of C which has a terminal object.

2.5 Mapping telescopes. If ∪n denotes the union of the categories n of 2.1, then a functor ∪n ⟶^{C} CAT is just a sequence C_{1} → C_{2} → C_{3} → ⋯ of categories. Show that the geometric realization of the category L = (∪n)∫C of Example 2.3.2 is homotopy equivalent to BC, where C is the colimit of the C_{n}. In particular, this shows that BL ≃ lim_{n→∞} BC_{n}. Hint: C_{n} ≃ n∫ C.

2.6 Suppose that F: C → D is pre-cofibered (definition 2.6.2).

(a) Show that F^{op}: C^{op} → D^{op} is pre-fibered. If F is cofibered, F^{op} is fibered.

(b) Derive the dual formulation of Quillen’s theorem B, using d\F, from 2.7.

(c) If each cobase-change functor f_{∗} is a homotopy equivalence, show that the geometric realization of F^{−1}(d) ⟶ C ⟶^{F} D is a homotopy fibration sequence for each d in D, and there is a long exact sequence:

⋯ → π_{i+1}(BD) ⟶^{∂} π_{i}BF^{−1}(d) ⟶ π_{i}(BC) ⟶^{F} π_{i}(BD) ⟶^{∂} · · · .

2.7 Let F: C → D be a cofibered functor (2.6.2). Construct a first quadrant double complex E^{0} in which E_{pq}^{0} is the free abelian group on the pairs (c_{0} → ⋯ → c_{q}, F (c_{q}) → d_{0} → ⋯ → d_{p}) of sequences of composable maps in C and D. By filtering the double complex by columns, show that the homology of the total complex Tot E^{0} is H_{q}(Tot E^{0}) ≅ H_{q}(C; ℤ). Then show that the row filtration yields a spectral sequence converging to H_{∗}(C; ℤ) with E_{pq}^{2} = H_{p}(D; H_{q}F^{−1}), the homology of D with coefficients in the functor d ↦ H_{q}(F^{−1}(d); ℤ) described in 2.5. 2.8 A lax functor M: I → CAT consists of functions assigning: (1) a category M(i) to each object i; (2) a functor f_{∗}: M(i) → M(j) to every map i ⟶^{f} j in I; (3) a natural transformation (id_{i})_{∗} ⇒ id_{M(i)} for each i; (4) a natural transformation (fg)_{∗} ⇒ f_{∗}g_{∗} for every pair of composable maps in I. This data is required to be “coherent” in the sense that the two transformations (fgh)_{∗} ⇒ f_{∗}g_{∗}h_{∗} agree, and so do the various transformations f_{∗} ⇒ f_{∗}. For example, a functor is a lax functor in which (3) and (4) are identities.

Show that the definitions of objects and morphisms in Example 2.3.2 define a category I∫ M, where the map ϕ′′ in the composition (f′f, ϕ′′) of (f, ϕ) and (f′, ϕ′) is (f′f)_{∗}(x) → f′_{∗}f_{∗}(x) → f′_{∗}(x′) → x′′. Show that the projection functor π: I∫ M → I is pre-cofibered. 2.9 Subdivision. If C is a category, its Segal subdivision Sub(C) is the category whose objects are the morphisms in C; a morphism from i: A → B to i′: A′ → B′ is a pair of maps (A′ ⟶ A, B ⟶ B′) so that i′ is A′ ⟶ A ⟶^{i} B ⟶ B′.

(a) Draw the Segal subdivisions of the unit interval 2 and the 2-simplex 3.

(b) Show that the source and target functors C^{op} ⟵ Sub(C) ⟶ C are homotopy equivalences. Hint: Use Quillen’s Theorem A and 2.2.2.

2.10 Given a simplicial set X, its Segal subdivision Sub(X) is the sequence of sets X_{1}, X_{3}, X_{5}, …, made into a simplicial set by declaring the face maps ∂′_{i}: X_{2n+1} → X_{2n−1} to be ∂_{i}∂_{2n+2−i} and σ′_{i}: X_{2n+1} → X_{2n+3} to be σ_{i}σ_{2n+2−i}.

If X is the nerve of a category C, show that Sub(X) is the nerve of the Segal subdivision category Sub(C) of Ex. 2.9. 2.11 If C is a category, its arrow category C/C has the morphisms of C as its objects, and a map (a, b): f → f′ in Ar(C) is a commutative diagram in C:

A ⟶^{f} B
a↓  ↓^{b}
A′ ⟶^{f′} B′

If f: A → B then the source s(f) = A and target t(f) = B of f define functors C/C → C.

Show that s is a fibered functor, and that t is a cofibered functor. Then show that both s and t are homotopy equivalences.

§3. Symmetric Monoidal Categories

The geometric realization BS of a symmetric monoidal category is an H-space with a homotopy-commutative, homotopy-associative product. To see this, recall from Definition I.5.1 that a symmetric monoidal category is a category S with a functor □: S × S → S which has a unit object “e” and is associative and is commutative, all up to coherent natural isomorphism. By 2.1(4) the geometric realization of □ is the “product” map (BS) × (BS) ≅ B(S × S) → BS. The natural isomorphisms s□e ≅ s ≅ e□s imply that the vertex e is an identity up to homotopy, i.e., that BS is an H-space. The other axioms imply that the product on BS is homotopy commutative and homotopy associative.

In many cases e is an initial object of S, and therefore the H-space BS is contractible by Example 2.2.2. For example, any additive category A is symmetric monoidal category (with □ = ⊕), and e = 0 is an initial object, so BA is contractible. Similarly, the category Sets_{f} of finite sets is symmetric monoidal (□ being disjoint union) by I.5.2, and e = ∅ is initial, so BSets_{f} is contractible.

Here is an easy way to modify S in order to get an interesting H-space.

Definition 3.1. Let iso S denote the subcategory of isomorphisms in S. It has the same objects as S, but its morphisms are the isomorphisms in S. Because iso S is also symmetric monoidal, B(iso S) is an H-space.

By Lemma 2.3, the abelian monoid π_{0}(iso S) is just the set of isomorphism classes of objects in S — the monoid S^{iso} considered in §II.5. In fact, iso S is equivalent to the disjoint union ∐ Aut_{S}(s) of the one-object categories Aut_{S}(s), and B(iso S) is homotopy equivalent to the disjoint union of the classifying spaces B Aut(s), s ∈ S^{iso}.

Examples 3.1.1. B(iso S) is often an interesting H-space.

(a) In the category Sets_{f} of finite sets, the group of automorphisms of any nelement set is isomorphic to the permutation group Σ_{n}. Thus the subcategory iso Sets_{f} is equivalent to ∐ Σ_{n}, the disjoint union of the one-object categories Σ_{n}. Thus the classifying space B(iso Sets_{f}) is homotopy equivalent to the disjoint union of the classifying spaces BΣ_{n}, n ≥ 0.

(b) The additive category P(R) of f.g. projective R-modules has 0 as an initial object, so BP(R) is a contractible space. However, its subcategory P = iso P(R) of isomorphisms is more interesting. The topological space BP is equivalent to the disjoint union of the classifying spaces B Aut(P) as P runs over the set of isomorphism classes of f.g. projective R-modules.

(c) Fix a ring R, and let F = F(R) be the category ∐ GL_{n}(R) whose objects are the based free R-modules {0, R, R^{2}, ⋯, R^{n}, ⋯ }. There are no maps in F between R^{m} and R^{n} if m ≠ n, and the self-maps of R^{n} form the group GL_{n}(R). This is a symmetric monoidal category: R^{m}□R^{n} = R^{m+n} by concatenation of bases. The space BF(R) is equivalent to the disjoint union of the classifying spaces BGL_{n}(R).

If R satisfies the Invariant Basis Property (I.1.1), then F(R) is a full subcategory of iso P(R). In this case, we saw in II.5.4.1 that F(R) is cofinal in iso P(R).

(d) Fix a commutative ring R, and let S = Pic(R) be the category of invertible R-modules and their isomorphisms. This is a symmetric monoidal category in which □ is tensor product and e is R; see II.5.2(5). In this case, S = iso S and S^{iso} is the Picard group Pic(R) discussed in §I.3. By Lemma I.3.3, Aut(L) = R^{×} for every L. Thus Pic(R) is equivalent to a disjoint union of copies of R^{×}, and B(Pic) is homotopy equivalent to the product Pic(R) × B(R^{×}).

(e) If F is a field, we saw in II.5.7 that the categories SBil(F) and Quad(F) = Quad^{+}(F) of symmetric inner product spaces and quadratic spaces are symmetric monoidal categories. More generally, let A be any ring with involution, and ϵ = ±1. Then the category Quad^{ϵ}(A) of nonsingular ϵ-quadratic A-modules is a symmetric monoidal category with □ = ⊕ and e = 0. See [B72, Bak] for more details.

The S^{−1}S Construction

In [GQ], Quillen gave a construction of a category S^{−1}S such that K(S) = B(S^{−1}S) is a “group completion” of BS (see 3.4 below), provided that every map in S is an isomorphism and every translation s□: Aut_{S}(t) → Aut_{S}(s□t) is an injection. The motivation for this construction comes from the construction of the universal abelian group completion of an abelian monoid given in Chapter II, §1.

Definition 3.2 (S^{−1}S). The objects of S^{−1}S are pairs (m, n) of objects of S. A morphism in S^{−1}S is an equivalence class of composites

(m_{1}, m_{2}) ⟶^{s□} (s□m_{1}, s□m_{2}) ⟶^{(f,g)} (n_{1}, n_{2}).

This composite is equivalent to

(m_{1}, m_{2}) ⟶^{t□} (t□m_{1}, t□m_{2}) ⟶^{(f′,g′)} (n_{1}, n_{2})

exactly when there is an isomorphism α: s ⟶^{≃} t in S so that composition with α□m_{i} sends f′ and g′ to f and g.

Explanation 3.2.1. There are two basic types of morphisms in S^{−1}S. The first type is a pair of maps (f_{1}, f_{2}): (m_{1}, m_{2}) → (n_{1}, n_{2}) with f_{i}: m_{i} → n_{i} in S, arising from the inclusion of S × S in S^{−1}S. The second type is a formal map s□: (m, n) → (s□m, s□n).

We shall say that “translations are faithful” in S if every translation Aut(s) → Aut(s□t) in S is an injection. In this case every map in S^{−1}S determines its object s up to unique isomorphism.

Remark 3.2.2. S^{−1}S is a symmetric monoidal category, with (m, n)□(m′, n′) = (m□m′, n□n′), and the functor S → S^{−1}S sending m to (e, m) is monoidal. Hence the natural map BS → B(S^{−1}S) is an H-space map, and π_{0}(S) → π_{0}(S^{−1}S) is a map of abelian monoids.

In fact π_{0}(S^{−1}S) is an abelian group, the inverse of (m, n) being (n, m). This follows from the existence of a morphism η in S^{−1}S from (e, e) to (m, n)□(n, m) = (m□n, n□m). Warning: η is not a natural transformation! See Ex. 3.3.

Definition 3.3. Let S be a symmetric monoidal category in which every morphism is an isomorphism. Its K-groups are the homotopy groups of B(S^{−1}S):

K_{n}^{□}(S) = π_{n}(BS^{−1}S).

It is sometimes convenient to write K^{□}(S) for the geometric realization B(S^{−1}S), and call it the K-theory space of S, so that K_{n}^{□}(S) = π_{n}K^{□}(S).

In order to connect this definition up with the definition of K_{0}^{□}(S) given in section II.5, we recall from 3.2.2 that the functor S → S^{−1}S induces a map of abelian monoids from π_{0}(S) = S^{iso} to π_{0}(S^{−1}S).

Lemma 3.3.1. The abelian group K_{0}^{□}(S) = π_{0}(S^{−1}S) is the group completion of the abelian monoid π_{0}(S) = S^{iso}. Thus definition 3.3 agrees with the definition of K_{0}^{□}(S) given in II.5.1.2.

Proof. Let A denote the group completion of π_{0}(S), and consider the function α(m, n) = [m] − [n] from the objects of S^{−1}S to A. If s ∈ S and f_{i}: m_{i} → n_{i} are morphisms in S then in A we have α(m, n) = α(s□m, s□n) and α(m_{1}, m_{2}) = [m_{1}]− [m_{2}] = [n_{1}]−[n_{2}] = α(n_{1}, n_{2}). By Lemma 2.3, α induces a set map π_{0}(S^{−1}S) → A. By construction, α is an inverse to the universal homomorphism A → π_{0}(S^{−1}S).

Group Completions

Group completion constructions for K-theory were developed in the early 1970’s by topologists studying infinite loop spaces. These constructions all apply to symmetric monoidal categories.

Any discussion of group completions depends upon the following well-known facts (see [Wh, III.7]). Let X be a homotopy commutative, homotopy associative H-space. Its set of components π_{0}X is an abelian monoid, and H_{0}(X; ℤ) is the monoid ring ℤ[π_{0}(X)]. Moreover, the integral homology H_{∗}(X; ℤ) is an associative graded-commutative ring with unit.

We say that a homotopy associative H-space X is group-like if it has a homotopy inverse; see [Wh, III.4]. Of course this implies that π_{0}(X) is a group. When X is a CW complex, the converse holds [Wh, X.2.2]: if the monoid π_{0}(X) is a group, then X is group-like.

For example, if S = iso S then π_{0}(BS) is the abelian monoid S^{iso} of isomorphism classes, and H_{0}(BS; ℤ) is the monoid ring ℤ[S^{iso}]. In this case, the above remarks show that BS is grouplike if and only S^{iso} is an abelian group under □.

Definition 3.4 (Group Completion). Let X be a homotopy commutative, homotopy associative H-space. A group completion of X is an H-space Y, together with an H-space map X → Y, such that π_{0}(Y) is the group completion of the abelian monoid π_{0}(X) (in the sense of §I.1), and the homology ring H_{∗}(Y; k) is isomorphic to the localization π_{0}(X)^{−1}H_{∗}(X; k) of H_{∗}(X; k) by the natural map, for all commutative rings k.

If X is a CW complex (such as X = BS), we shall assume that Y is also a CW complex. This hypothesis implies that the group completion Y is group-like.

It is possible to show directly that ℤ × BGL(R)^{+} is a group completion of BS when S = ∐ GL_{n}(R); see Ex. 3.9. We will see in theorems 3.6 and 3.7 below that the K-theory space B(S^{−1}S) is a group completion of BS, and that B(S^{−1}S) is homotopy equivalent to ℤ × BGL(R)^{+}.

Lemma 3.4.1. If X is a group-like H-space then X its own group completion, and any other group completion f: X → Y is a homotopy equivalence.

Proof. Since f is a homology isomorphism, it is an isomorphism on π_{0} and π_{1}. Therefore the map of basepoint components is a +–construction relative to the subgroup 1 of π_{1}(X), and Theorem 1.4 implies that X ≃ Y.

Segal’s ΩB Method 3.4.2. If X is a topological monoid, such as ∐ BGL_{n}(R) or ∐ BΣ_{n}, then we can form BX, the geometric realization of the (one-object) topological category X. In this case, ΩBX is an infinite loop space and the natural map X → ΩBX is a group completion. For example, if X is the one-object monoid ℕ then Bℕ ≃ S^{1}, π_{0}(ΩBℕ) is ℤ, and every component of ΩBℕ is contractible. See [Adams] for more details.

Machine Methods 3.4.3. (See [Adams].) If X isn’t quite a monoid, but the homotopy associativity of its product is nice enough, then there are constructions called “infinite loop space machines” which can construct a group completion Y of X, and give Y the structure of an infinite loop space. All machines produce the same infinite loop space Y (up to homotopy); see [MT]. Some typical machines are described in [Segal], and [May74].

The realization X = BS of a symmetric monoidal category S is nice enough to be used by infinite loop space machines. These machines produce an infinite loop space K(S) and a map BS → K(S) which is a group completion. Most infinite loop machines will also produce explicit deloopings of K(S) in the form of an Ω-spectrum K(S), the K-theory spectrum of S. The production of K(S) is natural enough that monoidal functors between symmetric monoidal categories induce maps of the corresponding spectra.

Actions on other categories

To show that B(S^{−1}S) is a group completion of BS, we need to fit the definition of S^{−1}S into a more general framework.

Definition 3.5. A monoidal category S is said to act upon a category X by a functor □: S × X → X if there are natural isomorphisms s□(t□x) ≅ (s□t)□x and e□x ≅ x for s, t ∈ S and x ∈ X, satisfying coherence conditions for the products s□t□u□x and s□e□x analogous to the coherence conditions defining S.

Definition 3.5.1. If S acts upon X, the category ⟨S, X⟩ has the same objects as X. A morphism from x to y in ⟨S, X⟩ is an equivalence class of pairs (s, s□x ⟶^{ϕ} y), where s ∈ S and ϕ is a morphism in X. Two pairs (s, ϕ) and (s′, ϕ′) are equivalent in case there is an isomorphism s ≅ s′ identifying ϕ′ with s′□x ≅ s□x ⟶^{ϕ} y.

We shall write S^{−1}X for ⟨S, S × X⟩, where S acts on both factors of S × X. Note that when X = S this definition recovers the definition of S^{−1}S given in 3.2 above. If S is symmetric monoidal, then the formula s□(t□x) = (s□t, x) defines an action of S on S^{−1}X.

For example, if every arrow in S is an isomorphism, then e is an initial object of ⟨S, S⟩ and therefore the space B⟨S, S⟩ is contractible.

We say that S acts invertibly upon X if each translation functor s□: X → X is a homotopy equivalence. For example, S acts invertibly on S^{−1}X (if S is symmetric) by the formula s□(t, x) = (s□t, x), the homotopy inverse of the translation (t, x) ↦ (s□t, x) being the translation (t, x) ↦ (t, s□x), because of the natural transformation (t, x) ↦ (s□t, s□x).

Now π_{0}S is a multiplicatively closed subset of the ring H_{0}(S) = ℤ[π_{0}S], so it acts on H_{∗}(X) and acts invertibly upon H_{∗}(S^{−1}X). Thus the functor X → S^{−1}X sending x to (0, x) induces a map

(3.5.2)  (π_{0}S)^{−1}H_{q}(X) → H_{q}(S^{−1}X).

Theorem 3.6 (Quillen). If every map in S is an isomorphism and translations are faithful in S, then the map (3.5.2) is an isomorphism for all X and q.

In particular, B(S^{−1}S) is a group completion of the H-space BS.

Proof. (See [GQ, p. 221].) By Ex. 3.4, the projection functor ρ: S^{−1}X → ⟨S, S⟩ is cofibered with fiber X. By Ex. 2.7 there is an associated spectral sequence E_{pq}^{2} = H_{p}(⟨S, S⟩; H_{q}(X)) ⇒ H_{p+q}(S^{−1}X). Localizing this at the multiplicatively closed subset π_{0}S of H_{0}(S) is exact, and π_{0}S already acts invertibly on H_{∗}(S^{−1}X) by Ex. 2.7, so there is also a spectral sequence E_{pq}^{2} = H_{p}(⟨S, S⟩; M_{q}) ⇒ H_{p+q}(S^{−1}X), where M_{q} = (π_{0}S)^{−1}H_{q}(X). But the functors M_{q} are morphism-inverting, so by Ex. 2.1 and the contractibility of ⟨S, S⟩, the group H_{p}(⟨S, S⟩; M_{q}) is zero for p ≠ 0, and equals M_{q} for p = 0. Thus the spectral sequence degenerates to the claimed isomorphism (3.5.2).

The final assertion is immediate from this and definition 3.4, given remark 3.2.2 and lemma 3.3.1.

Bass gave a classical definition of K_{1}(S) and K_{2}(S) in [B72]; we gave them implicitly in III.1.6.3 and III.5.6. We can now state these classical definitions, and show that they coincide with the K-groups defined in this section.

Corollary 3.6.1. If S = iso S and translations are faithful in S, then:

K_{1}(S) = \underrightarrow{lim}_{s∈S} H_{1}(Aut(s); ℤ),

K_{2}(S) = \underrightarrow{lim}_{s∈S} H_{2}([Aut(s), Aut(s)]; ℤ).

Proof. ([We81]) The localization of H_{q}(BS) = ⊕_{s∈S}H_{q}(Aut(s) at π_{0}(X) = S^{iso} is the direct limit of the groups H_{q}(Aut(s), taken over the translation category of all s ∈ S. Since π_{1}(X) = H_{1}(X; ℤ) for every H-space X, this gives the formula for K_{1}(S) = π_{1}B(S^{−1}S).

For K_{2} we observe that any monoidal category S is the filtered colimit of its monoidal subcategories having countably many objects. Since K_{2}(S) and Bass’ H_{2} definition commute with filtered colimits, we may assume that S has countably many objects. In this case the proof is relegated to Ex. 3.10, which is worked out in [We81].

Relation to the +–construction

Let S = F(R) = ∐ GL_{n}(R) be the monoidal category of free R-modules, as in example 3.1.1(c). In this section, we shall identify the +–construction on BGL(R) with the basepoint component of K(S) = B(S^{−1}S). As theorems 3.6 and 1.7 suggest, we need to find an acyclic map from BGL(R) to a connected component of B(S^{−1}S).

Any group map η from GL_{n}(R) to Aut_{S^{−1}S}(R^{n}, R^{n}) gives a map from BGL_{n}(R) to B(S^{−1}S). Consider the maps η = η_{n} defined by η_{n}(g) = (g, 1). Because

GL_{n}(R) ⟶^{η} Aut(R^{n}, R^{n})
□R↓  ^{□(R,R)}↓
GL_{n+1}(R) ⟶^{η} Aut(R^{n+1}, R^{n+1})

commutes, there is a natural transformation from η to η(□R). The resulting homotopy of maps BGL_{n}(R) → B(S^{−1}S) gives a map from the “mapping telescope” construction of BGL(R) to B(S^{−1}S); see Ex. 2.5. In fact, this map lands in the connected component Y_{S} of the identity in B(S^{−1}S). Since B(S^{−1}S) is an H-space, so is the connected component Y_{S} of the identity.

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BGL_{1} BGL_{2} BGL_{3} etc.  ⟶BS^{−1}S

Figure 1. The mapping telescope of BGL(R) and B(S^{−1}S).

Theorem 3.7. When S is ∐ GL_{n}(R),

B(S^{−1}S) ≃ ℤ × BGL(R)^{+}.

Proof. (Quillen) We shall show that the map BGL(R) → Y_{S} is acyclic. By theorem 1.7, this will induce an acyclic map BGL(R)^{+} → Y_{S}. This must be a homotopy equivalence by lemma 3.4.1, and the theorem will then follow.

Let e ∈ π_{0}BS be the class of R. By theorem 3.6, H_{∗}B(S^{−1}S) is the localization of the ring H_{∗}(BS) at π_{0}(S) = {e^{n}}. But this localization is the colimit of the maps H_{∗}(BS) → H_{∗}(BS) coming from the translation ⊕R: S → S. Hence H_{∗}B(S^{−1}S) ≅ H_{∗}(Y_{S}) ⊗ ℤ[e, e^{−1}], where Y_{S} denotes the basepoint component of B(S^{−1}S), and H_{∗}(Y_{S}) ≅ colim H_{∗}(BGL_{n}(R)) = H_{∗}(BGL(R)). This means that the map BGL(R) → Y_{S} is acyclic, as required.

Example 3.7.1. (Segal) Consider the symmetric monoidal category S = ∐ Σ_{n}, equivalent to the category Sets_{f} of example 3.1.1(a). The infinite symmetric group Σ is the union of the symmetric groups Σ_{n} along the inclusions □1 from Σ_{n} to Σ_{n+1}, and these inclusions assemble to give a map from the mapping telescope construction of BΣ to B(S^{−1}S), just as they did for GL(R). Moreover the proof of theorem 3.7 formally goes through to prove that B(S^{−1}S) ≃ K(Sets_{f}) is homotopy equivalent to ℤ × BΣ^{+}. This is the equivalence of parts (a) and (b) in the following result. We refer the reader to [BP71] and [Adams, §3.2] for the equivalence of parts (b) and (c).

The Barratt-Priddy-Quillen-Segal Theorem 3.7.2. The following three infinite loop spaces are the same:

(a) the group completion K(Sets_{f}) of BSets_{f};

(b) ℤ × BΣ^{+}, where Σ is the union of the symmetric groups Σ_{n}; and

(c) The infinite loop space Ω^{∞}S^{∞} = lim_{n→∞} Ω^{n}S^{n}. Hence the groups K_{n}(Sets_{f}) are the stable homotopy groups of spheres, π_{n}^{s}.

More generally, suppose that S has a countable sequence of objects s_{1}, … such that s_{n+1} = s_{n}□a_{n} for some a_{n} ∈ S, and satisfying the cofinality condition that for every s ∈ S there is an s′ and an n so that s□s′ ≅ s_{n}. In this case we can form the group Aut(S) = colim_{n→∞} Aut_{S}(s_{n}).

Theorem 3.7.3. Let S = iso S be a symmetric monoidal whose translations are faithful, and suppose the above condition is satisfied, so that the group Aut(S) exists. Then the commutator subgroup E of Aut(S) is a perfect normal subgroup, K_{1}(S) = Aut(S)/E, and the +–construction on B Aut(S) is the connected component of the identity in the group completion K(S). Thus

K(S) ≃ K_{0}(S) × B Aut(S)^{+}.

Proof. ([We81]) The assertions about E are essentially on p. 355 of [Bass]. On the other hand, the mapping telescope construction mentioned above gives an acyclic map from B Aut(S) to the basepoint component of B(S^{−1}S), and such a map is by definition a +–construction.

Cofinality

A monoidal functor f: S → T is called cofinal if for every t in T there is a t′ and an s in S so that t□t′ ≅ f(s); cf. II.5.3. For example, the functor F(R) → P(R) of example 3.1.1(c) is cofinal, because every projective module is a summand of a free one. For Pic(R), the one-object subcategory R^{×} is cofinal.

Cofinality Theorem 3.8. Suppose that f: S → T is cofinal. Then

(a) If T acts on X then S^{−1}X ≃ T^{−1}X.

(b) If Aut_{S}(s) ≅ Aut_{T} (fs) for all s in S then the basepoint components of K(S) and K(T) are homotopy equivalent. Consequently, for all n ≥ 1 we have K_{n}(S) ≅ K_{n}(T).

Proof. By cofinality, S acts invertibly on X if and only if T acts invertibly on X. Hence Ex. 3.5 yields

S^{−1}X ⟶^{≃} T^{−1}(S^{−1}X) ≅ S^{−1}(T^{−1}X) ⟵^{≃} T^{−1}X.

An alternate proof of part (a) is sketched in Ex. 3.7.

For part (b), let Y_{S} and Y_{T} denote the connected components of B(S^{−1}S) and B(T^{−1}T). Writing the subscript s ∈ S to indicate a colimit over the translation category 2.3.1 of π_{0}(S), and similarly for the subscript t ∈ T, theorem 3.6 yields:

H_{∗}(Y_{S}) = colim_{s∈S} H_{∗}(B Aut(s)) = colim_{s∈S} H_{∗}(B Aut(fs))
≅ colim_{t∈T} H_{∗}(B Aut(t)) = H_{∗}(Y_{T}).

Hence the connected H-spaces Y_{S} and Y_{T} have the same homology, and this implies that they are homotopy equivalent.

Note that K_{0}(F) = ℤ is not the same as K_{0}(P) = K_{0}(R) in general, although K_{n}(F) ≅ K_{n}(P) for n ≥ 1 by the Cofinality Theorem 3.8(b). By theorem 3.7 this establishes the following important result.

Corollary 3.8.1. Let S = iso P(R) be the category of f.g. projective R-modules and their isomorphisms. Then

B(S^{−1}S) ≃ K_{0}(R) × BGL(R)^{+}.

Let’s conclude with a look back at the other motivating examples in 3.1.1. In each of these examples, every morphism is an isomorphism and the translations are faithful, so the classifying space of S^{−1}S is a group completion of BS.

Example 3.9.1 (Stable homotopy). The “free R-module” on a finite set determines a functor from Sets_{f} to P(R), or from the subcategory ∐ Σ_{n} of Sets_{f} to ∐ GL_{n}(R). This functor identifies the symmetric group Σ_{n} maps with the permutation matrices in GL_{n}(R). Applying group completions, theorem 3.7 and 3.7.1 show that this gives a map from Ω^{∞}S^{∞} to K(R), hence maps π_{n}^{s} → K_{n}(R).

Example 3.9.2 (Picard groups). Let R be a commutative ring, and consider the symmetric monoidal category S = Pic(R) of Example II.5.2(5). Because π_{0}(S) is already a group, S and S^{−1}S are homotopy equivalent (by lemma 3.4.1). Therefore we get

K_{0}Pic(R) = Pic(R), K_{1}Pic(R) = U(R) and K_{n}Pic(R) = 0 for n ≥ 2.

The determinant functor from P = iso P(R) to Pic(R) constructed in §I.3 gives a map from K(R) = K(P) to KPic(R). Upon taking homotopy groups, this yields the familiar maps det: K_{0}(R) → Pic(R) of II.2.6 and det: K_{1}(R) → R^{×} of III.1.1.1.

Example 3.9.3 (L-theory). Let S = Quad^{ϵ}(A) denote the category of nonsingular ϵ-quadratic A-modules, where ϵ = ±1 and A is any ring with involution [B72, Bak]. The K-groups of this category are the L-groups _{ϵ}L_{n}(A) of Karoubi and others. For this category, the sequence of hyperbolic spaces H^{n} is cofinal (by Ex. II.5.10), and the automorphism group of H^{n} is the orthogonal group _{ϵ}O_{n}. The infinite orthogonal group _{ϵ}O = _{ϵ}O(A), which is the direct limit of the groups _{ϵ}O_{n}, is the group Aut(S) in this case. By theorem 3.7.3, we have

K(Quad^{ϵ}(A)) ≃ _{ϵ}L_{0} × B_{ϵ}O^{+}.

When A = ℝ, the classical orthogonal group O is _{+1}O. When A = ℂ and the involution is complex conjugation, the classical unitary group U is _{+1}O(ℂ). The description of K_{1}(S) and K_{2}(S) in 3.6.1 is given in [B72, p. 197]. For more bells and whistles, and classical details, we refer the reader to [Bak].

Example 3.9.4. When R is a topological ring (such as ℝ or ℂ), we can think of P(R) as a topological symmetric monoidal category. Infinite loop space machines (3.4.3) also accept topological symmetric monoidal categories, and we write K(R^{top}) for K(P(R)^{top}). This gives natural maps from the infinite loop spaces K(R^{δ}) to K(R^{top}). The naturality of these maps allows us to utilize infinite loop space machinery. As an example of the usefulness, we remark that

K(ℝ^{top}) ≃ ℤ × BO  and  K(ℂ^{top}) ≃ ℤ × BU.

EXERCISES

3.1 Let ℕ be the additive monoid {0, 1, ...}, considered as a symmetric monoidal category with one object. Show that ⟨ℕ, ℕ⟩ is the union ∪n of the ordered categories n, and that ℕ^{−1}ℕ is a poset, each component being isomorphic to ∪n.

3.2 Show that a sequence X_{0} → X_{1} → ⋯ of categories determines an action of ℕ on the disjoint union X = ∐ X_{n}, and that ⟨ℕ, X⟩ is the mapping telescope category ∪n∫ X of Ex. 2.5.

3.3 (Thomason) Let S be symmetric monoidal, and let ι: S^{−1}S → S^{−1}S be the functor sending (m, n) to (n, m) and (f_{1}, f_{2}) to (f_{2}, f_{1}). Show that there is no natural transformation 0 ⇒ id□ι. Hint: The obvious candidate is given in 3.2.2.

Thomason has shown that Bι is the homotopy inverse for the H-space structure on B(S^{−1}S), but for subtle reasons.

3.4 (Quillen) Suppose that S = isoS, and that the translations in S are faithful. Show that the projection S^{−1}X ⟶^{ρ} ⟨S, S⟩ is cofibered, where ρ(s, x) = s.

3.5 Let S = iso S be a monoidal category whose translations are faithful (3.2.1). Suppose that S acts invertibly upon a category X. Show that the functors X → S^{−1}X (x ↦ (s, x)) are homotopy equivalences for every s in S. If S acts upon a category Y, then S always acts invertibly upon S^{−1}Y, so this shows that S^{−1}Y ≃ S^{−1}(S^{−1}Y). Hint: Use exercises 2.6 and 3.4, and the contractibility of ⟨S, S⟩.

3.6 Suppose that every map in X is monic, and that each translation Aut_{S}(s) ⟶^{□x} Aut_{X}(s□x) is an injection. Show that the sequence S^{−1}S ⟶^{□x} S^{−1}X ⟶^{π} ⟨S, X⟩ is a homotopy fibration for each x in X, where π is projection onto the second factor. In particular, if ⟨S, X⟩ is contractible, this proves that S^{−1}S ⟶^{□x} S^{−1}X is a homotopy equivalence. Hint: Show that π and S^{−1}π: S^{−1}(S^{−1}X) → ⟨S, X⟩ are cofibered, and use the previous exercise.

3.7 Use exercises 3.4 and 3.5 to give another proof of the Cofinality Theorem 3.8(b).

3.8 If S is a symmetrical monoidal category, so is its opposite category S^{op}. Show that the group completions K(S) and K(S^{op}) are homotopy equivalent.

3.9 Fix a ring R and set S = ∐ GL_{n}(R). The maps BGL_{n}(R) → BGL(R) → {n} × BGL(R)^{+} assemble to give a map from BS to ℤ × BGL(R)^{+}. Use Ex. 1.9 to show that it is an H-space map. Then show directly that this makes ℤ×BGL(R)^{+} into a group completion of BS.

3.10 Let S be a symmetric monoidal category with countably many objects, so that the group Aut(S) exists and its commutator subgroup E is perfect, as in 3.7.3. Let F denote the homotopy fiber of the H-space map B Aut(S)^{+} → B(K_{1}S).

(a) Show that π_{1}(F) = 0 and H_{2}(F; ℤ) ≅ π_{2}(F) ≅ K_{2}(S).

(b) ([We81]) Show that the natural map BE → F induces H_{∗}(BE) ≅ H_{∗}(F), so that F = BE^{+}. Hint: Show that K_{1}S acts trivially upon the homology of BE and F, and apply the comparison theorem for spectral sequences.

(c) Conclude that K_{2}(S) ≅ H_{2}(E) ≅ \underrightarrow{lim}_{s∈S} H_{2}([Aut(s), Aut(s)]; ℤ).

3.11 If f: X → Y is a functor, we say that an action of S on X is fiberwise if S × X ⟶^{□} X ⟶^{f} Y equals the projection S × X → X followed by f.

(a) Show that a fiberwise action on X restricts to an action of S on each fiber category X_{y} = f^{−1}(y), and that f induces a functor S^{−1}X → Y whose fibers are the categories S^{−1}(X_{y}).

(b) If f is a fibered functor (2.6.2), we say that a fiberwise action is cartesian if the basechange maps commute with the action of S on the fibers. Show that in this case S^{−1}X → Y is a fibered functor.

§4. Quillen’s Q-construction

The higher K-theory groups of a small exact category A are defined to be the homotopy groups K_{n}(A) = π_{n+1}(BQA) of the geometric realization of a certain auxiliary category QA, which we now define. This category has the same objects as A, but morphisms are harder to describe. Here is the formal definition; we refer the reader to Ex. 4.1 for a more intuitive interpretation of morphisms in terms of subquotients.

Definition 4.1. A morphism from A to B in QA is an equivalence class of diagrams

(4.1.1)  A ↞^{j} B_{2} ↣^{i} B

where j is an admissible epimorphism and i is an admissible monomorphism in A. Two such diagrams are equivalent if there is an isomorphism between them which is the identity on A and B. The composition of the above morphism with a morphism B ↞ C_{2} ↣ C is A ↞ C_{1} ↣ C, where C_{1} = B_{2} ×_{B} C_{2}.

C_{1} ↣ C_{2} ↣ C
↓ ↓  ↓ ↓
A ↞ B_{2} ↣ B

Two distinguished types of morphisms play a special role in QA: the admissible monics A ↣ B (take B_{2} = A) and the oppositely oriented admissible epis A ↞ B (take B_{2} = B). Both types are closed under composition, and the composition of A ↞ B_{2} with B_{2} ↣ B is the morphism (4.1.1). In fact, every morphism in QA factors as such a composition in a way that is unique up to isomorphism.

Subobjects 4.1.2. Recall from [Mac] that (in any category) a subobject of an object B is an equivalence class of monics B_{2} ↣ B, two monics being equivalent if they factor through each other. In an exact category A, we call a subobject admissible if any (hence every) representative B_{2} ↣ B is an admissible monic.

By definition, every morphism from A to B in QA determines a unique admissible subobject of B in A. If we fix a representative B_{2} ↣ B for each subobject in A, then a morphism in QA from A to B is a pair consisting of an admissible subobject B_{2} of B and an admissible epi B_{2} ↠ A.

In particular, this shows that morphisms from 0 to B in QA are in 1-1 correspondence with admissible subobjects of B.

Isomorphisms in QA are in 1-1 correspondence with isomorphisms in A. To see this, note that every isomorphism i: A ≅ B in A gives rise to an isomorphism in

QA, represented either by A ↣^{i} B or by A ↞^{i^{−1}} B. Conversely, since the subobject determined by an isomorphism in QA must be the maximal subobject B ↣^{=} B, every isomorphism in QA arises in this way.

Remark 4.1.3. Some set-theoretic restriction is necessary for QA to be a category in our universe. It suffices for A to be well-powered, i.e., for each object of A to have a set of subobjects. We shall tacitly assume this, since we will soon need the stronger assumption that A is a small category.

We now consider the geometric realization BQA as a based topological space, the basepoint being the vertex corresponding to the object 0. In fact, BQA is a connected CW complex, because the morphisms 0 ↣ A in QA give paths in BQA from the basepoint 0 to every vertex A. (See Lemma 2.3.)

Proposition 4.2. The geometric realization BQA is a connected CW complex with π_{1}(BQA) ≅ K_{0}(A). The element of π_{1}(BQA) corresponding to [A] ∈ K_{0}(A) is represented by the based loop composed of the two edges 0 ↣ A and 0 ↞ A:

(4.2.1)  0 ↣ A ↠ 0.

Proof. Let T denote the family of all morphisms 0 ↣ A in QA. Since each nonzero vertex occurs exactly once, T is a maximal tree. By Lemma 2.4, π_{1}(BQA) has the following presentation: it is generated by the morphisms in QA, modulo the relations that [0 ↣ A] = 1 and [f] · [g] = [f ∘ g] for every pair of composable arrows in QA. Moreover, the element of π_{1}(BQA) corresponding to a morphism from A to B is the based loop following the edges 0 ↣ A → B ↢ 0.

Since the composition 0 ↣ B_{2} ↣ B is in T, this shows that [B_{2} ↣ B] = 1 in π_{1}(BQA). Therefore [A ↞ B_{2} ↣ B] = [A ↞ B_{2}]. Similarly, the composition 0 ↞ A ↞ B yields the relation [A ↞ B][0 ↞ A] = [0 ↞ B]. Since every morphism (4.1.1) factors, this shows that π_{1}(BQA) is generated by the morphisms [0 ↞ A].

If A ↣ B ↠ C is an exact sequence in A, then the composition 0 ↣ C ↞ B in QA is 0 ↞ A ↣ B. This yields the additivity relation

(4.2.2)  [0 ↞ B] = [C ↞ B][0 ↞ C] = [0 ↞ A][0 ↞ C]

in π_{1}(BQA), represented by the following picture in BQA:

0 ↣ A ↣ B
↓ ↓ ↗ ↓ ↓ ↘ ↘
0 ↣ C ↠ 0

Since every relation [f] · [g] = [f ∘ g] may be rewritten in terms of the additivity relation, π_{1}(BQA) is generated by the [0 ↞ A] with (4.2.2) as the only relation. Therefore K_{0}(A) ≅ π_{1}(BQA).

Definition 4.3. Let A be a small exact category. Then KA denotes the space ΩBQA, and we set

K_{n}(A) = π_{n}KA = π_{n+1}(BQA) for n ≥ 0.

Proposition 4.2 shows that this definition of K_{0}(A) agrees with the one given in chapter II. Note that any exact functor F: A → B induces a functor QA → QB, hence maps BQA → BQB and K_{n}(A) → K_{n}(B). Thus the space KA = Ω BQA and all the groups K_{n}(A) are functors from exact categories and exact functors to spaces and abelian groups, respectively. Moreover, isomorphic functors induce the same map on K-groups, because they induce isomorphic functors QA → QA′.

Remark 4.3.1. If an exact category A is not small but has a set of isomorphism classes of objects then we define K_{n}(A) to be K_{n}(A′), where A′ is a small subcategory equivalent to A. By Ex. 4.2 this is independent of the choice of A′. From now on, whenever we talk about the K-theory of a large exact category A we will use this device, assuming tacitly that we have replaced it by a small A′. For example, this is the case in the following definitions.

Definition 4.3.2. Let R be a ring with unit, and let P(R) denote the exact category of f.g. projective R-modules. We define the K-groups of R by K_{n}(R) = K_{n}P(R). For n = 0, lemma 4.2 shows that this agrees with the definition of K_{0}(R) in chapter II. For n ≥ 1, agreement with the +–construcion definition 1.1.1 will have to wait until section 5.

If R is noetherian, let M(R) denote the category of f.g. R-modules. Otherwise, M(R) is the category of pseudo-coherent modules defined in II.7.1.4. We define the G-groups of R by G_{n}(R) = K_{n}M(R). For n = 0, this also agrees with the definition in chapter II.

Similarly, if X is a scheme which is quasi-projective (over a commutative ring), we define K_{n}(X) = K_{n}VB(X). If X is noetherian, we define G_{n}(X) = K_{n}M(X). For n = 0, this agrees with the definition of K_{0}(X) and G_{0}(X) in chapter II.

Morita Invariance 4.3.3. Recall from II.2.7 that if two rings R and S are Morita equivalent then there are equivalences P(R) ≅ P(S) and M(R) ≅ M(S). It follows that K_{n}(R) ≅ K_{n}(S) and G_{n}(R) ≅ G_{n}(S) for all n.

Elementary Properties 4.3.4. Here are some elementary properties of the above definition.

If A^{op} denotes the opposite category of A, then Q(A^{op}) is isomorphic to QA by Ex 4.3, so we have K_{n}(A^{op}) = K_{n}(A). For example, if R is a ring then P(R^{op}) ≅ P(R)^{op} by P ↦ Hom_{R}(P, R), so we have K_{n}(R) ≅ K_{n}(R^{op}).

The product or direct sum A ⊕ A′ of two exact categories is exact by Example II.7.1.6, and Q(A ⊕ A′) = QA × QA′. Since the geometric realization preserves products by 2.1(4), we have BQ(A ⊕ A′) = BQA × BQA′ and hence K_{n}(A ⊕ A′) ≅ K_{n}(A) ⊕ K_{n}(A′). For example, if R_{1} and R_{2} are rings then P(R_{1} × R_{2}) ≅ P(R_{1}) ⊕ P(R_{2}) and we have K_{n}(R_{1} × R_{2}) ≅ K_{n}(R_{1}) ⊕ K_{n}(R_{2}).

Finally, suppose that i ↦ A_{i} is a functor from some small filtering category I to exact categories and exact functors. Then the filtered colimit A = \underrightarrow{lim} A_{i} is an exact category (Ex. II.7.9), and QA = \underrightarrow{lim} QA_{i}. Since geometric realization preserves filtered colimits by 2.1(3), we have BQA = \underrightarrow{lim} BQA_{i} and hence K_{n}(A) = \underrightarrow{lim} K_{n}(A_{i}). The K_{0} version of this result was given in chapter II, as 6.2.7 and 7.1.7.

For example, if a ring R is the filtered union of subrings R_{i} we have K_{n}(R) ≅ \underrightarrow{lim} K_{n}(R_{i}). However, i ↦ P(R_{i}) is not a functor. To fix this, we need to replace the category P(R_{i}) by the equivalent category P′(R_{i}) whose objects are idempotent matrices over R_{i}; P(R) is equivalent to the category P′(R) = \underrightarrow{lim} P_{i}. Alternatively one could use the Kleisli rectification, which is described in Ex. 4.5.

EXERCISES

4.1 Admissible subquotients. Let B be an object in an exact category A. An admissible layer in B is a pair of subobjects represented by a sequence B_{1} ↣ B_{2} ↣ B of admissible monics, and we call the quotient B_{2}/B_{1} an admissible subquotient of B. Show that a morphism A → B in QA may be identified with an isomorphism j: B_{2}/B_{1} ≅ A of A with an admissible subquotient of B, and that composition in QA arises from the fact that a subquotient of a subquotient is a subquotient.

4.2 If two exact categories A and A′ are equivalent (and the equivalence respects exactness), show that QA and QA′ are equivalent. If both are small categories, conclude that K_{n}(A) ≅ K_{n}(A′) for all n. 4.3 If A is an exact category, so is its opposite category A^{op} (see Example II.7.1.5). Show that Q(A^{op}) is isomorphic to QA. 4.4 Let B be an object in an exact category A. Show that the comma category QA/B is equivalent to the poset of admissible layers of B in the sense of Ex. 4.1. If P is an exact subcategory of A and i denotes the inclusion QP ⊂ QA, show that i/B is equivalent to the poset of admissible layers of B with B_{2}/B_{1} ∈ P. 4.5 Kleisli rectification. Let I be a filtering category, and let I → CAT be a lax functor in the sense of Ex. 2.8. Although the family of exact categories QA(i) is not filtering, the family of homotopy groups K_{n}A(i) is filtering. The following trick allows us make K-theoretic sense out of the phantom category A = \underrightarrow{lim} A(i).

Let A_{i} be the category whose objects are pairs (A_{j}, j ⟶^{f} i) with A_{j} in A(j)

and f a morphism in I. A morphism from (A_{j}, j ⟶^{f} i) to (A_{k}, k ⟶^{g} i) is a pair (j ⟶^{h} k, θ_{j}) where f = gh in I and θ_{j} is an isomorphism h_{∗}(A_{j}) ≅ A_{k} in A(k). Clearly A_{i} is equivalent to A(i), and i ↦ A_{i} is a functor. Thus if A denotes \underrightarrow{lim} A_{i} we have K_{n}A = \underrightarrow{lim} K_{n}A(i).

4.6 (Gersten) Suppose given a surjective homomorphism ϕ: K_{0}A → G.

(a) Show that there is a functor ψ: QA → G sending the morphism (4.1.1) of QA to ϕ[ker(j)], where G is regarded as a category with one object ∗. Using 4.2, show that the map π_{1}(QA) → π_{1}(G) is just ϕ.

(b) Show that the hypotheses of Theorem B are satisfied by ψ, so that B(ψ/∗) is the homotopy fiber of BQA → BG.

(c) Let B denote the full subcategory of all B in A with ϕ[B] = 0 in G. Use Theorem A to show that QB → ψ^{−1}(∗) is a homotopy equivalence.

(d) Suppose in addition that for every A in A there is an A′ such that ϕ[A′] = −ϕ[A], so that K_{0}B is the subgroup ker(ϕ) of K_{0}A by II.7.2. Use Theorem A to show that ψ^{−1}(∗) ≃ ψ/∗. This proves that BQB is the homotopy fiber of BQA → BG. Conclude that K_{n}B ≅ K_{n}A for all n ≥ 1.

§5. The “+ = Q” Theorem

Suppose that A is an additive category. One way to define the K-theory of A is to consider the symmetric monoidal category S = iso A (where □ = ⊕) and use the S^{−1}S construction: K_{n}^{⊕}A = π_{n}B(S^{−1}S) and K^{⊕}A = K(S) = B(S^{−1}S).

Another way is to suppose that A has the structure of an exact category and form the Q-construction on A with the S^{−1}S construction on S. Comparing the definitions of K_{0}^{⊕}A and K_{0}A in II.5.1.2 and II.7.1, we see that the K_{0} groups are not isomorphic in general, unless perhaps every exact sequence splits in A, i.e., unless A is a split exact category in the sense of II.7.1.2.

Here is the main theorem of this section.

Theorem 5.1 (Quillen). If A is a split exact category and S = iso A, then ΩBQA ≃ B(S^{−1}S). Hence K_{n}(A) ≅ K_{n}(S) for all n ≥ 0.

In fact, B(S^{−1}S) is the group completion of BS by Ex. 5.1. In some circumstances (see 3.7, 3.7.3 and 3.8.1), the S^{−1}S construction is a +–construction. In these cases, theorem 5.1 shows that the Q–construction is also a +–construction. For A = P(R), this yields the “+ = Q” theorem:

Corollary 5.2 (+ = Q). For every ring R,

ΩBQP(R) ≃ K_{0}(R) × BGL(R)^{+}.

Hence K_{n}(R) ≅ K_{n}P(R) for all n ≥ 0.

Definition 5.3. Given an exact category A, we define the category EA as follows. The objects of EA are admissible exact sequences in A. A morphism from E′ : (A′ ↣ B′ ↠ C′) to E : (A ↣ B ↠ C) is an equivalence class of diagrams of the following form, where the rows are exact sequences in A:

(5.3.1)  \begin{aligned}E′ : & A′ & ↣ & B′ & ↠ & C′ \\ & α↑ ∧ & & || & & ↑ ↑ \\ ↓ & A & ↣ & B′ & ↠ & C′′ \\ & || & & ∨ ↓β & & ∨ ↓ \\ E : & A & ↣ & B & ↠ & C. \end{aligned}

Two such diagrams are equivalent if there is an isomorphism between them which is the identity at all vertices except for the C′′ vertex.

Notice that the right column in (5.3.1) is just a morphism φ in QA from C′ to C, so the target C = t(A ↣ B ↠ C) is a functor t: EA → QA.

In order to improve legibility, it is useful to write E_{C} for the fiber category t^{−1}(C).

Fiber categories 5.4. If we fix φ as the identity map of C = C′, we see that the fiber category E_{C} = t^{−1}(C) of exact sequences with target C has for its morphisms all pairs (α, β) of isomorphisms fitting into a commutative diagram:

A′ ↣ B′ ↠ C
α ↑≅ ≅↓ β ||
A ↣ B ↠ C.

In particular, every morphism in E_{C} is an isomorphism.

Example 5.4.1. The fiber category E_{0} = t^{−1}(0) is equivalent to S = iso A. To see this, consider the functor from iso A to E_{0} sending A to the trivial sequence

A↣^{id}A ↠ 0. This functor is a full embedding. Moreover, every object of E_{0} is naturally isomorphic to such a trivial sequence, whence the claim.

Lemma 5.5. For any C in A, E_{C} is a symmetric monoidal category, and there is a faithful monoidal functor S → E_{C} sending A to A ↣ A ⊕ C ↠ C.

Proof. Given E_{i} = (A_{i} ↣ B_{i} ↠ C) in E_{C}, set E_{1} ∗ E_{2} equal to

(5.5.1)  A_{1} ⊕ A_{2} ↣ (B_{1} ×_{C} B_{2}) ↠ C.

This defines a symmetric product on E_{C} with identity e: 0 ↣ C ↠ C. It is now straightforward to check that S → E_{C} is a monoidal functor, and that it is faithful.

Remark 5.5.2. If A is split exact then every object of E_{C} is isomorphic to one coming from S. In particular, the category ⟨S, E_{C}⟩ is connected. This fails if A has a non-split exact sequence.

Proposition 5.6. If A is split exact, each S^{−1}S → S^{−1}E_{C} is a homotopy equivalence.

Proof. By Ex. 3.6 and 5.1, S^{−1}S → S^{−1}E_{C} → ⟨S, E_{C}⟩ is a fibration, so it suffices to prove that L = ⟨S, E_{C}⟩ is contractible. First, observe that the monoidal product on E_{C} induces a monoidal product on L, so BL is an H-space (as in 3.1). We remarked in 5.5.2 that L is connected. By [Wh, X.2.2], BL is group-like, i.e., has a homotopy inverse.

For every exact sequence E, there is a natural transformation δ_{E}: E → E ∗ E in L, given by the diagonal.

A ↣  B  ↠ C
↓  ↓  ||
A ⊕ A ↣ B ×_{C} B ↠ C

Now δ induces a homotopy between the identity on BL and muliplication by 2. Using the homotopy inverse to subtract the identity, this gives a homotopy between zero and the identity of BL. Hence BL is contractible.

We also need a description of how E_{C} varies with C.

Lemma 5.7. For each morphism φ: C′ → C in QA, there is a canonical functor φ^{∗}: E_{C} → E_{C′} and a natural transformation η_{E}: φ^{∗}(E) → E from φ^{∗} to the inclusion of E_{C} in EA.

In fact, t is a fibered functor with base-change φ^{∗} (Ex. 5.2). It follows (from 2.6.4) that C ↦ E_{C} is a contravariant functor from QA to CAT.

Proof. Choose a representative C′ ↞ C′′ ↣ C for φ and choose a pullback B′ of B and C′′ along C. This yields an exact sequence A ↣ B′ ↠ C′′ in A. (Why?) The composite B′ ↠ C′′ ↠ C′ is admissible; if A′ is its kernel then set

φ^{∗}(A ↣ B ↠ C) = (A′ ↣ B′ ↠ C′).

Since every morphism in E_{C} is an isomorphism, it is easy to see that φ^{∗} is a functor, independent (up to isomorphism) of the choices made. Moreover, the construction yields a diagram (5.3.1), natural in E; the map β is an admissible monic because A ↣ B′ ⟶^{β} B′ is. Hence (5.3.1) constitutes the natural map η_{E} from E to φ^{∗}(E).

Now the direct sum of sequences defines an operation ⊕ on EA, and S acts on EA via the inclusion of S in EA given by 5.4.1. That is, A′□(A ↣ B ↠ C) is the sequence A′ ⊕ A ↣ A′ ⊕ B ↠ C. Since t(A′□E) = t(E) we have an induced map T = S^{−1}t: S^{−1}EA → QA. This is also a fibered functor (Ex. 5.2).

Theorem 5.8. If A is a split exact category and S = iso A, then the sequence S^{−1}S → S^{−1}EA ⟶^{T} QA is a homotopy fibration.

Proof. We have to show that Quillen’s Theorem B applies, i.e., that the basechanges φ^{∗} of 5.7 are homotopy equivalences. It suffices to consider φ of the form 0 ↣ C and 0 ↞ C. If φ is 0 ↣ C, the composition of the equivalence S^{−1}S → S^{−1}E_{C} of 5.6 with φ^{∗} is the identity by Ex. 5.5, so φ^{∗} is a homotopy equivalence.

Now suppose that φ is 0 ↞ C. The composition of the equivalence S^{−1}S → S^{−1}E_{C} of 5.6 with φ^{∗} sends A to A ⊕ C by Ex. 5.5. Since there is a natural transformation A → A ⊕ C in S^{−1}S, this composition is a homotopy equivalence. Hence φ^{∗} is a homotopy equivalence.

Proof of theorem 5.1. This will follow from theorem 5.8, once we show that S^{−1}EA is contractible. By Ex. 5.3, EA is contractible. Any action of S on a contractible category must be invertible (3.5.1). By Ex. 3.5 and Ex. 5.1, EA → S^{−1}EA is a homotopy equivalence, and therefore S^{−1}EA is contractible.

EXERCISES

5.1 If A is an additive category, S = iso A is equivalent to the disjoint union of one-object categories Aut(A), one for every isomorphism class in A. Show that the translations Aut(A) → Aut(A⊕B) are injections. Then conclude using theorem 3.6 that B(S^{−1}S) is the group completion of the H-space BS = ∐ Aut(A).

5.2 Show that the target functor t: EA → QA is a fibered functor in the sense of definition 2.6.2, with base-change φ^{∗} given by 5.7. Then show that the action of S on EA is cartesian (Ex. 3.11), so that the induced functor S^{−1}EA → QA is also fibered, with fiber S^{−1}S over 0.

5.3 Let iQA denote the subcategory of QA whose objects are those of A but whose morphisms are admissible monomorphisms. Show that the category EA of 5.3 is equivalent to the subdivision category Sub(iQA) of Ex. 2.9. Conclude that the category EA is contractible.

5.4 Show that Quillen’s Theorem B can not apply to EA → QA unless A ≅ 0. Hint: Compare π_{0}S to K_{0}A.

5.5 If φ is the map 0 ↣ C, resp. 0 ↞ C, show that φ^{∗}: E_{C} → E_{0} ≅ S sends A ↣ B ↠ C to A, resp. to B.

5.6 Describe E′A = (EA)^{op}, which by Ex. 2.6 and 5.2 is cofibered over (QA)^{op}. Use E′A to prove the + = Q Theorem 5.1. Hint: There is a new action of S. Use pushout instead of pullback in (5.5.1) to prove the analogue of proposition 5.6.

5.7 Finite Sets. Let Sets_{f} denote the category of pointed finite sets, as in 3.1.1(a). We say that a sequence A ↣ B ↠ C is exact just in case the first map is an injection and the second map identifies C with B/A.

(a) Copy the Q-construction 4.1 to form a category QSets_{f}.

(b) Show that there is an extension category E′Sets_{f}, defined as in Ex. 5.6, which is cofibered over (QSets_{f})^{op} with S = iso Sets_{f} as the fiber over the basepoint.

(c) Modify the proof of the + = Q theorem to prove that ΩBQSets_{f} ≃ S^{−1}S.

§6. Waldhausen’s wS. construction

Our last construction of K-theory applies to Waldhausen categories, i.e., “categories with cofibrations and weak equivalences.” Unfortunately, this will occur only after a lengthy list of definitions, and we ask the reader to be forgiving.

Recall from chapter II, section 9 that a category with cofibrations is a category C with a distinguished zero object ‘0’ and a subcategory co(C) of morphisms in C called “cofibrations” (indicated with feathered arrows ↣). Every isomorphism in C is to be a cofibration, and so are the unique arrows 0 ↣ A for every object A in C. In addition, the pushout C ↣ B ∪_{A} C of any cofibration A ↣ B is a cofibration. (See Definition II.9.1 more precise statements.) These axioms imply that two constructions make sense: the coproduct B ⨿ C = B ∪_{0} C of any two objects, and every cofibration A ↣ B fits into a cofibration sequence A ↣ B ↠ B/A, where B/A is the cokernel of A ↣ B. The following is a restatement of Definition II.9.1.1:

Definition 6.1. A Waldhausen category C is a category with cofibrations, together with a family w(C) of morphisms in C called “weak equivalences” (indicated with decorated arrows ⟶^{∼}). Every isomorphism in C is to be a weak equivalence, and weak equivalences are to be closed under composition (so we may regard w(C) as a subcategory of C). In addition, a “Glueing axiom” (W3) must be satisfied, saying that the pushout of weak equivalences is a weak equivalence.

A functor F : A → C between two Waldhausen categories is called an exact functor if it preserves all the relevant structure: zero, cofibrations, weak equivalences and the pushouts along a cofibration.

A Waldhausen subcategory A of a Waldhausen category C is a subcategory which is also a Waldhausen category in such a way that: (i) the inclusion A ⊆ C is an exact functor, (ii) the cofibrations in A are the maps in A which are cofibrations in C and whose cokernel lies in A, and (iii) the weak equivalences in A are the weak equivalences of C which lie in A.

In order to describe Waldhausen’s wS. construction for K-theory, we need a sequence of Waldhausen categories S_{n}(C). S_{0}(C) is the zero category, and S_{1}(C) is the category C, but whose objects A are thought of as the cofibrations 0 ↣ A. The category S_{2}(C) is the extension category E of II.9.3. For convenience, we repeat its definition here.

Extension Categories 6.2. The objects of the extension category S_{2}(C) are the cofibration sequences A_{1} ↣ A_{2} ↠ A_{12} in C. A morphism E → E′ in S_{2}(C) is a commutative diagram:

E :  A_{1} ↣ A_{2} ↠ A_{12}
↓ ↓ ↓ ↓
E′ :  A′_{1} ↣ A′_{2} ↠ A′_{12}

We make S_{2}(C) in to a Waldhausen category as follows. A morphism E → E′ in S_{2}(C) is a cofibration if A_{1} → A′_{1}, A_{12} → A′_{12} and A′_{1} ∪_{A_{1}} A_{2} → A′_{2} are cofibrations in C. A morphism in S_{2}(C) is a weak equivalence if its component maps A_{i} → A′_{i} (i = 1, 2, 12) are weak equivalences in C.

The following technically convenient axiom is often imposed on C. It says that weak equivalences are closed under extensions.

Extension axiom 6.2.1. Suppose that f: E → E′ is a map between cofibration sequences, as in 6.2. If the source and quotient maps of f (A → A′ and C → C′) are weak equivalences, so is the total map of f (B → B′).

Definition 6.3. (S.C) If C is a Waldhausen category, let S_{n}C be the category whose objects A. are sequences of n cofibrations in C:

A.: 0 = A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n}

together with a choice of every subquotient A_{ij} = A_{j}/A_{i} (0 < i < j ≤ n). These choices are to be compatible in the sense that there is a commutative diagram:

(6.3.0)  A_{n−1,n} ; ↑ ↑ ; ⋯; A_{23} ↣ ⋯ ↣ A_{2n} ; ↑ ↑  ↑ ↑ ; A_{12} ↣ A_{13} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; A_{1} ↣ A_{2} ↣ A_{3} ↣ ⋯ ↣ A_{n}

The conventions A_{0j} = A_{j} and A_{jj} = 0 will be convenient at times. A morphism A. → B. in S_{n}C is a natural transformation of sequences.

If we forget the choices of the subquotients A_{ij} we obtain the higher extension category E_{n}(C) constructed in II.9.3.2. Since we can always make such choices, it follows that the categories S_{n}C and E_{n}(C) are equivalent. By Ex. II.9.4, E_{n}(C) is a Waldhausen category, so S_{n}C is also a Waldhausen category. Here are the relevant definitions for S_{n}, translated from the definitions II.9.3.2 for E.

A weak equivalence in S_{n}C is a map A. → B. such that each A_{i} → B_{i} (hence, each A_{ij} → B_{ij}) is a weak equivalence in C. A map A. → B. is a cofibration when for every 0 ≤ i < j < k ≤ n the map of cofibration sequences

A_{ij} ↣ A_{ik} ↠ A_{jk}
↓ ↓ ↓
B_{ij} ↣ B_{ik} ↠ B_{jk}

is a cofibration in S_{2}(C)(C).

The reason for including choices in the definition of the categories S_{n}(C) is that we can form a simplicial Waldhausen category.

Definition 6.3.1. For each n ≥ 0, the exact functor ∂_{0} : S_{n}(C) → S_{n−1}(C) is defined deletion of the bottom row of (6.3.0). That is, i.e., ∂_{0} is defined by the formula

∂_{0}(A.): 0 = A_{11} ↣ A_{12} ↣ A_{13} ↣ ⋯ ↣ A_{1n}

together with the choices ∂_{0}(A.)_{ij} = A_{i+1,j+1}. By Ex. 6.1, ∂_{0}(A.) is in S_{n−1}(C).

For 0 < i ≤ n we define the exact functors ∂_{i}: S_{n}(C) → S_{n−1}(C) by omitting the row A_{i∗} and the column containing A_{i} in (6.3.0), and reindexing the A_{jk} as needed. Similarly, we define the exact functors s_{i}: S_{n}(C) → S_{n+1}(C) by duplicating A_{i}, and reindexing with the normalization A_{i,i+1} = 0.

By Ex. 6.2, the S_{n}C fit together to form a simplicial Waldhausen category S.C, and the subcategories wS_{n}C of weak equivalences fit together to form a simplicial category wS.C. Hence their geometric realizations B(wS_{n}C) fit together to form a simplicial topological space BwS.C, and we write |wS.C| for the realization of BwS.C. Since S_{0}(C) is trivial, |wS.C| is a connected space.

Recall from chapter II, 9.1.2, that K_{0}(C) is defined as the group generated by the set of weak equivalence classes [A] of objects of C with the relations that [B] = [A] + [B/A] for every cofibration sequence

A ↣ B ↠ B/A.

Proposition 6.4. If C is a Waldhausen category then π_{1}|wS.C| ≅ K_{0}(C).

Proof. If X. is any simplicial space with X_{0} a point, then |X.| is connected and π_{1}|X.| is the free group on π_{0}(X_{1}) modulo the relations d_{1}(x) = d_{2}(x)d_{0}(x) for every x ∈ π_{0}(X_{2}). For X. = BwS.C, π_{0}(BwS_{1}C) is the set of weak equivalence classes of objects in C, π_{0}(BwS_{2}C) is the set of equivalence classes of cofibration sequences, and the maps ∂_{i}: S_{2}(C) → S_{1}(C) of 6.3.1 send A ↣ B ↠ B/A to B/A, B and A, respectively.

Definition 6.5. If C is a small Waldhausen category, its algebraic K-theory space K(C) = K(C, w) is the loop space

K(C) = Ω|wS.C|.

The K-groups of C are defined to be its homotopy groups:

K_{i}(C) = π_{i}K(C) = π_{i+1}|wS.C| if i ≥ 0.

Remark 6.5.1. Since the subcategory wC is closed under coproducts in C by axiom (W3), the coproduct gives an H-space structure to |wS.C| via the map

|wS.C| × |wS.C| ≅ |wS.C × wS.C| ⟶^{⨿} |wS.C|.

Simplicial Model 6.5.2. Suppose that C is a small Waldhausen category in which the isomorphisms iC are the weak equivalences. Let s_{n}C denote the set of objects of S_{n}C; as n varies, we have a simplicial set s.C. Waldhausen proved in [W1126, 1.4] that the inclusion |s.C| → |iS.C| is a homotopy equivalence. Therefore Ω|s.C| is a simplicial model for the space K(C).

Remark 6.5.3. In fact K(C) is an infinite loop space. To see this, note that we can apply the S. construction to each S_{n}C, obtaining a bisimplicial Waldhausen category S.S.C. Iterating this construction, we can form the multisimplicial Waldhausen categories S.^{n}C = S.S. · · · S.C and the multisimplicial categories wS.^{n}C of weak equivalences. Waldhausen points out on p. 330 of [W1126] that |wS.^{n}C| is the loop space of |wS.^{n+1}C|, and that the sequence of spaces

Ω|wS.C|, Ω|wS.S.C|, …, Ω|wS.^{n}C|, …

forms a connective Ω-spectrum KC, called the K-theory spectrum of C. Many authors think of the K-theory of C in terms of this spectrum. This does not affect the K-groups, because:

π_{i}(KC) = π_{i}K(C) = K_{i}(C), i ≥ 0.

An exact functor F induces a map F_{∗}: K(B) → K(C) of spaces, and spectra, and of their homotopy groups K_{i}(B) → K_{i}(C).

Exact Categories 6.6. We saw in II.9.1.3 that any exact category A becomes a Waldhausen category in which the cofibration sequences are just the admissible exact sequences, and the weak equivalences are just the isomorphisms. We write i(A) for the family of isomorphisms, so that we can form the K-theory space K(A) = Ω|iS.A|. Waldhausen proved in [W1126, 1.9] that there is a homotopy equivalence between |iS.A| and BQA, so that this definition is consistent with the definition of K(A) in definition 4.3. His proof is given in exercises 6.5 and 6.6 below.

Another important example of a Waldhausen category is R_{f}(X), introduced in II.9.1 and Ex. II.9.1. The so-called K-theory of spaces refers to the corresponding K-theory spaces A(X).

Example 6.7 (A(∗)). Recall from II.9.1.4 that the category R_{f} = R_{f}(∗) of finite based CW complexes is a Waldhausen category in which the family hR_{f} of weak equivalences is the family of weak homotopy equivalences. This category is saturated (II.9.7.1) and satisfies the extension axiom 6.2.1. Following Waldhausen [W1126], we write A(∗) for the space K(R_{f}) = Ω|hS.R_{f}|. In particular, A_{0}(∗) = K_{0}R_{f} = ℤ by II.9.1.5.

Example 6.7.1 (A(X)). More generally, let X be a CW complex. The category R(X) of CW complexes Y obtained from X by attaching cells, and having X as a retract, is a Waldhausen category in which cofibrations are cellular inclusions (fixing X) and weak equivalences are homotopy equivalences (see Ex. II.9.1). Consider the Waldhausen subcategory R_{f}(X) of those Y obtained by attaching only finitely many cells. Following Waldhausen [W1126], we write A(X) for the space K(R_{f}(X)) = Ω|hS.R_{f}(X)|. Thus A_{0}(X) = K_{0}R_{f}(X) is ℤ by Ex. II.9.1.

Similarly, we can form the Waldhausen subcategory R_{fd}(X) of those Y which are finitely dominated. We write A^{fd}(X) for K(R_{fd}(X)) = Ω|hS.R_{fd}(X)|. Note that A_{0}^{fd}(X) = K_{0}R_{fd}(X) is ℤ[π_{1}(X)] by Ex. II.9.1.

Cylinder Functors

When working with Waldhausen categories, it is often technically convenient to have mapping cylinders. Recall from Ex. 2.10 that the category C/C of arrows in C has the morphisms of C as its objects, and a map (a, b): f → f′ in Ar(C) is a commutative diagram in C:

(6.8.0)  \begin{matrix}A & ⟶^{f} & B \\ ^{a}↓ & & ↓^{b} \\ A′ & ⟶^{f′} & B′ \end{matrix}

The source s(f) = A and target t(f) = B of f define functors s, t: C/C → C.

Definition 6.8 (Cylinders). Let C be a Waldhausen category. A (mapping) cylinder functor on C is a functor T from the category C/C of arrows in C to the category C, together with natural transformations j_{1} : s ⇒ T, j_{2} : t ⇒ T and p: T ⇒ t so that for every f: A → B the diagram

A ⟶^{j_{1}} T (f) ⟵^{j_{2}} B
f ↘ ↓ p ↙=
B

commutes in C. The following conditions must also hold:

(i) T (0 ↣ A) = A, with p and j_{2} the identity map, for all A ∈ C. (ii) j_{1} ⨿ j_{2} : A ⨿ B ↣ T (f) is a cofibration for all f: A → B. (iii) Given a map (a, b): f → f′ in Ar(C), i.e., a commutative square (6.8.0), if a

and b are weak equivalences in C then so is T (f) → T (f′). (iv) Given a map (a, b): f → f′ in Ar(C), if a and b are cofibrations in C, then so

is T (f) → T (f′), and the following map, induced by condition (ii), is also a cofibration in C.

A′ ⨿_{A} T (f) ⨿_{B} B′ → T (f′)

We often impose the following extra axiom on the weak equivalences of C.

Cylinder Axiom 6.8.1. All maps p : T (f) → B are weak equivalences in C.

Suppose C has a cylinder functor T. The cone of an object A is cone(A) = T (A ↠ 0), and the suspension of A is ΣA = cone(A)/A. The cylinder axiom implies that cone(A) ⟶^{∼} 0 is a weak equivalence. Since A ↣ cone(A) ↠ ΣA is a cofibration sequence it follows from the description of K_{0}(C) in II.9.1.2 that [ΣA] = −[A] in K_{0}(C). (Cf. Lemma II.9.2.1.) In fact, the Additivity Theorem (see Chapter V below) implies that the map Σ: K(C) → K(C) is a homotopy inverse with respect to the H-space structure on K(C), because Σ_{∗} + 1 = cone_{∗} = 0.

The name ‘cylinder functor’ comes from the following two paradigms.

Example 6.8.2. The Waldhausen categories R_{f}(∗) and R_{f}(X) of examples 6.7 and 6.7.1 have a cylinder functor: T (f) is the usual (based) mapping cylinder of f. By construction, the mapping cylinder satisfies the cylinder axiom 6.8.1. Because of this paradigm, j_{1} and j_{2} are sometimes called the front and back inclusions.

Example 6.8.3. Let Ch be the Waldhausen category of chain complexes and quasi-isomorphisms constructed from an abelian (or exact) category C; see II.9.2. The mapping cylinder of f: A. → B. is the usual mapping cylinder chain complex, with

T (f)_{n} = A_{n} ⊕ A_{n−1} ⊕ B_{n}.

The suspension functor Σ(A.) = A.[−1] here is the shift operator: Σ(A.)_{n} = A_{n−1}.

Example 6.8.4. Exact categories usually do not have cylinder functors. This is reflected by the fact that for some A ∈ A there may be no B such that [A ⊕ B] = 0 in K_{0}(A). However, the Waldhausen category Ch^{b}(A) of bounded chain complexes does have a cylinder functor, and we used it to prove that K_{0}(A) ≅ K_{0}Ch^{b}(A) in II.9.2.2. In fact, K(A) ≃ K(Ch^{b}(A) by the Gillet-Waldhausen theorem presented in chapter V. Thus many results requiring mapping cylinders in Waldhausen Ktheory can be translated into results for Quillen K-theory.

Cofinality

Let A be a Waldhausen subcategory of B. We say that A is strictly cofinal in B if for every B in B there is an A in A so that A ∧ B is in A. The following result was proven by Waldhausen in [W1126, 1.5.9]. We will prove a different cofinality result in chapter VI.

Strict Cofinality Theorem 6.9. If A is a strictly cofinal Walhausen subcategory of B, then K(A) → K(B) is a homotopy equivalence. In particular, K_{n}(A) ≅ K_{n}(B) for all n.

EXERCISES

6.1 Show that for every 0 ≤ i < j < k ≤ n the diagram

A_{ij} ↣ A_{ik} ↠ A_{jk}

is a cofibration sequence, and that this gives an exact functor from S_{n}C to S_{2}C.

6.2 Show that each functor ∂_{i}: S_{n}C → S_{n−1}C is exact. Then show that S.C is a simplicial category.

6.3 Let F, F′: A → B be a pair of exact functors. A natural transformation η: F → F′ is called a weak equivalence from F to F′ if each F (A) ⟶^{∼} F′(A) is a weak equivalence in B. Show that a weak equivalence induces a homotopy between the two maps K(A) → K(B). Hint: Show that the maps wS_{n}A → wS_{n}B are homotopic in a compatible way.

6.4 If X. is any simplicial space with X_{0} a point, the formula for |X.| gives a map from the reduced suspension ΣX_{1} to |X.|. Using the Freudenthal map X_{1} → ΩΣX_{1} this gives a map X_{1} → Ω|X.| and hence maps π_{i}(X_{1}) → π_{i+1}|X.|. When X. = BwS.C, this yields a map from X_{1} = Bw(C) to K(C).

(a) Show that every weak self-equivalence α: A ⟶^{∼} A determines an element [α] of K_{1}(C). If β is another weak self-equivalence, show that [α][β] = [α ∧ β].

(b) If A is an exact category, considered as a Waldhausen category, show that the resulting map B(iso A) → K(A) induces a map from the group K_{1}^{⊕}A of 3.6.1 to K_{1}(A).

6.5 (Waldhausen) Let A be a small exact category. In this exercise we produce a map from |iS.A| ≃ |s.A| to BQA, where s.A is defined in 6.5.2.

(a) Show that an object A. of iS_{3}A determines a morphism in QA from A_{12} to A_{3}.

(b) Show that an object A. of iS_{5}A determines a sequence A_{23} → A_{14} → A_{5} of row morphisms in QA.

(c) Recall from Ex. 2.10 that the Segal subdivision Sub(s.A) is homotopy equivalent to s.A. Show that (a) and (b) determine a simplicial map Sub(s.A) → QA. Composing with |iS.A| ≃ Sub(s.A), this yields a map from |iS.A| to BQA.

6.6 We now show that the map |iS.A| → BQA constructed in the previous exercise is a homotopy equivalence. Let iQ_{n}A denote the category whose objects are the degree n elements of the nerve of QA, i.e., sequences A_{0} → ⋯ → A_{n} in QA, and whose morphisms are isomorphisms.

(a) Show that iQ.A is a simplicial category, and that the nerve of QA is the simplicial set of objects. Waldhausen proved in [W1126, 1.6.5] that BQA → |iQ.A| is a homotopy equivalence.

(b) Show that for each n there is an equivalence of categories Sub(iS_{n}A) ⟶^{∼} iQ_{n}A, where Sub(iS_{n}A) is the Segal subdivision category of Ex. 2.9. Then show that the equivalences form a map of simplicial categories Sub(iS.A) → iQ.A. This map must be a homotopy equivalence, because it is a homotopy equivalence in each degree.

(c) Show that the map of the previous exercise fits into a diagram

|s.A| ⟵^{≃} |Sub(s.A)| ⟶ BQA
≃↓  ^{≃}↓  ^{≃}↓
|iS.A| ⟵^{≃} |Sub(iS.A)| ⟶^{≃} |iQ.A|.

Conclude that the map |iS.A| → BQA of Ex. 6.5 is a homotopy equivalence. 6.7 Let Ch(C) be the Waldhausen category of chain complexes in an exact category C, as in 6.8.3. Show that Ch(C) and Ch^{b}(C) satisfy the Saturation axiom II.9.7.1, the Extension axiom 6.2.1, and the Cylinder Axiom 6.8.1. 6.8 Finite Sets. Let Sets_{f} be the Waldhausen category of finite pointed sets, where the cofibrations are the injections and the weak equivalences are the isomorphisms. Show that the argument of Ex. 6.5 and 6.6 may be used, together with theorem 3.7.2 and Ex. 5.7, to prove that the Waldhausen K-theory space K(Sets_{f}) is ℤ × Σ^{+} ≃ Ω^{∞}Σ^{∞}. Thus K_{n}(Sets_{f}) ≅ π_{n}^{s} for all n.

§7. The Gillet-Grayson construction

Let A be an exact category. Following Grayson and Gillet [GG], we define a simplicial set G. = G.A as follows.

Definition 7.1. If A is a small exact category, G. is the simplicial set defined as follows. The set G_{0} of vertices consists of all pairs of objects (A, B) in A. The set G_{1} of edges consists of all pairs of short exact sequences with the same cokernel:

(7.1.0)  A_{0} ↣ A_{1} ↠ A_{01}, B_{0} ↣ B_{1} ↠ A_{01}.

The degeneracy maps G_{1} → G_{0} send (7.1.0) to (A_{1}, B_{1}) and (A_{0}, B_{0}), respectively.

The set G_{n} consists of all pairs of triangular commutative diagrams in A of the form

(7.1.1) A_{n−1,n} ; ↑ ↑ ; ⋯; A_{12} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑ ; A_{01} ↣ A_{02} ↣ ⋯ ↣ A_{0n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n} A_{n−1,n} ; ↑ ↑ ; ⋯; A_{12} ↣ ⋯ ↣ A_{1n} ; ↑ ↑  ↑ ↑ ; A_{01} ↣ A_{02} ↣ ⋯ ↣ A_{0n} ; ↑ ↑  ↑ ↑  ↑ ↑ ; B_{0} ↣ B_{1} ↣ B_{2} ↣ ⋯ ↣ B_{n}

so that each sequence A_{i} ↣ A_{j} ↠ A_{ij} and B_{i} ↣ B_{j} ↠ A_{ij} is exact. As in the definition of S.A (6.3.1), the face maps ∂_{i}: G_{n} → G_{n−1} are obtained by deleting the row A_{i∗} and the columns containing A_{i} and B_{i}, while the degeneracy maps σ_{i}: G_{n} → G_{n+1} are obtained by duplicating A_{i} and B_{i}, and reindexing.

Omitting the choices A_{ij} for the cokernels, we can abbreviate (7.1.1) as:

(7.1.2)  A_{0} ↣ A_{1} ↣ A_{2} ↣ ⋯ ↣ A_{n} ; B_{0} ↣ B_{1} ↣ B_{2} ↣ ⋯ ↣ B_{n} .

Remark 7.1.3. |G.| is a homotopy commutative and associative H-space. Its product |G.| × |G.| → |G.| arises from the simplicial map G. × G. → G. whose components G_{n} × G_{n} → G_{n} are termwise ⊕.

Note that for each isomorphism A ≅ A′ in A there is an edge in G_{1} from (0, 0) to (A, A′), represented by (0 ↣ A ↠ A, 0 ↣ A′ ↠ A). Hence (A, A′) represents zero in the group π_{0}|G.|.

Lemma 7.2. There is a group isomorphism π_{0}|G.| ≅ K_{0}(A).

Proof. As in 2.3, π_{0}|G.| is presented as the set of elements (A, B) of G_{0}, modulo the equivalence relation that for each edge (7.1.0) we have

(A_{1}, B_{1}) = (A_{0}, B_{0}).

It is an abelian group by 7.1.3, with operation (A, B) ⊕ (A′, B′) = (A ⊕ A′, B ⊕ B′). Since (A⊕B, B ⊕A) represents zero in π_{0}|G.|, it follows that (B, A) is the inverse of (A, B). From this presentation, we see that there is a map K_{0}(A) → π_{0}|G.| sending [A] to (A, 0), and a map π_{0}|G.| → K_{0}(A), sending (A, B) to [A] − [B]. These maps are inverses to each other.

(7.3) We now compare G. with the loop space of the simplicial set s.A of 6.5.2. If we forget the bottom row of either of the two triangular diagrams in (7.1.1), we get a triangular commutative diagram of the form (6.3.0), i.e., an element of s_{n}A. The resulting set maps G_{n} → s_{n}A fit together to form a simplicial map ∂_{0} : G. → s.A.

Path Spaces 7.3.1. Recall from [WHomo, 8.3.14] that the path space P X. of a simplicial set X. has P X_{n} = X_{n+1}, its ith face operator is the ∂_{i+1} of X., and its ith degeneracy operator is the σ_{i+1} of X.. The forgotten face maps ∂_{0} : X_{n+1} → X_{n} form a simplicial map P X. → X., and π_{0}(P X.) ≅ X_{0}. In fact, σ_{0} induces a canonical simplicial homotopy equivalence from P X. to the constant simplicial set X_{0}; see [WHomo, Ex. 8.3.7]. Thus P X. is contractible exactly when X_{0} is a point.

Now there are two maps G_{n} → s_{n+1}A, obtained by forgetting one of the two triangular diagrams (7.1.1) giving an element of G_{n}. The face and degeneracy maps of G. are defined so that these yield two simplicial maps from G. to the path space P. = P (s.A). Clearly, either composition with the canonical map P. → s.A yields the map ∂_{0} : G. → s.A. Thus we have a commutative diagram

(7.3.2)  \begin{matrix}G. & ⟶ & P. \\ ↓ & & ↓ \\ P. & ⟶ & s.A. \end{matrix}

Since s_{0}A is a point, the path space |P.| is canonically contractible. Therefore this diagram yields a canonical map |G.| → Ω|s.A|. On the other hand, we saw in 6.6 that |s.A| ≃ BQA, so Ω|s.A| ≃ ΩBQA = K(A).

We cite the following result from [GG, 3.1]. Its proof uses simplicial analogues of Quillen’s theorems A and B.

Theorem 7.4. (Gillet-Grayson) Let A be a small exact category. Then the map of (7.3) is a homotopy equivalence:

|G.| ≃ Ω|s.A| ≃ K(A).

Hence π_{i}|G.| = K_{i}(A) for all i ≥ 0.

Example 7.5. A double s.e.s. in A is a pair ℓ of short exact sequences in A on the same objects:

ℓ: A ↣^{f} B ↠^{g} C, A ↣^{f′} B ↠^{g′} C.

Thus ℓ is an edge (in G_{1}) from (A, A) to (B, B). To ℓ we attach the element [ℓ] of K_{1}(A) = π_{1}|G.| given by the following 3-edged loop.

(A, A) ⟶^{ℓ} (B, B)
e_{A} ↖ ↗ e_{B}
(0, 0)

where e_{A} denotes the canonical double s.e.s. (0 ↣ A ↠ A, 0 ↣ A ↠ A).

The following theorem was proven by A. Nenashev in [Nen].

Nenashev’s Theorem 7.6. K_{1}(A) may be described as follows.

(a) Every element of K_{1}(A) is represented by the loop [ℓ] of a double s.e.s.;

(b) K_{1}(A) is presented as the abelian group with generators the double s.e.s. in A, subject to two relations:

(i) If E is a short exact sequence, the loop of the double s.e.s. (E, E) is zero; (ii) for any diagram of six double s.e.s. (7.6.1) such that the “first” diagram commutes, and the “second” diagram commutes, then

[r_{0}] − [r_{1}] + [r_{2}] = [c_{0}] − [c_{1}] + [c_{2}],

where r_{i} is the i^{th} row and c_{i} is the i^{th} column of (7.6.1).

(7.6.1)  \begin{matrix}A′ & ⇉ & A & ⇉ & A′′ \\ ↓↓ & & ↓↓ & & ↓↓ \\ B′ & ⇉ & B & ⇉ & B′′ \\ ↓↓ & & ↓↓ & & ↓↓ \\ C′ & ⇉ & C & ⇉ & C′′. \end{matrix}

Example 7.6.2. If α is an automorphism of A, the class [α] ∈ K_{1}(A) is the class of the double s.e.s. (0 ↣ A ↠^{α} A, 0 ↣ A ↠^{=} A).

If β is another automorphism of A, the relation [αβ] = [α] + [β] comes from relation (ii) for

0 ⇉  0  ⇉  0
↓↓  ↓↓  ↓↓
0 ⇉ A ⇉^{α}_{1} A
↓↓  1 ↓↓ 1  β ↓↓ 1
0 ⇉ A ⇉^{αβ}_{1} A.

EXERCISES

7.1 Verify that condition 7.6(i) holds in π_{1}|G.|. 7.2 Show that omitting the choice of quotients A_{ij} from the definition of G.A yields a homotopy equivalent simplicial set G′.A. An element of G′_{n}A is a diagram (7.1.2) together with a compatible family of isomorphisms A_{j}/A_{i} ≅ B_{j}/B_{i}. 7.3 Consider the involution on G. which interchanges the two diagrams in (7.1.1). We saw in 7.2 that it induces multiplication by −1 on K_{0}(A). Show that this involution is an additive inverse map for the H-space structure 7.1.3 on |G.|. 7.4 If α: A ≅ A is an isomorphism, use relation (ii) in Nenashev’s presentation 7.6 to show that [α^{−1}] ∈ K_{1}(A) is represented by the loop of the double s.e.s.:

A →^{α} A ; A →_{=} A

7.5 If A is a split exact category, use Nenashev’s presentation 7.6 to show that K_{1}(A) is generated by automorphisms (7.6.2).

§8. Karoubi-Villamayor K-theory

Following Gersten, we say that a functor F from rings (or rings without unit) to sets is homotopy invariant if F (R) ≅ F (R[t]) for every R. Similarly, a functor F from rings to CW complexes (spaces) is called homotopy invariant if for every ring R the natural map R → R[t] induces a homotopy equivalence F (R) ≃ F (R[t]). Note that each homotopy group π_{n}F (R) also forms a homotopy invariant functor.

Of course, this notion may be restricted to functors defined on any subcategory of rings which is closed under polynomial extensions and contains the evaluations as well as the inclusion R ⊂ R[t]. For example, we saw in II.6.5 and 7.8.3 that G_{0}(R) is a homotopy invariant functor defined on noetherian rings (and schemes) and maps of finite Tor-dimension.

Conversely, recall from III.3.1 that R is called F -regular if F (R) ≅ F (R[t_{1}, ..., t_{n}]) for all n. Clearly, any functor F from rings to sets becomes homotopy invariant when restricted to the subcategory of F -regular rings. For example, we see from II.7.7 that K_{0} becomes homotopy invariant when restricted to regular rings. The Fundamental Theorem in chapter V implies that the functors K_{n} are also homotopy invariant when restricted to regular rings.

There is a canonical way to make F into a homotopy invariant functor.

Strict homotopization 8.1. Let F be a functor from rings to sets. Its strict homotopization [F] is defined as the coequalizer of the evaluations at t = 0, 1: F (R[t]) ⇉ F (R). In fact, [F] is a homotopy invariant functor and there is a universal transformation F (R) → [F](R); see Ex. 8.1. Moreover, if F takes values in groups then so does [F]; see Ex. 8.2.

Example 8.1.1. Recall that a matrix is called unipotent if it has the form 1 + ν for some nilpotent matrix ν. Let Uni(R) denote the subgroup of GL(R) generated by the unipotent matrices. This is a normal subgroup of GL(R), because the unipotent matrices are closed under conjugation. Since every elementary matrix e_{ij}(r) is unipotent, this contains the commutator subgroup E(R) of GL(R).

We claim that [E]R = [Uni]R = 1 for every R. Indeed, if 1 + ν is unipotent, (1 + tν) is a matrix in Uni(R[t]) with ∂_{0}(1 + tν) = 1 and ∂_{1}(1 + tν) = (1 + ν). Since Uni(R) is generated by these elements, [Uni]R must be trivial. The same argument applies to the elementary group E(R).

We now consider GL(R) and its quotient K_{1}(R). A priori, [GL]R → [K_{1}]R is a surjection. In fact, it is an isomorphism.

Lemma 8.2. Both [GL]R and [K_{1}]R are isomorphic to GL(R)/Uni(R).

Definition 8.2.1. For each ring R, we define KV_{1}(R) to be GL(R)/Uni(R). Thus KV_{1}(R) is the strict homotopization of K_{1}(R) = GL(R)/E(R).

Proof. The composite Uni(R) → GL(R) → [GL]R is trivial, because it factors through [Uni]R = 1. Hence [GL]R (and [K_{1}]R) are quotients of GL(R)/Uni(R). By Higman’s trick III.3.4.1, if g ∈ GL(R[t]) is in the kernel of ∂_{0} then g ∈ Uni(R[t]) and hence ∂_{1}(g) ∈ Uni(R). Hence ∂_{1}(NGL(R)) = Uni(R). Hence GL(R)/Uni(R) is a strictly homotopy invariant functor; universality implies that the induced maps [GL]R → [K_{1}]R → GL(R)/Uni(R) must be isomorphisms.

To define the higher Karoubi-Villamayor groups, we introduce the simplicial ring R[Δ^{.}], and use it to define the notion of homotopization. The simplicial ring R[Δ^{.}] will also play a critical role in the last chapter, when we construct higher Chow groups and Motivic cohomology.

Definition 8.3. For each ring R the coordinate rings of the standard simplices form a simplicial ring R[Δ^{.}]. It may described by the diagram

R ⇇ R[t_{1}] ←^{←}_{←} R[t_{1}, t_{2}] ⇇ ⇇ ⋯ R[t_{1}, …, t_{n}] ⋯

with R[Δ^{n}] = R[t_{0}, t_{1}, ⋯, t_{n}]/ (∑ t_{i} = 1) ≅ R[t_{1}, ⋯, t_{n}]. The face maps are given by: ∂_{i}(t_{i}) = 0; ∂_{i}(t_{j}) is t_{j} for i < j and t_{j−1} for i > j. Degeneracies are given by: σ_{i}(t_{i}) = t_{i} + t_{i+1}; σ_{i}(t_{j}) is t_{j} for i < j and t_{j+1} for i > j.

Definition 8.4 (Homotopization). Let F be a functor from rings to CW complexes. Its homotopization F^{h}(R) is the geometric realization of the simplicial space F (R[Δ^{.}]). Thus F^{h} is also a functor from rings to CW complexes, and there is a canonical map F (R) → F^{h}(R).

Lemma 8.4.1. F^{h} is a homotopy invariant functor. Moreover, if F is homotopy invariant then F (R) ≃ F^{h}(R) for all R.

Proof. We claim that the inclusion R[Δ^{.}] ⊂ R[x][Δ^{.}] is a simplicial homotopy equivalence, split by evaluation at x = 0. For this, we define ring maps h_{i}: R[x][Δ^{n}] → R[x][Δ^{n+1}] by: h_{i}(f) = σ_{i}(f) if f ∈ R[Δ^{n}] and h_{i}(x) = x(t_{i+1} + ⋯ + t_{n}). These maps define a simplicial homotopy between the identity map of R[x][Δ^{.}] and the composite

R[x][Δ^{.}] ⟶^{x=0} R[Δ^{.}] ⊂ R[x][Δ^{.}].

Applying F gives a simplicial homotopy equivalence between F^{h}(R[Δ^{.}]) and F^{h}(R[x][Δ^{.}]). Geometric realization converts this into a topological homotopy equivalence between F^{h}(R) and F^{h}(R[x]).

Finally, if F is homotopy invariant then the map from the constant simplicial space F (R) to F (R[Δ^{.}]) is a homotopy equivalence in each degree. It follows (see [Wa78]) that their realizations F (R) and F^{h}(R) are homotopy equivalent.

It is easy to see that F → F^{h} is universal (up to homotopy equivalence) for functors from F to homotopy invariant functors. A proof of this fact is left to Ex. 8.3.

Lemma 8.5. Let F be a functor from rings to CW complexes. Then π_{0}(F^{h}) is the strict homotopization [F_{0}] of the functor F_{0}(R) = π_{0}F (R).

Proof. For any simplicial space X., the group π_{0}(|X.|) is the coequalizer of ∂_{0}, ∂_{1} : π_{0}(X_{1}) ⇉ π_{0}(X_{0}). In this case π_{0}(X_{0}) = π_{0}F (R) and π_{0}(X_{1}) = π_{0}F (R[t]).

Applying the functor GL gives us a simplicial group GL. = GL(R[Δ^{.}]).

Corollary 8.5.1. KV_{1}(R) ≅ π_{0}(GL.).

Definition 8.6. For n ≥ 1, we set KV_{n}(R) = π_{n−1}(GL.) = π_{n}(BGL.).

Corollary 8.6.1. The abelian groups KV_{n}(R) are homotopy invariant, i.e.,

KV_{n}(R) ≅ KV_{n}(R[t]) for every n ≥ 1.

Gersten observed that BGL. ≃ BGL(R[Δ^{.}])^{+}. In this case, the standard spectral sequence for a simplicial space E_{pq}^{1} = π_{p}(X_{q}) ⇒ π_{p+q}|X.| becomes the first quadrant spectral sequence (for p ≥ 1, q ≥ 0):

(8.6.2)  E_{pq}^{1} = K_{p}(R[Δ^{q}]) ⇒ KV_{p+q}(R).

If R is regular, then each simplicial group K_{p}(R[Δ^{.}]) is constant (by the Fundamental Theorem in chapter V). Thus the spectral sequence degenerates at E^{2}:

Theorem 8.7. (Gersten) If R is regular, then K_{p}(R) ≅ KV_{p}(R) for all p ≥ 1. For general R, there is a natural edge map K_{p}(R) → KV_{p}(R).

We now quickly develop the key points in KV -theory.

Definition 8.8. We say that a ring map f: A → B is a GL-fibration if

GL(A[t_{1}, ..., t_{n}]) × GL(B) → GL(B[t_{1}, ..., t_{n}])

is onto for every n. Note that we do not require A and B to have a unit.

Remark 8.8.1. Any GL-fibration must be onto. That is, B ≅ A/I for some ideal I of A. To see this, consider the (1, 2) entry α_{12} of a preimage of the elementary matrix e_{12}(bt). Since f(α_{12}) = bt, evaluation at t = 1 gives an element of A mapping to b ∈ B. However, not every surjection is a GL-fibration; see Ex. 8.5(b).

Proposition 8.9. If A → B is a GL-fibration with kernel I, there is a long exact sequence

KV_{n+1}(B) ⟶ KV_{n}(I) → KV_{n}(A) → KV_{n}(B) ⟶ ⋯
KV_{1}(I) → KV_{1}(A) → KV_{1}(B) → K_{0}(I) → K_{0}(A) → K_{0}(B).

Proof. Let G_{n} ⊂ GL(B[Δ^{n}]) denote the image of GL(A[Δ^{n}]). Then there is an exact sequence of simplicial groups

(8.9.1)  1 → GL(I[Δ^{.}]) → GL(A[Δ^{.}]) → G. → 1.

Now any short exact sequence of simplicial groups is a fibration sequence, meaning there is a long exact sequence of homotopy groups. Since the quotient GL(B[Δ^{.}])/G. is a constant simplicial group, it is now a simple matter to splice the long exact sequences together to get the result. We have indicated the details in Ex. 8.3.

Here is an application of this result. Since R[x] → R has a section, it is a GLfibration. By homotopy invariance, it follows that KV_{n}(xR[x]) = 0 for all n ≥ 1. (Another proof is given in Ex. 8.4.)

Definition 8.10. For any ring R (with or without unit), we define ΩR to be the ideal (x^{2} − x)R[x] of R[x].

Since ΩR is the kernel of x = 1: xR[x] → R, and this map is a GL-fibration by Ex. 8.7, we have the following corollary of 8.9.

Corollary 8.10.1. For all R, KV_{1}(R) is isomorphic to the kernel of the map K_{0}(ΩR) → K_{0}(xR[x]), and KV_{n}(R) ≅ KV_{n−1}(ΩR) for all n ≥ 2.

This shows that we can define KV_{n}(R) as KV_{1}(Ω^{n−1}R) for all n ≥ 2.

The following definition is due to Karoubi and Villamayor.

Definition 8.11. A positive homotopy K-theory consists of a sequence of functors K_{n}^{h}, n ≥ 1, on the category of rings without unit, together with natural connecting maps δ_{n}: K_{n+1}^{h}(R/I) → K_{n}^{h}(I) and δ_{0} : K_{1}^{h}(R/I) → K_{0}(I), defined for every GL-fibration R → R/I, satisfying the following axioms:

(1) The functors K_{n}^{h} are homotopy invariant;

(2) For every GL-fibration R → R/I the resulting sequence is exact:

K_{n+1}^{h}(R/I) ⟶^{δ} K_{n}^{h}(I) → K_{n}^{h}(R) → K_{n}^{h}(R/I) ⟶^{δ} K_{n−1}^{h}(I) →
K_{1}^{h}(R) → K_{1}^{h}(R/I) ⟶^{δ} K_{0}(I) → K_{0}(R) → K_{0}(R/I).

Theorem 8.11.1. There is a unique positive homotopy K-theory (up to isomorphism), namely K_{n}^{h} = KV_{n}.

Proof. The fact that KV_{n} form a positive homotopy K-theory is given by 8.6.1 and 8.9. The axioms imply that any other positive homotopy K-theory must satisfy the conclusion of 8.10.1, and so must be isomorphic to KV -theory.

EXERCISES

8.1 Let F be a functor from rings to sets. Show that [F] is a homotopy invariant functor, and that every natural transformation F (R) → G(R) to a homotopy invariant functor G factors uniquely through F (R) → [F](R). 8.2 If F is a functor from rings to groups, let NF (R) denote the kernel of the map t = 0: F (R[t]) → F (R). Show that the image F_{0}(R) of the induced map t = 1: NF (R) → F (R) is a normal subgroup of F (R), and that [F]R = F (R)/F_{0}(R). Thus [F]R is a group. 8.3 Let F and G be functors from rings to CW complexes, and assume that G is homotopy invariant. Show that any natural transformation F (R) → G(R) factors through maps F^{h}(R) → G(R) such that for each ring map R → S the map F^{h}(R) → F^{h}(S) → G(S) is homotopy equivalent to F^{h}(R) → F^{h}(S) → G(S). 8.4 Let R = R_{0}⊕R_{1}⊕⋯ be a graded ring. Show that for every homotopy invariant functor F on rings we have F (R_{0}) ≃ F (R). In particular, if F is defined on rings without unit then F (xA[x]) ≃ F (0) for every A. Hint: Copy the proof of III.3.3.2. 8.5 GL-fibrations. Let f: A → B be a GL-fibration with kernel I.

(a) Show that tA[t] → tB[t] and ΩA → ΩB are GL-fibrations.

(b) Show that ℤ → ℤ/4 is not a GL-fibration, but GL(ℤ) → GL(ℤ/4) is onto.

(c) If B is a regular ring (with unit), show that every surjection A → B is a GL-fibration. Hint: K_{1}(B) ≅ K_{1}(B[t]). 8.6 Let f: A → B is a GL-fibration with kernel I, and define G_{.} as in the proof of Proposition 8.9. Show that GL(B[Δ^{.}])/G. is a constant simplicial group. Use this to show that π_{i}(G_{.}) = KV_{i+1}(B) for all i > 0, but that the cokernel of π_{0}(G_{.}) → π_{0}GL(B[Δ^{.}]) is the image of K_{1}(B) in K_{0}(I) under the map of III.2.3. Then combine this with the long exact sequence of homotopy groups for (8.9.1) to finish the proof of 8.9. 8.7 Show that the map tA[t] → A, f(t) ↦ f(1), is a GL-fibration with kernel ΩA.

§9. Mod ℓ K-theory

In addition to the usual K-groups K_{i}(C) of a category C, it is often useful to study its “mod ℓ K-groups” K_{i}(C; ℤ/ℓ), where ℓ is a positive integer. In this section we quickly recount the basic construction from mod ℓ homotopy theory. Basic properties of mod ℓ homotopy theory may be found in [N].

Recall [N] that if m ≥ 2 the mod ℓ Moore space P^{m}(ℤ/ℓ) is the space formed from the sphere S^{m−1} by attaching an m-cell via a degree ℓ map. The suspension of P^{m}(ℤ/ℓ) is the Moore space P^{m+1}(ℤ/ℓ), and as m varies this yields a suspension spectrum, called the Moore spectrum P^{∞}(ℤ/ℓ).

Definition 9.1. If m ≥ 2, the mod ℓ homotopy “group” π_{m}(X; ℤ/ℓ) of a based topological space X is defined to be the pointed set [P^{m}(ℤ/ℓ), X] of based homotopy classes of maps from the Moore space P^{m}(ℤ/ℓ) to X.

For a general space X, π_{2}(X; ℤ/ℓ) isn’t even a group, but the π_{m}(X; ℤ/ℓ) are always groups for m ≥ 3 and abelian groups for m ≥ 4. However, if X = ΩY then π_{2}(X; ℤ/ℓ) is a group and we can define π_{1}(X; ℤ/ℓ) as π_{2}(Y; ℤ/ℓ); this is independent of the choice of U by Ex. 9.1. More generally, if X = Ω^{k}Y_{k} for k >> 0 and P^{m} = P^{m}(ℤ/ℓ) then the formula

π_{m}(X; ℤ/ℓ) = [P^{m}, X] = [P^{m}, Ω^{k}Y_{k}] ≅ [P^{m+k}, Y_{k}] = π_{m+k}(Y_{k}; ℤ/ℓ)

shows that we can ignore these restrictions on m, and that π_{m}(X; ℤ/ℓ) is an abelian group for all m ≥ 0 (or even negative m, as long as k > 2 + |m|).

In particular, if X is an infinite loop space then abelian groups π_{m}(X; ℤ/ℓ) are defined for all m ∈ ℤ, using the explicit sequence of deloopings of X provided by the given structure on X.

If m ≥ 2, the cofibration sequence S^{m−1} ⟶^{ℓ} S^{m−1} ⟶ P^{m}(ℤ/ℓ) defining P^{m}(ℤ/ℓ) induces an exact sequence of homotopy groups

π_{m}(X) ⟶^{ℓ} π_{m}(X) → π_{m}(X; ℤ/ℓ) ⟶^{∂} π_{m−1}(X) ⟶^{ℓ} π_{m−1}(X).

It is convenient to adopt the notation that if A is an abelian group then _{ℓ}A denotes the subgroup of all elements a of A such that ℓ · a = 0. This allows us to restate the above exact sequence in a concise fashion.

Universal Coefficient Sequence 9.2. For all m ≥ 3 there is a natural short exact sequence

0 → (π_{m}X) ⊗ ℤ/ℓ → π_{m}(X; ℤ/ℓ) ⟶^{∂} _{ℓ}(π_{m−1}X) → 0.

This sequence is split exact (but not naturally) when ℓ ≢ 2 mod 4.

For π_{2}, the sequence 9.2 of pointed sets is also exact in a suitable sense; see [N, p. 3]. However this point is irrelevant for loop spaces, so we ignore it.

Example 9.2.2. When ℓ = 2, the sequence need not split. For example, it is known that π_{m+2}(S^{m}; ℤ/2) = ℤ/4 for m ≥ 3, and that π_{2}(BO; ℤ/2) = ℤ/4; see [AT65].

Now suppose that C is either a symmetric monoidal category, or an exact category, or a Waldhausen category, so that the K-theory space K(C) is defined, and is an infinite loop space.

Definition 9.3. The mod ℓ K-groups of C are defined to be the abelian group:

K_{m}(C; ℤ/ℓ) = π_{m}(K(C); ℤ/ℓ), m ∈ ℤ.

If m ≥ 2 this definition states that K_{m}(C; ℤ/ℓ) = [P^{m}(ℤ/ℓ), K(C)]. Because K(C) ≃ Y, we can define K_{1}(C; ℤ/ℓ) in a way that is independent of the choice of Y. However, the groups K_{0}(C; ℤ/ℓ) and K_{m}(C; ℤ/ℓ) for m < 0 depend not only upon K(C), but also upon the choice of the deloopings of K(C) in the underlying K-theory spectrum KC. In fact, the literature is not consistent about K_{m}(C; ℤ/ℓ) when m < 2, even for K_{1}(R; Z/ℓ).

By Theorem 9.2, the mod ℓ K-groups are related to the usual K-groups.

Universal Coefficient Theorem 9.4. There is a short exact sequence

0 → K_{m}(C) ⊗ ℤ/ℓ → K_{m}(C; ℤ/ℓ) → _{ℓ}K_{m−1}(C) → 0

for every m ∈ ℤ, C, and ℓ. It is split exact (not naturally) unless ℓ ≡ 2 mod 4.

In particular, for C = P(R) we have an exact sequence

0 → K_{m}(R) ⊗ ℤ/ℓ → K_{m}(R; ℤ/ℓ) → _{ℓ}K_{m−1}(R) → 0

Ex. 9.2 shows that the splitting in 9.4 is not natural in R.

Example 9.4.1. (ℓ = 2) Since the isomorphism Ω^{∞}Σ^{∞} → ℤ × BO factors through K(ℤ) and K(ℝ), the universal coefficient theorem and 9.2.2 show that

K_{2}(ℤ; ℤ/2) ≅ K_{2}(ℝ; ℤ/2) ≅ π_{2}(BO; ℤ/2) = ℤ/4.

It turns out that for ℓ = 2 the sequence for K_{m}(R; ℤ/2) is split whenever multiplication by [−1] ∈ K_{1}(ℤ) is the zero map from K_{m−1}(R) to K_{m}(R). For example, this is the case for the finite fields 𝔽_{q}, an observation made in [Br].

Example 9.4.2 (Bott elements). Suppose that R contains a primitive ℓ^{th} root of unity ζ. The Universal Coefficient Theorem 9.4 provides an element β ∈ K_{2}(R; ℤ/ℓ), mapping to ζ ∈ _{ℓ}K_{1}(R). This element is called the Bott element, and it plays an important role in the product structure of the ring K_{∗}(R; ℤ/ℓ).

Remark 9.4.3. A priori, β depends not only upon ζ but also upon the choice of the splitting in 9.4. One way to choose β is to observe that the inclusion of μ_{ℓ} in GL_{1}(R) induces a map Bμ_{ℓ} → BGL(R) → BGL(R)^{+} and therefore a set function μ_{ℓ} → K_{2}(R; ℤ/ℓ). A posteriori, it turns out that this is a group homomorphism unless ℓ ≡ 2 (mod 4).

Example 9.5. Let k be the algebraic closure of the field 𝔽_{p}. Quillen’s compution of K_{∗}(𝔽_{q}) in 1.9.1 shows that K_{n}(k) = 0 for m even (m ≥ 2), and that K_{m}(k) = ℚ/ℤ[\frac{1}{p}] for m odd (m ≥ 1). It follows that if ℓ is prime to p then:

K_{m}(k; ℤ/ℓ) = \{ \begin{matrix}ℤ/ℓ & if m is even, m ≥ 0 \\ 0 & otherwise. \end{matrix}

In fact, K_{∗}(k; ℤ/ℓ) is the polynomial ring ℤ/ℓ[β] on the Bott element β ∈K_{2}(k; ℤ/ℓ), under the K-theory product we will discuss in the next chapter. See [Br] for more details.

Here is another way to define mod ℓ homotopy groups, and hence K_{∗}(C; ℤ/ℓ). Suppose that X is a loop space, and let F denote the homotopy fiber of the map X → X which is multiplication by ℓ.

Proposition 9.6. There are isomorphisms π_{m}(X; ℤ/ℓ) ≅ π_{m−1}(F) for all m ≥ 2.

Proof. (Neisendorfer) Let map(A, X) be the space of pointed maps. If S = S^{k} is the k-sphere then the homotopy groups of map(S^{k}, X) are the homotopy groups of X (reindexed by k), while if P = P^{k}(ℤ/ℓ) is a mod ℓ Moore space, the homotopy groups of map(P, X) are the mod ℓ homotopy groups of X (reindexed by k).

Now applying map(−, X) to a cofibration sequence yields a fibration sequence, and applying map(A, −) to a fibration sequence yields a fibration sequence; this may be formally deduced from the axioms (SM0) and (SM7) for any model structure, which hold for spaces. Applying map(−, X) to S^{k} → S^{k} → P^{k+1}(ℤ/ℓ) shows that map(P, X) is the homotopy fiber of map(S^{k}, X) → map(S^{k}, X). Applying map(S^{k}, −) to F → X → X shows that map(S^{k}, F) is also the homotopy fiber, and is therefore homotopy equivalent to map(P, X). Taking the homotopy groups yields the result.

EXERCISES

9.1 Suppose that X is a loop space. Show that π_{1}(X; ℤ/ℓ) is independent of the choice of Y such that X ≃ ΩY. This shows that K_{1}(R; ℤ/ℓ) and even K_{1}(C; ℤ/ℓ) 9.2 Let R be a Dedekind domain with fraction field F. Show that the kernel of the map K_{1}(R; ℤ/ℓ) → K_{1}(F; ℤ/ℓ) is SK_{1}(R)/ℓ. Hence it induces a natural map

ℓ Pic(R) ⟶^{ρ} F^{×}/F^{×ℓ}R^{×}.

Note that F^{×}/R^{×} is a free abelian group by I.3.6, so the target is a free ℤ/ℓ-module for every integer ℓ. Finally, use I.3.6 and I.3.8.1 to give an elementary description of ρ.

In particular, If R is the ring of integers in a number field F, the Bass-Milnor-Serre Theorem III.2.5 shows that the extension K_{1}(R; ℤ/ℓ) of_{ℓ} Pic(R) by R^{×} injects into F^{×}/F^{×ℓ}.