ALGEBRAIC K-THEORY OF SYMMETRIC MONOIDAL

ΓS_{∗}-CATEGORIES WITH WEAK EQUIVALENCES

John Rognes

October 22–29 1998

Abstract. These are notes for the third talk on Dundas’ preprint “The cyclotomic trace for symmetric monoidal categories”.

1. More about Γ-spaces

1.1 Special and very special Γ-spaces. A Γ-space is a functor F: Γ^{op} → S_{∗} such that F (S^{0}) = ∗. We say that F is special if the natural map

F (X ∨ Y) ⟶ F (X) × F (Y)

is a homotopy equivalence for all (finite) pointed sets X and Y. Then applying π_{0} to the diagram

F (S^{0}) × F (S^{0}) ⟵^{≃} F (S^{0} ∨ S^{0}) ⟶^{∇} F (S^{0})

induces a commutative monoid structure on π_{0}F (S^{0}). We say that F is very special if this abelian monoid is in fact a group, i.e., has inverses.

If F is a special Γ-space, then the associated prespectrum {n ↦ F (S^{n})} is a semi-Ω-spectrum, in the sense that the adjoint structure maps

F (S^{n}) ⟶ ΩF (S^{n+1})

are homotopy equivalences for all n ≥ 1. For a proof, see [Segal] or [Bousfield– Friedlander]. The idea is to show that the cofiber sequence X → CX → ΣX induces a simplicial fiber sequence, whose realization F (X) → F (CX) → F (ΣX) is also a fiber sequence when F (X) is connected. Since F (CX) is contractible, this gives the result for X = S^{n}, n ≥ 1.

If F is a very special Γ-space, then the associated prespectrum {n ↦ F (S^{n})} is a Ω-spectrum, in the sense that the adjoint structure maps

F (S^{n}) ⟶ ΩF (S^{n+1})

are homotopy equivalences for all n ≥ 0. The proof for n = 0 uses that F (S^{0}) is a grouplike monoid.

Typeset by AMS-TEX

1.2 Closed model category structures on ΓS_{∗}. A closed model category is a category C equipped with three subcategories cC, fC and wC whose morphisms are called cofibrations, fibrations and weak equivalences in C, respectively, subject to axioms reminiscent of the properties of cofibrations, fibrations and homotopy equivalences in the category of spaces and maps. In particular C has initial and terminal objects ∅ and ∗, respectively. An object c ∈ obC is called cofibrant if the unique morphism ∅ → c is a cofibration, and c is called fibrant if the unique morphism c → ∗ is a fibration.

The category S_{∗} of pointed simplicial sets (= spaces) is a closed model category, with degreewise injections as cofibrations, Kan fibrations as fibrations, and homotopy equivalences as weak equivalences. A Kan fibration E → B is a map with the right lifting property with respect to all inclusions Λ_{k}^{n} → Δ^{n} for 0 ≤ k ≤ n. Here Λ_{k}^{n} ⊂ ∂Δ^{n} is the kth horn given by omitting the kth face from the boundary of Δ^{n}. All spaces are cofibrant, while the fibrant spaces are the Kan complexes, i.e., the spaces having the extension property with respect to all inclusions Λ_{k}^{n} → Δ^{n}.

The category ΓS_{∗} has several useful closed model category structures. We now describe two: the strict Q-structure and the stable Q-structure. In both cases the cofibrations are the Q-cofibrations, i.e., the maps A → X having the left lifting property with respect to all pointwise fibrations E → B which are pointwise equivalences. This means that for each k_{+} ∈ obΓ^{op} the map of spaces E(k_{+}) → B(k_{+}) is a fibration and a homotopy equivalence.

In the strict Q-structure the weak equivalences are the pointwise equivalences. In the stable Q-structure the weak equivalences are the stable equivalences, i.e., the maps inducing isomorphisms between the homotopy groups of the associated (pre-)spectra.

In the strict Q-structure, the fibrations are the pointwise fibrations which are pointwise equivalences. These have the right lifting property with respect to all maps which are (strict) Q-cofibrations and strict (= pointwise) weak equivalences. The strictly fibrant objects are those which are pointwise fibrant, i.e., Kan.

In the stable Q-structure the fibrations are the maps having the right lifting property with respect to all maps which are (stable) Q-cofibrations and stable weak equivalences. In this case the stably fibrant objects are the very special Γ-spaces which are pointwise fibrant.

2. Algebraic K-theory

2.1 Segal’s construction. Let (C, □, e) be a symmetric monoidal category. It comes equipped with isomorphisms

a_{xyz}: (x□y)□z ≅ x□(y□z)
c_{xy}: x□y ≅ y□x
l_{y}: e□y ≅ y
r_{x}: x□e ≅ x

for all x, y, z ∈ obC, which satisfy a number of coherence diagrams. It is often convenient to suppress the direction of these isomorphisms, by using the same notation for a_{xyz} and its inverse, etc.

Building on an idea of Segal, we now define a functor H̄C from Γ^{op} taking values in (symmetric monoidal) categories, i.e., a so-called Γ-category.

ALGEBRAIC K-THEORY 3

For k_{+} ∈ obΓ^{op} we let (H̄C)(k_{+}) be a category of certain k-dimensional cubical diagrams in C, and maps between such cubical diagrams. Precisely, let Pk_{+} be the (discrete) category of pointed subsets of k_{+}, with only the identity morphisms. An object (c, γ) of (H̄C)(k_{+}) is a functor c: Pk_{+} → C such that c(∗) = e, equipped with coherent isomorphisms

γ_{A,B}: c(A ∨ B) ⟶^{≅} c(A)□c(B)

in C for all pointed subsets A, B ⊆ k_{+} with A ∩ B = ∗. Coherence means that the following diagrams commute:

[figure: c(A ∨ B ∨ C); γ_{A∨B,C}; ≅; c(A ∨ B)□c(C); γ_{A,B}□id; ≅; (c(A)□c(B))□c(C); a; ≅; c(A ∨ B ∨ C); γ_{A,B∨C}; ≅; c(A)□c(B ∨ C); id□γ_{B,C}; ≅; c(A)□(c(B)□c(C)) .]

Here a = a_{c(A),c(B),c(C)}.

[figure: c(A ∨ B); γ_{A,B}; ≅; c(A)□c(B); c; ≅; c(B ∨ A); γ_{B,A}; ≅; c(B)□c(A) .]

Here c = c_{c(A),c(B)}.

[figure: c(∗ ∨ B); γ_{∗,B}; ≅; c(∗)□c(B); c(B); l; ≅; e□c(B) .]

Here l = l_{c(B)} (or its inverse).

[figure: c(A ∨ ∗); γ_{A,∗}; ≅; c(A)□c(∗); c(A); r; ≅; c(A)□e .]

Here r = r_{c(A)} (or its inverse).

A morphism f: (c, γ) → (d, δ) in (H̄C)(k_{+}) is a natural transformation f: c → d of functors Pk_{+} → C which is compatible with the structural isomorphisms, i.e., such that the diagram

\xymatrix{c(A ∨ B) \ar[r]^{γ_{A,B}}_{≅} \ar[d]^{f_{A∨B}} & c(A)□c(B) \ar[d]^{f_{A}□f_{B}} \\ d(A ∨ B) \ar[r]^{δ_{A,B}}_{≅} & d(A)□d(B)}

commutes for all pointed subsets A, B ⊆ k_{+} with A ∩ B = ∗.

We think of the objects c(A) for pointed subsets A ⊆ k_{+} as located at the corners of a k-dimensional cube. Corresponding to the k subsets {0, i} ⊂ k_{+} for 1 ≤ i ≤ k there are k objects

c({0, 1}), …, c({0, k})

at the corners adjacent to the minimal one. If A = {i_{1}, …, i_{n}}_{+} there is an isomorphism

c(A) ≅ c({0, i_{1}})□ … □c({0, i_{n}})

induced by the structural isomorphisms γ. Hence the k objects displayed above determine the cube c: Pk_{+} → C up to isomorphism. Also a morphism f = {f_{A} | A ∈ obPk_{+}} is determined by its value on these corners, since f_{A∨B} = δ_{A,B}^{−1}(f_{A}□f_{B})γ_{A,B} allows us to recover each f_{A} from the f_{{0,i}} for 1 ≤ i ≤ k.

When k = 0 note that (H̄C)(0_{+}) is the trivial category (one object and one morphism), while when k = 1 there is an isomorphism of categories (H̄C)(1_{+}) ≅ C.

Next we explain how k_{+} ↦ (H̄C)(k_{+}) defines a functor from Γ^{op}. Given a pointed function f: k_{+} → l_{+}, we obtain a functor

(H̄C)(f): (H̄C)(k_{+}) → (H̄C)(l_{+})

taking a k-dimensional cubical diagram (c, γ) to an l-dimensional cubical diagram (d, δ), given as follows. For pointed subsets B ⊆ l_{+}, let

f^{!}(B) = f^{−1}(B \ ∗) ∪ ∗ ⊆ k_{+} .

Then d(B) = c(f^{!}(B)) and δ_{A,B} = γ_{f^{!}(A),f^{!}(B)} defines (d, δ) = (H̄C)(f)(c, γ).

There are k projection maps p_{i}: k_{+} → 1_{+} given by p_{i}(j) = 0 for j ≠ i and p_{i}(i) = 1. They induce functors (H̄C)(p_{i}): (H̄C)(k_{+}) → (H̄C)(1_{+}) for 1 ≤ i ≤ k.

Lemma. H̄C is a special Γ-category, i.e., the product of functors

∏_{i=1}^{k}(H̄C)(p_{i}): (H̄C)(k_{+}) ⟶ ∏_{i=1}^{k}(H̄C)(1_{+})

is an equivalence of categories.

2.2 Algebraic K-theory of a symmetric monoidal category.

Let C be a (small) category. Its nerve category NC is the simplicial category

[q] ↦ N_{q}C

which in degree q is the category of functors [q] → C and natural transformations between these. Taking object sets degreewise defines its traditional nerve as a simplicial set, here denoted obNC.

Let (C, □, e) be a (small) symmetric monoidal category. The algebraic K-theory of C is the special Γ-space

K(C) = obNH̄C .

Example. Let R be an associative ring, and let iP(R) be the category of finitely generated projective R-modules and R-module isomorphisms. Its algebraic Ktheory

K(R) = obNH̄iP(R)

is a special Γ-space with underlying space

K(R)(S^{0}) = obN(H̄iP(R))(1_{+}) ≅ obNiP(R) ≃ ∐_{P} B Aut_{R}(P) ,

i.e., the traditional nerve of iP(R) as a simplicial set. Its underlying infinite loop space is

Ω^{∞}K(R) ⟵^{≃} ΩK(R)(S^{1}) ⟵^{ι} K(R)(S^{0}) ,

i.e., the group completion of this underlying space.

Example. Let M be an abelian monoid, viewed as a symmetric monoidal category M with obM = M and only identity morphisms. Then

(H̄M)(k_{+}) ≅ M × ⋯ × M

(k factors on the right), and obH̄M is isomorphic to the Eilenberg–Mac Lane Γspace HM.

2.3 Enrichment. Each category (H̄C)(k_{+}) is again a symmetric monoidal category, with operation ¯□ acting pointwise on cubical diagrams as in (c ¯□d)(A) = c(A)□d(A) for all A ∈ obPk_{+}. Hence the construction H̄ acts on symmetric monoidal categories, and may be iterated.

Also, if C is enriched, say in ΓS_{∗}, then so is each category (H̄C)(k_{+}), and H̄C is a functor from Γ^{op} to (symmetric monoidal) ΓS_{∗}-categories. This does not affect the objects of (H̄C)(k_{+}), but replaces the morphism sets (H̄C)(k_{+})(c, d) by morphism Γ-spaces.

When C is a ΓS_{∗}-category we let UC denote its underlying Ens-category, with the same object class obUC = obC, but with morphism sets

UC(c, d) = ΓS_{∗}(𝕊, \underline{C}(c, d))_{0} ≅ \underline{C}(c, d)(S^{0})_{0} .

Then C defines a functor

\underline{C}: UC^{op} × UC ⟶ ΓS_{∗}

given on objects by \underline{C}: (c, d) ↦ \underline{C}(c, d). Morphisms f: c′ → c and g: d → d′ in UC are represented by maps f: 𝕊 → \underline{C}(c′, c) and g: 𝕊 → \underline{C}(d, d′) in ΓS_{∗}, and \underline{C} takes (f, g) to the composite

f^{∗}g_{∗} = g_{∗}f^{∗}: \underline{C}(c, d) ≅ 𝕊 ∧ \underline{C}(c, d) ∧ 𝕊 ⟶^{f∧1∧g} \underline{C}(c′, c) ∧ \underline{C}(c, d) ∧ \underline{C}(d, d′) ⟶^{∘} \underline{C}(c′, d′) .

3. Nerve categories

3.1 The nerve of a ΓS_{∗}-category. Let C be a ΓS_{∗}-category. We will define its nerve category NC as a simplicial ΓS_{∗}-category:

NC: [q] ↦ N_{q}C .

In degree q, N_{q}C has as objects the functors c: [q] → UC, i.e., the diagrams

c_{0} → c_{1} → ⋯ → c_{q}

in the underlying Ens-category UC. Thus obN_{q}C = obN_{q}UC. The problem is to define the morphism Γ-spaces N_{q}\underline{C}(c, d).

Let c, d: [q] → UC be objects in N_{q}C. The morphism Γ-space should be the Γ-space of diagrams

\xymatrix{c_{0} \ar[r] \ar[d]^{f_{0}} & c_{1} \ar[r] \ar[d]^{f_{1}} & … \ar[r] & c_{q} \ar[d]^{f_{q}} \\ d_{0} \ar[r] & d_{1} \ar[r] & … \ar[r] & d_{q}}

in some appropriate sense. Here the horizontal arrows are morphisms in UC, while the vertical arrows belong to C. Their composites must be interpreted in the sense of section 2.3, namely using the associated functor \underline{C}: UC^{op} × UC → ΓS_{∗}. For each arrow i → j in [q] we then want the various composite maps c_{i} → d_{j} to agree, as elements in \underline{C}(c_{i}, d_{j}).

To make this precise, consider the following diagram in ΓS_{∗}:

(∗)  \xymatrix{ &  &  & \underline{C}(c_{0}, d_{0}) \ar[d]^{d(0→1)_{∗}} \\  &  & \underline{C}(c_{1}, d_{1}) \ar[r]^{c(0→1)^{∗}} \ar[d]^{d(1→2)^{∗}} & \underline{C}(c_{0}, d_{1}) \ar[d]^{d(1→2)_{∗}} \\  &  & ⋮ \ar[d]^{d(q−1→q)_{∗}} & ⋮ \ar[d]^{d(q−1→q)_{∗}} \\ \underline{C}(c_{q}, d_{q}) \ar[r]^{c(q−1→q)^{∗}} & … \ar[r]^{c(1→2)^{∗}} & \underline{C}(c_{1}, d_{q}) \ar[r]^{c(0→1)^{∗}} & \underline{C}(c_{0}, d_{q}) .}

Its categorical limit in ΓS_{∗} is contained in the product ∏_{i=0}^{q} \underline{C}(c_{i}, d_{i}), given pointwise as the subspace of tuples (f_{i})_{i=0}^{q} such that c(i → j)^{∗}(f_{j}) = d(i → j)_{∗}(f_{i}) in \underline{C}(c_{i}, d_{j}). This is the morphism Γ-space we want.

Hence we define

N_{q}\underline{C}(c, d) = lim_{i→j∈obT [q]} \underline{C}(c_{i}, d_{j})

pointwise as a Γ-space. Here T [q] is the twisted arrow category of [q], appearing as the category indexing the diagram (∗) above. This defines the ΓS_{∗}-category N_{q}C, and the association [q] ↦ N_{q}C is contravariantly functorial in [q] as usual, defining the nerve category of C as a simplicial ΓS_{∗}-category NC.

We have obNC = obNUC as simplicial sets.

3.2 The homotopy nerve of a ΓS_{∗}-category. The categorical limit used in defining the nerve category NC does in general not preserve homotopy equivalences. This will be a problem later, and one way to alleviate the difficulty is to replace the categorical limit by a homotopy limit, in the sense of [Bousfield–Kan].

We define the homotopy nerve category hoNC of a ΓS_{∗}-category C as the simplicial ΓS_{∗}-category

[q] ↦ hoN_{q}C

where ob hoN_{q}C = obN_{q}UC is the set of functors c: [q] → UC, with morphism Γ-spaces

hoN_{q}\underline{C}(c, d) = holim_{i→j∈obT [q]} \underline{C}(c_{i}, d_{j}) .

There is a canonical map

κ: lim_{i→j∈obT [q]} \underline{C}(c_{i}, d_{j}) ⟶ holim_{i→j∈obT [q]} \underline{C}(c_{i}, d_{j})

inducing a ΓS_{∗}-functor κ: NC → hoNC. If C is such that the diagram (∗) consists of pointwise equivalences of pointwise fibrant Γ-spaces, then κ is a pointwise equivalence, and the nerve category and homotopy nerve categories will be pointwise equivalent as ΓS_{∗}-categories.

3.3 Nerve categories of ΓS_{∗}-categories with weak equivalences. Let wUC be a subcategory of UC. We call the morphisms in wUC weak equivalences. Let N^{w}C be the full simplicial ΓS_{∗}-subcategory of NC with objects in degree q the functors c: [q] → UC that factorize through the subcategory wUC. Likewise let hoN^{w}C be the full simplicial ΓS_{∗}-subcategory of hoNC with objects in degree q the functors c: [q] → UC that factorize through the subcategory wUC.

The nerve and homotopy nerve constructions discussed in this chapter are functorial with respect to ΓS_{∗}-functors, and ΓS_{∗}-natural transformations induce simplicial homotopies, much as for the traditional nerve construction.

4. Enriched algebraic K-theory

4.1 The algebraic K-theory category. A symmetric monoidal ΓS_{∗}-category with weak equivalences is a symmetric monoidal ΓS_{∗}-category (C, □, e) together with a subcategory wUC ⊆ UC which is closed under □, and contains all isomorphisms.

(Thus (wUC, □, e) is a symmetric monoidal subcategory of (UC, □, e) with the □-operation induced from C.)

We define the algebraic K-theory of C with respect to wUC to be the objects of the simplicial ΓS_{∗}-category

K(C, w) = hoN^{w}H̄C .

Thus obK(C, w) is a simplicial special Γ-space. In degree [q] it is the Γ-space of functors c: [q] → UH̄C ≅ H̄UC that factor through the subcategory wH̄UC = H̄wUC. Its underlying space obK(C, w)(S^{0}) is the traditional nerve of the category (H̄wUC)(1_{+}) ≅ wUC. Hence there is a natural map

ι: obNwUC ≅ obK(C, w)(S^{0}) ⟶ ΩobK(C, w)(S^{1}) ≃ Ω^{∞}obK(C, w)

to the underlying infinite loop space of obK(C, w).

4.2 The Dennis trace map. For any ΓS_{∗}-category C we have defined its topological Hochschild homology T HH(C) as a simplicial Γ-space, and there is a Dennis trace map

obC ⟶ T HH(C)(S^{0})_{0} ⊂ T HH(C)(S^{0}) .

Now suppose C is a symmetric monoidal ΓS_{∗}-category with weak equivalences wUC. Then K(C, w) is a simplicial ΓS_{∗}-category, so we may apply the Dennis trace map construction degreewise. This defines a natural transformation

tr: obK(C, w) ⟶ T HH(K(C, w))(S^{0})_{0} ⊂ T HH(K(C, w))(S^{0})

from the algebraic K-theory of C with respect to w to the topological Hochschild homology of K(C, w).

5. Strong equivalences

5.1 Topological Hochschild homology and strong equivalences. Let C be a symmetric monoidal ΓS_{∗}-category with weak equivalences wUC. We say that C is a symmetric monoidal ΓS_{∗}-category with strong equivalences if each morphism f: c → d in wUC induces pointwise equivalences of pointwise fibrant Γ-spaces

f_{∗}: \underline{C}(b, c) ⟶ \underline{C}(b, d)

for all b ∈ obUC.

Proposition. [Dundas, 4.7] Let C be a symmetric monoidal ΓS_{∗}-category with strong equivalences wUC. Then

T HH(C) ⟶^{≃} T HH(hoN^{w}C) .

Here C is identified with the 0-simplices of hoN^{w}C.

The proof uses that all vertical maps in (∗) are pointwise equivalences, so the homotopy limit defining hoN^{w}\underline{C}(c, d) is homotopy equivalent to the bottom left Γspace \underline{C}(c_{q}, d_{q}). This makes it possible to replace the morphism Γ-spaces in hoN^{w}C by the morphism Γ-spaces in C, and then an explicit simplicial homotopy at the level of T HH replaces the underlying Ens-category of hoN^{w}C by the underlying Ens-category of C.

Proposition. [Dundas, 4.9] Let C be a symmetric monoidal ΓS_{∗}-category with weak equivalences wUC. Then

∏_{i=1}^{k}(p_{i})_{∗}: T HH((H̄C)(k_{+})) ⟶^{≃} ∏_{i=1}^{k} T HH(C)

is a pointwise equivalence, and T HH(H̄C) is a very special Γ-space (in both directions).

Here T HH(H̄C) is a biΓ-space, i.e., a functor Γ^{op} ×Γ^{op} → S_{∗}, with one direction coming from T HH and the ΓS_{∗}-enrichment in C, the other direction coming from Segal’s construction H̄ and the symmetric monoidal structure on C.

The proof uses the equivalence (H̄C)(k_{+}) → C^{×k} of ΓS_{∗}-categories and the (not so obvious) fact that T HH preserves products, to factorize the map above into the two pointwise equivalences

T HH((H̄C)(k_{+})) ⟶^{≃} T HH(C^{×k})

and

T HH(C^{×k}) ⟶^{≃} T HH(C)^{×k} .

Let k_{+}, l_{+} ∈ obΓ^{op}. For a fixed k_{+} the assembly map

T HH(C)(l_{+}) ∧ k_{+} ⟶ T HH((H̄C)(k_{+}))(l_{+})

is a stable equivalence viewed as a map of Γ-spaces in l_{+}, because T HH((H̄C)(k_{+})) is special in k_{+}. We can express this as:

Corollary. There is a stable equivalence

(Σ^{∞}T HH(C))(X) ⟶^{≃} T HH((H̄C)(X))

for each X ∈ Γ^{op}.

To combine these two results we can think of H̄C as a (symmetric monoidal) ΓS_{∗}-category with strong equivalences.

Corollary. [Dundas, 4.10] Let C be a symmetric monoidal ΓS_{∗}-category with strong equivalences wUC. Then the composite

(Σ^{∞}T HH(C))(X) ⟶ T HH((H̄C)(X)) ⟶ T HH((K(C, w))(X))

is a stable equivalence for each X ∈ Γ^{op}.

5.2 The trace map and strong equivalences. Let C be a symmetric monoidal ΓS_{∗}-category with strong equivalences wUC. The results above combine to define a trace map of infinite loop spaces

Ω^{∞}obK(C, w) ⟶^{tr} Ω^{∞}T HH(K(C, w))

⟵^{≃} Ω^{∞}T HH(H̄C)

⟵^{≃} Ω^{∞}T HH(C)

where the ‘wrong way’ maps are homotopy equivalences. The diagram is natural in C and w.

Given an 𝕊-algebra A, i.e., a monoid in (ΓS_{∗}, ∧, 𝕊), it remains to construct a symmetric monoidal ΓS_{∗}-category F_{A} with strong equivalences such that

ΩobK(F_{A}, w)(S^{1}) ≃ K_{0}^{f}(π_{0}(A)) × BĜL(A)^{+}

and such that T HH(A) ≃ T HH(F_{A}). We leave this for another lecture.