STABLE BUNDLES OVER RIG CATEGORIES

NILS A. BAAS, BJØRN IAN DUNDAS, BIRGIT RICHTER AND JOHN ROGNES

Abstract. The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology.

The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories) the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to K(R) ≃ ℤ × |BGL(R)|^{+}, where GL(R) is the monoidal category of weakly invertible matrices over R.

The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If π_{0}R is a ring this is almost formal, and our approach is to replace R by a ring completed version, R̄, provided by [BDRR1] with HR ≃ HR̄ and π_{0}R̄ the ring completion of π_{0}R. The remaining step is then to show that “stable R-bundles” and “stable R̄-bundles” are the same, which is done by a hands-on contraction of a custom-built model for the difference between BGL(R) and BGL(R̄).

1. Introduction and main result

In telescopic complexity 0, 1 and ∞ there are cohomology theories that possess a geometric definition: de Rham cohomology of manifolds is given in terms of differential forms, cohomology classes in real and complex K-theory are classes of virtual vector bundles, and complex cobordism has a geometric definition per se. In order to understand phenomena of intermediate telescopic complexity, it is desirable to have geometric interpretations for such cohomology theories as well.

In [BDR] it was conjectured that virtual 2-vector bundles provide a geometric interpretation of a cohomology theory of telescopic complexity 2 which qualifies as a form of elliptic cohomology. More precisely, it was conjectured that the algebraic K-theory of a commutative rig category R is equivalent to the algebraic K-theory of the ring spectrum associated with R. The case of virtual 2-vector bundles arises when R is the category of finite dimensional complex vector spaces, with ⊕ and ⊗_{C} as sum and multiplication. This, together with the analysis of the K-theory of complex topological K-theory due to Ausoni and the fourth author [AR, A], and the Quillen–Lichtenbaum conjecture for the integers, gives the desired relation to elliptic cohomology.

In this paper we prove the conjecture from [BDR].

Let R be a rig category (also known as a bimonoidal category), i.e., a category with two operations ⊕ and ⊗ satisfying the axioms of a rig (ring without negative elements) up to coherent natural isomorphisms. In analogy with Quillen’s definition of the algebraic K-theory space K(A) = ΩB(∐_{n} BGL_{n}(A)) of a ring A, the algebraic K-theory of R was defined in [BDR] as K(R) = ΩB(∐_{n} |BGL_{n}(R)|) ≃ ℤ × |BGL(R)|^{+} where B and GL_{n} are versions of the bar construction and the general linear group appropriate for rig categories.

On the other hand, forgetting the multiplicative structure, R has an underlying symmetric monoidal category, and so it makes sense to speak about its K-theory spectrum HR with respect to ⊕. The K-theory spectrum construction HR is a direct extension of the usual Eilenberg–Mac Lane construction, and can, since R is a rig category, be endowed with the structure of a strict ring spectrum, for instance through the model given by Elmendorf and Mandell in [EM]. Hence, we may speak about its algebraic K-theory space K(HR). We prove that, under certain mild restrictions on R, there is an equivalence

K(R) ≃ K(HR) .

Date: September 10, 2009.

2000 Mathematics Subject Classification. Primary 19D23, 55R65; Secondary 19L41, 18D10.

Key words and phrases. Algebraic K-theory, topological K-theory, 2-vector bundles, elliptic cohomology, bimonoidal categories, bipermutative categories.

The first author would like to thank the Institute for Advanced Study, Princeton, for their hospitality and support during his stay in the spring of 2007. Part of the work was done while the second author was on sabbatical at Stanford University, whose hospitality and stimulating environment is gratefully acknowledged. The third author thanks the SFB 676 for support and the topology group in Sheffield for stimulating discussions on the subject.

In the special situation where R is a ring (i.e., R is discrete as a category and has negative elements), this is the standard assertion that the K-theory of a ring is equivalent to the K-theory of its associated Eilenberg-Mac Lane spectrum. The key difficulty in establishing the equivalence above lies in proving that the lack of negative elements makes no difference for algebraic K-theory, even for rig categories.

More precisely, we prove the following result:

Theorem 1.1. Let (R, ⊕, 0_{R}, ⊗, 1_{R}) be a small topological rig category satisfying the following conditions:

(1) All morphisms in R are isomorphisms, i.e., R is a groupoid.

(2) For every object X ∈ R the translation functor X ⊕ (−) is faithful. Then |BGL(R)| and BGL(HR) are weakly equivalent. Hence, the algebraic K-theory space of R as

a rig category,

K(R) = ΩB( ∐_{n⩾0} |BGL_{n}(R)|) ≃ ℤ × |BGL(R)|^{+} ,

is weakly equivalent to the algebraic K-theory space of the strict ring spectrum associated to R,

K(HR) = ΩB( ∐_{n⩾0} BGL_{n}(HR)) ≃ ℤ × BGL(HR)^{+} .

Addendum 1.2. In particular, if R is the category of finite dimensional complex vector spaces, the theorem states that stable 2-vector bundles, in the sense of [BDR] are classified by BGL(ku), where ku = HR is the connective complex K-theory spectrum with π_{∗}ku = ℤ[u], |u| = 2.

In contrast to K(HR), which is built in a two-stage process, the K-theory of the (strictly) bimonoidal category R is built using both monoidal structures at once, so in this sense K(R) is a model that is easier to understand and handle than K(HR).

The conditions (1) and (2) on R are not restrictive for the applications we have in mind, and are associated with the fact that in [BDRR1] we chose to work with variants of the Grayson–Quillen model for K-theory. Probably, the restrictions can be removed if one uses another technological platform.

Among those rig categories that satisfy the requirements of Theorem 1.1 are the following ‘standard’ ones, usually considered in the context of K-theory constructions.

• If R is the discrete category (having only identity morphisms) with objects the elements of a ring with unit, R, then HR is the Eilenberg-Mac Lane spectrum HR.

• The sphere spectrum S is the algebraic K-theory spectrum of the small rig category of finite sets E. The objects of E are the finite sets n = {1, …, n} with n ⩾ 0. Here the convention is that 0 is the empty set. Morphisms from n to m are only non-trivial for n = m and in this case they consist of the symmetric group on n letters. The algebraic K-theory of S is equivalent to Waldhausen’s A-theory of a point A(∗) [W], and so gives information about diffeomorphisms of high dimensional disks. Thus we obtain that

A(∗) ≃ K(S) ≃ K(E) ≃ ℤ × |BGL(E)|^{+} .

• For a commutative ring A we consider the following small rig category of finitely generated free A-modules, F(A). Objects of F(A) are the finitely generated free A-modules A^{n} for n ⩾ 0. The set of morphisms from A^{n} to A^{m} is empty unless n = m, and the morphisms from A^{n} to itself are the A-module automorphisms of A^{n}, i.e., GL_{n}(A). Our result allows us to identify the two-fold iterated algebraic K-theory of A, K(K(A)), with ℤ × |BGL(F(A))|^{+}.

• The case that started the project is the 2-category of 2-vector spaces of Kapranov and Voevodsky [KV], viewed as modules over the rig category V of complex (Hermitian) vector spaces. Here V has objects ℂ^{n} for n ⩾ 0, and the automorphism space of ℂ^{n} is the unitary group U(n). This identifies K(HV) = K(ku) with K(V) ≃ ℤ × |BGL(V)|^{+}, which was called the K-theory of the 2-category of complex 2-vector spaces in [BDR]. Ausoni’s calculations [A] show that K(ku_{p}) has telescopic complexity 2 for every prime ⩾ 5, and thus qualifies as a form of elliptic cohomology.

• Replacing the complex numbers by the reals yields an identification of K(ko) with the K-theory of the 2-category of real 2-vector spaces.

• Considering other subgroups of GL_{n}(ℂ) or GL_{n}(ℝ) as morphisms on a category with objects n = {1, …, n} with n ⩾ 0 gives a large variety of K-theory spectra that are in the range of our result. For a sample of such species have a look at [M2, pp. 161–167].

1.1. The spine of the argument giving a proof of Theorem 1.1. Although the proof contains some lengthy technical lemmas, it is possible to give the main flow of the argument in a few paragraphs, referring away the hard parts.

Remember that the group-like monoid GL_{n}(HR) is defined by the pullback

\xymatrix{GL_{n}(HR) \ar[r] \ar[d] & hocolim_{m∈I} Ω^{m}M_{n}(HR(S^{m})) \ar[d] \\ GL_{n}(π_{0}HR) \ar[r] & M_{n}(π_{0}HR).}

where I is a strictly monoidal skeleton of the category of finite sets and injective functions, making GL_{n}(HR) a group-like monoid (here we have written HR in the form of a simplicial functor with associated symmetric spectrum m ↦ HR(S^{m})).

Lemma 1.3. If R̄ is a ring category, i.e., a rig category with π_{0}R̄ a ring, then

|GL_{n}(R̄)| ⟶^{∼} GL_{n}(HR̄)

is a homotopy equivalence.

Here, GL_{n}(R̄) denotes the components of the monoidal category of n × n-matrices over R̄, M_{n}(R̄), corresponding to GL_{n}(π_{0}R̄) (see Definition 2.4).

Proof. By assumption π_{0}R̄ is isomorphic to its group completion Gr(π_{0}R̄) = π_{0}HR̄, so it is enough to show that |M_{n}(R̄)| and hocolim_{m∈I} Ω^{m}M_{n}(HR̄(S^{m})) are equivalent. Both are n^{2}-fold products, so it suffices to show that |R̄| and hocolim_{m∈I} Ω^{m}HR̄(S^{m}) are equivalent. All the structure maps Ω^{m}HR̄(S^{m}) → Ω^{m′}HR̄(S^{m′}) are equivalences for m → m′ an injection of nonempty finite sets, and since R̄ is already group-like |R̄| ≃ HR̄(S^{0}) maps by an equivalence to ΩHR̄(S^{1}). □

We know from [BDRR1] that there is a chain of rig categories

R ⟵^{∼} ZR ⟶ R̄

such that • R ← ZR becomes a weak equivalence upon realization,

• HZR → HR̄ is a stable equivalence and

• π_{0}R̄ is a ring. Consider the commutative diagram

\xymatrix{|BGL(R)| \ar[d] & |BGL(ZR)| \ar[r] \ar[l]_{∼} \ar[d] & |BGL(R̄)| \ar[d]^{∼} \\ BGL(HR) & BGL(HZR) \ar[l]_{∼} \ar[r]^{∼} & BGL(HR̄),}

where the definition of GL of rig categories is given in Section 2 and the bar construction is recalled in Section 3. Both constructions preserve equivalences.

• The leftwards pointing horizontal maps are weak equivalences since R ← ZR is,

• the bottom rightwards pointing arrow is a weak equivalence since HZR → HR̄ is a stable equivalence and

• the right vertical map is a weak equivalence by Lemma 1.3. To show that the left vertical arrow is a weak equivalence, it therefore suffices to prove that the

upper right hand horizontal map is a weak equivalence. By Proposition 3.7, the homotopy fiber of BGL(ZR) → BGL(R̄) is given by the one-sided bar construction B(∗, GL(ZR), GL(R̄)), and so we have reduced the problem to giving a contraction of the associated space.

Under the assumptions of Theorem 1.1 we know from [BDRR1] that there is a chain of weak equivalences

(−R)R ⟵^{∼} Z(−R)R ⟶^{∼} R̄

of ZR-modules. Here, (−R)R is the Grayson-Quillen model [G], also discussed in Section 4.1, and so by Remark 2.5 we get weak equivalences

B(∗, GL(R), GL((−R)R)) ⟵^{∼} B(∗, GL(ZR), GL(Z(−R)R)) ⟶^{∼} B(∗, GL(ZR), GL(R̄)) .

This means that the somewhat complicated construction of R̄ from [BDRR1] may be safely forgotten once we know it exists.

Just one simplification remains: in Lemma 4.2 we display a weak equivalence T R → (−R)R of Rmodules. The R-modules T R and (−R)R are generalizations of how to construct the integers from the natural numbers by considering equivalence classes of pairs of natural numbers.

We are left with showing that B(∗, GL(R), GL(T R)) is contractible. This is done through a concrete contraction. It is an elaboration of the following path

[\begin{matrix}a − b & 0 \\ 0 & 1 \end{matrix}] → [\begin{matrix}a − b & b \\ 0 & 1 \end{matrix}] ← [\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}] , a, b ∈ ℕ

in B(∗, GL_{2}(ℕ), GL_{2}(ℤ)) = {[q] ↦ GL_{2}(ℕ)^{×q} × GL_{2}(ℤ)}, with 1-simplices given by

([\begin{matrix}1 & b \\ 0 & 1 \end{matrix}] , [\begin{matrix}a − b & 0 \\ 0 & 1 \end{matrix}]) and ([\begin{matrix}a & b \\ 1 & 1 \end{matrix}] , [\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}]) ∈ GL_{2}(ℕ) × GL_{2}(ℤ)

respectively, showing that the inclusion of B(∗, GL_{1}(ℕ), GL_{1}(ℤ)) in B(∗, GL_{2}(ℕ), GL_{2}(ℤ)) is null-homo-topic.

1.2. Plan. The structure of the paper is as follows. In Section 2 we discuss the monoidal category GL_{n}(R) of (weakly) invertible matrices over a strictly bimonoidal category R. Section 3 recalls the definition of the bar construction of monoidal categories as in [BDR] and introduces a version with coefficients in a module. In Section 4 we construct the promised contraction of B(∗, GL(R), GL(T R)), thus completing the proof of the main theorem.

This paper has circulated in preprint form under the title “Two-vector bundles define a form of elliptic cohomology”, which, while not highlighting the nuts and bolts of the paper perhaps better represented the reason for writing (or reading) it. The old preprint [BDRR] also contained the main result of the paper [BDRR1]

Graeme Segal constructed a ring completion of the rig category of complex vector spaces [S2, p. 300] (see also the Appendix of [S1]). His model is a topological category consisting of certain spaces of bounded chain complexes and spaces of quasi-isomorphisms. One can probably build a variant of his model that could be implemented in our proof of Theorem 1.1.

A piece of notation: if C is any small category, then the expression X ∈ C is short for “X is an object of C” and likewise for morphisms and diagrams. Displayed diagrams commute unless the contrary is stated explicitly. For basics on bipermutative and rig categories we refer to [BDRR1] Section 2.

2. Weakly invertible matrices

Let R be a strictly bimonoidal category.

Definition 2.1. The category of n × n-matrices over R, M_{n}(R), is defined as follows. The objects of M_{n}(R) are matrices X = (X_{i,j})_{i,j=1}^{n} of objects of R and morphisms from X = (X_{i,j})_{i,j=1}^{n} to Y = (Y_{i,j})_{i,j=1}^{n} are matrices F = (F_{i,j})_{i,j=1}^{n} where each F_{i,j} is a morphism in R from X_{i,j} to Y_{i,j}.

Lemma 2.2. For a strictly bimonoidal category (R, ⊕, 0_{R}, c_{⊕}, ⊗, 1_{R}) the category M_{n}(R) is a monoidal category with respect to the matrix multiplication bifunctor

M_{n}(R) × M_{n}(R) ⟶^{·} M_{n}(R)
(X_{i,j})_{i,j=1}^{n} · (Y_{i,j})_{i,j=1}^{n} = (Z_{i,j})_{i,j=1}^{n} with Z_{i,j} = ⨁_{k=1}^{n} X_{i,k} ⊗ Y_{k,j} .

The unit of this structure is given by the unit matrix object E_{n} which has 1_{R} ∈ R as diagonal entries and 0_{R} ∈ R in the other places.

The property of R being bimonoidal gives π_{0}R the structure of a rig, and its (additive) group completion Gr(π_{0}R) = (−π_{0}R)π_{0}R is a ring.

Definition 2.3. We define the weakly invertible n × n-matrices over π_{0}R, GL_{n}(π_{0}R), to be the n × nmatrices over π_{0}R that are invertible as matrices over Gr(π_{0}R).

Note that we can define GL_{n}(π_{0}R) by the pullback square

[figure: GL_{n}(π_{0}R); GL_{n}(Gr(π_{0}R)); M_{n}(π_{0}R); M_{n}(Gr(π_{0}R))]

Definition 2.4. The category of weakly invertible n × n-matrices over R, GL_{n}(R), is the full subcategory of M_{n}(R) with objects all matrices X = (X_{i,j})_{i,j=1}^{n} ∈ M_{n}(R) whose matrix of π_{0}-classes [X] = ([X_{i,j}])_{i,j=1}^{n} is contained in GL_{n}(π_{0}R).

Matrix multiplication is of course compatible with the property of being weakly invertible. Thus, the category GL_{n}(R) inherits a monoidal structure from M_{n}(R).

However, even if our base category is not bimonoidal it still makes sense to talk about matrices and even weakly invertible matrices, as long as π_{0} of that category is a rig.

Remark 2.5. If M is an R-module, matrix multiplication makes the category M_{n}(M) into a module over the monoidal category GL_{n}(R). For our applications the following situation will be particularly important: let M → N be a map of R-modules, where the map of π_{0}R-modules π_{0}M → π_{0}N comes from a rig map under π_{0}R. Then we get a map GL_{n}(M) → GL_{n}(N) of GL_{n}(R)-modules which induces a weak equivalence upon realization if M → N does.

There is a canonical stabilization functor GL_{n}(R) → GL_{n+1}(R) which is induced by taking the block sum with E_{1} ∈ GL_{1}(R). Let GL(R) be the sequential colimit of the categories GL_{n}(R).

3. The one-sided bar construction

In this section we review some well-known facts about the one-sided bar construction of monoidal categories.

Definition 3.1. Let (M, ·, 1) be a monoidal category and T a left M-module. The one-sided bar construction B(∗, M, T ) is the simplicial category whose q-simplices B_{q}(∗, M, T ) is the following category: consider the ordered set [q]_{+} = [q] ⊔ {∞}, i.e., in addition to the numbers 0 < 1 < ⋯ < q there is a greatest element ∞. An object a in B_{q}(∗, M, T ) consists of the following data.

(1) For each 0 ⩽ i < j ⩽ q there is an object a_{ij} ∈ M, and for each 0 ⩽ i ⩽ q an object a_{i∞} ∈ T .

(2) For each 0 ⩽ i < j < k ⩽ ∞ there is an isomorphism

a_{ijk}: a_{ij} · a_{jk} → a_{ik}

(in M if k < ∞ and in T if k = ∞) such that if 0 ⩽ i < j < k < l ⩽ ∞, the following diagram commutes

\xymatrix{(a_{ij} · a_{jk}) · a_{kl} \ar[d]_{a_{ijk}·id} \ar[rr]^{struct. iso.} &  & a_{ij} · (a_{jk} · a_{kl}) \ar[d]^{id·a_{jkl}} \\ a_{ik} · a_{kl} \ar[r]^{a_{ikl}} & a_{il} & a_{ij} · a_{jl}. \ar[l]_{a_{ijl}}}

A morphism f: a → b consists of morphisms f_{ij}: a_{ij} → b_{ij} (in M if j < ∞ and in T if j = ∞) such that if 0 ⩽ i < j < k ⩽ ∞

f_{ik}a_{ijk} = b_{ijk}(f_{ij} · f_{jk}): a_{ij} · a_{jk} → b_{ik} .

The simplicial structure is gotten as follows: if ϕ: [q] → [p] ∈ Δ the functor ϕ^{∗}: B_{p}(∗, M, T ) → B_{q}(∗, M, T ) is obtained by precomposing with ϕ_{+} = ϕ⊔{∞}. So for instance d_{1}(a) is gotten by deleting all entries with indices containing 1 from the data giving a. In order to allow for degeneracy maps s_{i}, we use the convention that all objects of the form a_{ii} are the unit of the monoidal structure, and all isomorphisms of the form a_{iik} and a_{ikk} are identities.

A good way to think about this comes from the discrete case (M is a monoid and T is a M-set). Then an object a ∈ B_{q}(∗, M, T ) is uniquely given by the “diagonal” (a_{01}, a_{12}, …, a_{q−1 q}, a_{q∞}), and B(∗, M, T ) is isomorphic to the nerve of the category with objects T and morphisms a_{1∞} → a_{01} · a_{1∞} = a_{0∞} corresponding to (a_{01}, a_{1∞}).

Example 3.2. (1) If T is the one-point category ∗, then B(∗, M, ∗) is isomorphic to the bar construction BM of [BDR].

(2) If F: M → M′ is a lax monoidal functor, then M′ may be considered as an M-module, and we write without further ado B(∗, M, M′) for the corresponding bar construction (with F suppressed). In case F is an isomorphism, B(∗, M, M′) is contractible.

We think of elements of B_{q}(∗, M, T ) in terms of triangular arrays of objects, suppressing the isomorphisms, so that a typical element in B_{2}(∗, M, T ) is written

a_{01} a_{02} a_{0∞}
a_{12} a_{1∞}
a_{2∞}

with d_{1} given by

\begin{aligned}a_{02} & a_{0∞} \\ & a_{2∞}. \end{aligned}

The one-sided bar construction is functorial in “natural modules”. A natural module is a pair (M, T ) where M is a monoidal category and T is an M-module. A morphism (M, T ) → (M′, T′) consists of a pair (F, G) where F: M → M′ is a lax monoidal functor and G: T → F^{∗}T′ is a map of M-modules, where F^{∗}T′ is T′ endowed with the M-module structure given by restricting along F.

Lemma 3.3. For each q there is an equivalence of categories between B_{q}(∗, M, T ) and the product category M^{×q} × T .

Proof. The equivalence is given by the forgetful functor

F: B_{q}(∗, M, T ) → M^{×q} × T

sending a to the “diagonal” F (a) = (a_{01}, …, a_{q−1 q}, a_{q∞}). The inverse is gotten by sending (a_{1}, …, a_{q}, a_{∞}) to the a with a_{ij} = a_{i+1} · (⋯ (a_{j−1} · a_{j}) ⋯) and a_{ijk} given by the structural isomorphisms. □

Corollary 3.4. Let (F, G): (M, T ) → (M′, T′) be a map of natural modules such that F and G are equivalences of categories. Then the induced map

B(∗, F, G): B(∗, M, T ) → B(∗, M′, T′)

is a degreewise equivalence of simplicial categories.

Remark 3.5. The same result holds, if instead of equivalences of categories we consider weak equivalences.

Usually M^{×q} × T is not functorial in [q], but if (M, T ) is strict, the monoidal structure gives a simplicial category

B^{strict}(∗, M, T ) = {[q] ↦ M^{×q} × T } .

In this situation Lemma 3.3 reads:

Corollary 3.6. Let M be a strict monoidal category and T a strict M-module. Then there is a degreewise equivalence between the simplicial categories B(∗, M, T ) and B^{strict}(∗, M, T ).

Proposition 3.7. Let F: M → G be a strong monoidal functor such that the monoidal structure on G induces a group structure on π_{0}G. Then

B(∗, M, G) ⟶ BM
↓  ↓
B(∗, G, G) ⟶ BG

is homotopy cartesian, meaning that it induces a homotopy cartesian diagram upon applying the nerve functor in every degree. The (nerve of the) lower left hand corner is contractible.

Proof. By [JS] there is a diagram of monoidal categories

stM ⟶^{∼} M
stF ↓  ^{F} ↓
stG ⟶^{∼} G

such that the horizontal maps are monoidal equivalences, and stF is a strict monoidal functor between strict monoidal categories. Together with Corollaries 3.4 and 3.6 this tells us that we may just as well consider the strict situation, and use the strict bar construction. However, note that the nerve of the strict monoidal category stM is a simplicial monoid, and that reversal of priorities gives a natural isomorphism

B(∗, NstM, NstG) ≅ NB^{strict}(∗, stM, stG),

so that our statement reduces to the statement that

B(∗, NstM, NstG) → B(NstM) → B(NstG)

is a fiber sequence up to homotopy, which is a classical result [M1] given that NstG is group-like. □

4. Contracting the one-sided bar construction

4.1. A model for K-theory of R as an R-module. In order to construct concrete homotopies, we offer a slight variant of the Grayson–Quillen model where morphisms are not entire equivalence classes. The price is as usual that the resulting object is a two-category. Since there was some confusion about this point while the paper was still at a preprint stage, we emphasize that this is not the construction of Thomason [Th1, 4.3.2] and Jardine [J].

Definition 4.1. Let (M, ⊕, 0_{M}, τ_{M}) be a permutative category written additively. Let T M be the following 2-category. The objects of T M are pairs (A^{+}, A^{−}) =: A of objects in M, thought of as plus and minus objects in M. Given two objects A, B ∈ T M, the category of morphisms T M(A, B) has objects the pairs (X, α) where X is an object in M and α is a pair of morphisms α^{±}: A^{±} ⊕ X → B^{±} in M. A morphism from (X, α) to (Y, β) is an isomorphism ϕ: X → Y such that β(1⊕ϕ) = α. Composition T M(B, C)×T M(A, B) → T M(A, C) is given by sending ((Y, β), (X, α)) to the pair consisting of X ⊕Y and the composite maps

A^{±} ⊕ (X ⊕ Y) = (A^{±} ⊕ X) ⊕ Y ⟶^{α^{±}⊕id} B^{±} ⊕ Y ⟶^{β^{±}} C^{±} .

Composition on morphisms is simply given by addition. Composition is strictly associative because M is permutative; if M is merely symmetric monoidal, standard modifications are necessary. Symmetry allows for a symmetric monoidal structure on T M: if we define (A^{+}, A^{−}) ⊕ (B^{+}, B^{−}) := (A^{+} ⊕ B^{+}, A^{−} ⊕ B^{−}), we need the symmetry in order to turn that prescription into a bifunctor.

The Grayson-Quillen model for the K-theory of M is the category (−M)M with the same objects as T M and with morphism sets the path components (−M)M(A, B) = π_{0}T M(A, B). If all morphisms in M are isomorphisms and if additive translation in M is faithful, (−M)M is shown in [G] to be a group completion of M. Under these hypotheses, there is at most one morphism between two given objects (X, α) and (Y, β) in T M(A, B). Consequently the morphism spaces are homotopy discrete: the projection T M(A, B) → π_{0}T M(A, B) is a weak equivalence. In the case of a topological category, we interpret π_{0} as the coequalizer of the source and target maps from the morphism space to the object space. The assignment that is the identity on objects and otherwise is induced by the projection T M(A, B) → π_{0}T M(A, B) gives a 2-functor T M → (−M)M (considering (−M)M as a 2-category with only identity 2-morphism).

Lemma 4.2. Let M be a permutative category with all morphisms in M isomorphisms and faithful additive translation. The 2-functor T M → (−M)M is a weak equivalence and the standard inclusion M → T M is a group completion.

Note that if R is a rig category, T R will not be a rig category (essentially because of the non-strict symmetry in quadratic terms, as in [Th2, p. 572]), but it will still be an R-module:

Lemma 4.3. Let (R, ⊕, 0_{R}, c_{⊕}, ⊗, 1_{R}) be a strictly bimonoidal category. The map

R × T R → T R

given on objects by (A, (B^{+}, B^{−})) ↦ (A ⊗ B^{+}, A ⊗ B^{−}), and on morphisms by sending ϕ: A → B ∈ R and (X, α) ∈ T R(C, D) to the pair consisting of A ⊗ X and the map

A ⊗ C^{±} ⊕ A ⊗ X ⟶ A ⊗ (C^{±} ⊕ X) ⟶^{ϕ⊗α^{±}} B ⊗ D^{±}

(where the first map is the left distributivity isomorphism) induces an R-module structure on T R.

We consider T R as a simplicial category by taking the nerve of each category of morphisms; thus in simplicial degree ℓ, the objects of T_{ℓ}R are the objects of T R. The morphisms in T_{ℓ}R from (A^{+}, A^{−}) to (B^{+}, B^{−}) consist of objects X^{0}, …, X^{ℓ}, a one-morphism α^{±}: A^{±} ⊕ X^{0} → B^{±}, and isomorphisms ϕ^{l}: X^{l} → X^{l−1} for l = 1, …, ℓ. Note that l and ℓ are different symbols. The simplicial structure is given by composing and forgetting ϕ^{l}’s and inserting identity maps.

4.2. Subdivisions. We will use the following variant of edgewise subdivision to make room for an explicit simplicial contraction, whose construction begins in Subsection 4.4. Consider the shear functor z: Δ × Δ → Δ × Δ given by sending (S, T) to (T ⊔ S, T) where T ⊔ S is the disjoint union with the ordering obtained from T and S with the extra declaration that every object in S is greater than every object in T. If B is a bisimplicial object, we let z^{∗}B = B ∘ z. The standard inclusion S → T ⊔ S gives a natural transformation η in Δ × Δ from the identity to z, and hence a natural transformation in bisimplicial sets η^{∗}: z^{∗} → id. Let Ens denote the category of sets and functions.

Lemma 4.4. For any bisimplicial set X the map η^{∗}: z^{∗}X → X becomes a weak equivalence upon realization.

Proof. The diagonal of z^{∗}X is equal to the evaluation of X on the opposite of the composite

Δ ⟶^{S↦(S,S)} Δ × Δ ⟶^{(S,T)↦(S⊔S,T)} Δ × Δ ,

so since a map of bisimplicial sets is an equivalence if it is one in every (vertical) degree, it is enough to know that for each fixed T ∈ Δ the natural map {S ↦ X(S ⊔ S, T)} → {S ↦ X(S, T)} is a weak equivalence. But this is a standard weak equivalence from the (second) edgewise subdivision, which is known to be homotopic to a homeomorphism after realization. See [BHM, Lemma 1.1] and the proof of [BHM, Proposition 2.5]. □

Vertices in z^{∗}(Δ[p] × Δ[q]) (where products of simplicial sets are viewed as bisimplicial sets, and vertices are (0, 0)-simplices) are for instance indexed by tuples ((a, b), c) where 0 ⩽ a ⩽ b ⩽ p and 0 ⩽ c ⩽ q.

Here are pictures of z^{∗}(Δ[2] × Δ[0]) and z^{∗}(Δ[2] × Δ[1]):

\xymatrix{((2, 2), 0) \ar[r] \ar[dr] & ((1, 2), 0) \ar[d] \ar[r] & ((1, 1), 0) \ar[d] \\  & ((0, 2), 0) \ar[r] \ar[dr] & ((0, 1), 0) \ar[d] \\  &  & ((0, 0), 0)}

[figure: ((2, 2), 1); ((1, 2), 1); ((1, 1), 1); ((0, 2), 1); ((0, 1), 1); ((2, 2), 0); ((1, 2), 0); ((1, 1), 0); ((0, 0), 1); ((0, 2), 0); ((0, 1), 0); ((0, 0), 0) .]

Note that for any bisimplicial set X

X_{(p,q)} ≅ Hom_{bisimp. sets}(Δ[p] × Δ[q], X) = ∫_{([s],[t])} Ens(Δ([s], [p]) × Δ([t], [q]), X_{(s,t)})

as a categorical end. Therefore, the right adjoint of z^{∗}, z_{∗}, is given by

(z_{∗}X)_{(p,q)} = Hom_{bisimp. sets}(Δ[p] × Δ[q], z_{∗}X) ≅ Hom_{bisimp. sets}(z^{∗}(Δ[p] × Δ[q]), X)

and thus

z_{∗}X = {[p], [q] ↦ {bisimp. maps z^{∗}(Δ[p] × Δ[q]) → X}}
= {[p], [q] ↦ ∫_{([s],[t])} Ens(Δ([t] ⊔ [s], [p]) × Δ([t], [q]), X_{(s,t)})} .

Let η_{∗}: X → z_{∗}X be the natural transformation associated with η. Notice that η_{∗} maps X_{(0,q)} isomorphically to (z_{∗}X)_{(0,q)} for all q ⩾ 0, so (z_{∗}X)_{(0,q)} ≅ X_{(0,q)}.

Lemma 4.5. In the homotopy category with respect to maps that are weak equivalences on the diagonal, η_{∗}: X → z_{∗}X is a split monomorphism.

Proof. By formal considerations the diagram

\xymatrix{z^{∗}X \ar[r]^{z^{∗}η_{∗}} \ar[dr]_{η^{∗}} & z^{∗}z_{∗}X \ar[d]^{adjunction} \\  & X}

commutes, and η^{∗} is a weak equivalence on the diagonal. Hence z^{∗}η_{∗} (and so η_{∗}) is a split monomorphism in the homotopy category. □

4.3. The bar construction on matrices. Let R be a strictly bimonoidal category such that all morphisms are isomorphisms and each translation functor is faithful.

Consider the one-sided bar construction B(∗, GL_{n}(R), GL_{n}(T R)). In the following, 0 and 1 are short for zero resp. unit matrices over R of varying size. Viewing T R as a simplicial category we get that B(∗, GL_{n}(R), GL_{n}(T R)) is a bisimplicial category. We are going to show that

B(∗, GL(R), GL(T R)) ≅ colim_{n} B(∗, GL_{n}(R), GL_{n}(T R))

is contractible, and it is enough to show that B(∗, GL(R), GL(T_{ℓ}R)) is contractible for every ℓ.

To ease readability, we will abandon the cumbersome ⊕ and ⊗ in favor of the more readable + and · — reminding us of the matrix nature of our efforts.

Fix ℓ once and for all, and let B^{n} = B(∗, GL_{n}(R), GL_{n}(T_{ℓ}R)). An object in B_{q}^{n} is a collection m_{ij} of n×n matrices in R for 0 ⩽ i < j ⩽ q and for each 0 ⩽ i ⩽ q a matrix m_{i∞} in T_{ℓ}R, together with suitably compatible structural isomorphisms m_{ijk}: m_{ij} · m_{jk} → m_{ik}. The matrices are drawn from the “weakly invertible components”. The matrices m_{i∞} and the structural isomorphisms relating these need special attention. Each entry is in T_{ℓ}R, so m_{i∞} can be viewed as a pair m_{i∞}^{±} of matrices, and the structural isomorphism m_{ij∞}: m_{ij} · m_{j∞} → m_{i∞} is a tuple (m_{ij∞}^{±}, ϕ_{ij∞}^{1}, …, ϕ_{ij∞}^{ℓ}), where the ϕ_{ij∞}^{l}: x_{ij∞}^{l} → x_{ij∞}^{l−1} ∈ M_{n}(R) for l = 1, …, ℓ are matrices of isomorphisms, and the m_{ij∞}^{±}: m_{ij} · m_{j∞}^{±} + x_{ij∞}^{0} → m_{i∞}^{±} are isomorphisms.

The commutativity of

\xymatrix{(m_{ij} · m_{jk}) · m_{k∞} \ar[rr]^{≅} \ar[d]_{m_{ijk}·id} &  & m_{ij} · (m_{jk} · m_{k∞}) \ar[d]^{id·m_{jk∞}} \\ m_{ik} · m_{k∞} \ar[r]_{m_{ik∞}} & m_{i∞} & m_{ij} · m_{j∞} \ar[l]^{m_{ij∞}}}

says that two morphisms from (m_{ij} · m_{jk}) · m_{k∞} agree: one is an isomorphism with source (m_{ij} · m_{jk}) · m_{k∞} +x_{ik∞}^{l}, the other one is an isomorphism with source (m_{ij} ·m_{jk})·m_{k∞} +m_{ij} ·x_{jk∞}^{l} +x_{ij∞}^{l}. Therefore we obtain the following equality.

Lemma 4.6. In the situation above one has the identity

x_{ik∞}^{l} = m_{ij} · x_{jk∞}^{l} + x_{ij∞}^{l}

for l = 0, 1, …, ℓ, and the diagram

[figure: m_{ij} · (m_{jk} · m_{k∞}^{±}) + x_{ik∞}^{0}; m_{ij} · (m_{jk} · m_{k∞}^{±}) + m_{ij} · x_{jk∞}^{0} + x_{ij∞}^{0}; id·m_{jk∞}^{±}+id; (m_{ij} · m_{jk}) · m_{k∞}^{±} + x_{ik∞}^{0}; ≅; m_{ijk}·id+id; m_{ij} · m_{j∞}^{±} + x_{ij∞}^{0}; m_{ij∞}^{±}; m_{ik} · m_{k∞}^{±} + x_{ik∞}^{0}; m_{ik∞}^{±}; m_{i∞}^{±}]

commutes.

Here the map id · m_{jk∞}^{±} already incorporates the distributivity isomorphism, as specified in Lemma 4.3.

A morphism α: m → m̃ in B_{q}^{n} consists of an n × n matrix of maps α_{ij}: m_{ij} → m̃_{ij} in R for 0 ⩽ i < j ⩽ q, and of morphisms (α_{i∞}^{±}, ψ_{i∞}^{1}, …, ψ_{i∞}^{ℓ}): m_{i∞}^{±} → m̃_{i∞}^{±} in T_{ℓ}R for 0 ⩽ i ⩽ q, all compatible with the structure maps of m and m̃. Here α_{i∞}^{±} is a map m_{i∞}^{±} +ξ_{i∞}^{0} → m̃_{i∞}^{±}, and the ψ_{i∞}^{l} for l = 1, …, ℓ are maps ξ_{i∞}^{l} → ξ_{i∞}^{l−1} of matrices in R.

The condition α_{i∞}m_{ij∞} = m̃_{ij∞}(α_{ij} · α_{j∞}) allows us to draw the following conclusion.

Lemma 4.7. In the situation above one has the identity

x_{ij∞}^{l} + ξ_{i∞}^{l} = m_{ij} · ξ_{j∞}^{l} + x̃_{ij∞}^{l}

for each l = 0, …, ℓ.

4.4. Start of the proof that B(∗, GL(R), GL(T_{ℓ}R)) is contractible. We will show that colim_{n} B^{n} = B(∗, GL(R), GL(T_{ℓ}R)) is contractible by showing that each matrix stabilization functor in: B^{n} → B^{2n} is trivial in the homotopy category. Here in(m) = [ \begin{matrix}m & 0 \\ 0 & 1 \end{matrix} ].

We regard the simplicial categories B^{n} and B^{2n} as bisimplicial sets, by way of their respective nerves NB^{n} and NB^{2n}. To be precise, the (p, q)-simplices of NB^{2n} are N_{p}B_{q}^{2n}. By Lemma 4.5 it then suffices to show that the composite map inc = η_{∗} ∘ in: NB^{n} → z_{∗}NB^{2n} is trivial in the homotopy category. As remarked above, z_{∗}(NB^{2n})_{(0,q)} ≅ (NB^{2n})_{(0,q)} = N_{0}B_{q}^{2n}, so the subdivision operator z_{∗} does not make any difference before we start to consider positive-dimensional simplices (p > 0) in the nerve direction.

Seeing that the image lies in a single path component is easy: if m ∈ N_{0}B_{0}^{n} = ob GL_{n}(T_{ℓ}R) then there is a path

[\begin{matrix}m & 0 \\ 0 & 1 \end{matrix}] → [\begin{matrix}m & m^{−} \\ 0 & 1 \end{matrix}] → [\begin{matrix}m & m^{−} \\ (1, 1) & 1 \end{matrix}] ← [\begin{matrix}1 & 0 \\ (0, 1) & 1 \end{matrix}] .

The first arrow represents the one-simplex in the bar direction given by the matrix multiplication

[\begin{matrix}1 & m^{−} \\ 0 & 1 \end{matrix}] · [\begin{matrix}m & 0 \\ 0 & 1 \end{matrix}] = [\begin{matrix}m & m^{−} \\ 0 & 1 \end{matrix}] ∈ GL_{2n}(T_{ℓ}R) .

The second arrow represents the one-simplex in the nerve direction induced by the map 0 = (0, 0) → (1, 1) ∈ T_{0}R ⊂ T_{ℓ}R. The third map represents the one-simplex in the bar direction given by multiplication by

[\begin{matrix}m^{+} & m^{−} \\ 1 & 1 \end{matrix}] ∈ GL_{2n}(R) .

The rest of this section extends this path to a full homotopy, from inc via maps jnc and knc to a constant map lnc.

4.5. The homotopic maps inc and jnc. Recall that ℓ ⩾ 0 is fixed, B^{n} = B(∗, GL_{n}(R), GL_{n}(T_{ℓ}R)) is the simplicial category given by the one-sided bar construction, and NB^{n}: [p], [q] ↦ N_{p}B_{q}^{n} is the bisimplicial set given by its degreewise nerve. We let inc: NB^{n} → z_{∗}NB^{2n} be the composite of the matrix stabilization map in: NB^{n} → NB^{2n} and the natural map η_{∗}: NB^{2n} → z_{∗}NB^{2n}.

There is another map jnc: NB^{n} → z_{∗}NB^{2n} which is homotopic to inc. On N_{0}B_{q}^{n} it is easy to describe: if m ∈ N_{0}B_{q}^{n}, we declare that X(m) is given by

X(m)_{ij} = \begin{cases}[\begin{matrix}1 & m_{i∞}^{−} \\ 0 & 1 \end{matrix}] & if i < j = ∞ \\ [\begin{matrix}1 & x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] & if i < j < ∞ \end{cases}

and let

jnc(m) = X(m) · inc(m) ∈ N_{0}B_{q}^{2n} = z_{∗}(NB^{2n})_{(0,q)} .

Here

jnc(m)_{ij} = \begin{cases}[\begin{matrix}m_{i∞} & m_{i∞}^{−} \\ 0 & 1 \end{matrix}] & if i < j = ∞ \\ [\begin{matrix}m_{ij} & x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] & if i < j < ∞ \end{cases}

with jnc(m)_{ijk}: jnc(m)_{ij} · jnc(m)_{jk} → jnc(m)_{ik} being the isomorphisms induced by m_{ijk} as follows: for k < ∞ we use the identity x_{ik∞}^{0} = m_{ij} · x_{jk∞}^{0} + x_{ij∞}^{0} from Lemma 4.6 and obtain

[\begin{matrix}m_{ijk} & id \\ id & id \end{matrix}] : [\begin{matrix}m_{ij} & x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] · [\begin{matrix}m_{jk} & x_{jk∞}^{0} \\ 0 & 1 \end{matrix}] = [\begin{matrix}m_{ij} · m_{jk} & m_{ij} · x_{jk∞}^{0} + x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] → [\begin{matrix}m_{ik} & x_{ik∞}^{0} \\ 0 & 1 \end{matrix}]

and for k = ∞ we use the string of isomorphisms

[\begin{matrix}x_{ij∞}^{ℓ} & 0 \\ 0 & 0 \end{matrix}] → … → [\begin{matrix}x_{ij∞}^{0} & 0 \\ 0 & 0 \end{matrix}]

together with the isomorphism

[\begin{matrix}m_{ij∞} & m_{ij∞}^{−} \\ id & id \end{matrix}]: [\begin{matrix}m_{ij} & x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] · [\begin{matrix}m_{j∞} & m_{j∞}^{−} \\ 0 & 1 \end{matrix}] + [\begin{matrix}x_{ij∞}^{0} & 0 \\ 0 & 0 \end{matrix}]

= [\begin{matrix}m_{ij} · m_{j∞} + x_{ij∞}^{0} & m_{ij} · m_{j∞}^{−} + x_{ij∞}^{0} \\ 0 & 1 \end{matrix}] → [\begin{matrix}m_{i∞} & m_{i∞}^{−} \\ 0 & 1 \end{matrix}] .

We notice that the T_{ℓ}-direction does not add any complications but notational. This continues to be true in general, so we simplify notation by considering only the case ℓ = 0.

The relevant complications arise when one starts moving in the nerve direction. As the construction of the map jnc is quite involved, we will give some examples first. The impatient reader can skip this part and restart reading in Subsection 4.6 where the formula in the general case is given.

As an illustration, let ℓ = 0, p = 2 and q = 0 so that

m  (m^{0} ⟵^{(ξ^{1},α^{1})} m^{1} ⟵^{(ξ^{2},α^{2})} m^{2}) ∈ N_{2}B_{0}^{n} = N_{2}GL_{n}(T_{0}R) .

Then jnc(m) is captured by the picture

[figure: [ \begin{matrix}m^{2} & (m^{2})^{−} \\ 0 & 1 \end{matrix} ]; ^{[ \begin{matrix}1 & ξ^{2} \\ 0 & 1 \end{matrix} ]}; _{[ \begin{matrix}1 & ξ^{2}+ξ^{1} \\ 0 & 1 \end{matrix} ]}; [ \begin{matrix}m^{2} & (m^{2})^{−}+ξ^{2} \\ 0 & 1 \end{matrix} ]; _{[ \begin{matrix}1 & ξ^{1} \\ 0 & 1 \end{matrix} ]}; [ \begin{matrix}m^{1} & (m^{1})^{−} \\ 0 & 1 \end{matrix} ]; _{[ \begin{matrix}1 & ξ^{1} \\ 0 & 1 \end{matrix} ]}; _{[ \begin{matrix}m^{2} & (m^{2})^{−}+ξ^{2}+ξ^{1} \\ 0 & 1 \end{matrix} ]}; _{[ \begin{matrix}m^{1} & (m^{1})^{−}+ξ^{1} \\ 0 & 1 \end{matrix} ]}; _{[ \begin{matrix}m^{0} & (m^{0})^{−} \\ 0 & 1 \end{matrix} ]}]

where the bar direction is written in the “g ⟶^{m} mg” form, and the unlabeled arrows correspond to the nerve direction, with entries consisting of the appropriate α’s.

An even more complicated example, essentially displaying all the complexity of the general case: ℓ = 0, p = q = 1, and α: m^{1} → m^{0} ∈ N_{1}B_{1}^{n} with (m^{u})_{01∞}^{±}: (m^{u})_{01} · (m^{u})_{1∞}^{±} + (x^{u})_{01∞}^{0} → (m^{u})_{0∞}^{±} (for u = 0, 1 running in the nerve direction), α_{01} : m_{01}^{1} → m_{01}^{0} and α_{i∞}^{±}: (m^{1})_{i∞}^{±} + ξ_{i∞} → (m^{0})_{i∞}^{±}.

Then jnc(m) is the map from

z^{∗}(Δ[1] × Δ[1]) = \xymatrix{((0, 0), 0) & ((0, 1), 0) \ar[l] & ((1, 1), 0) \ar[l] \\ ((0, 0), 1) \ar[u] & ((0, 1), 1) \ar[l] \ar[u] & ((1, 1), 1) \ar[l] \ar[u] \ar[ul]}

sending the ((0, 1), 0) ← ((1, 1), 0) ← ((1, 1), 1) simplex to

\begin{aligned}[ \begin{matrix}1 & ξ_{0∞} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{01}^{1} & x_{01∞}^{1}+ξ_{0∞} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{0∞}^{1} & (m^{1})_{0∞}^{−}+ξ_{0∞} \\ 0 & 1 \end{matrix} ] \\ & [ \begin{matrix}m_{01}^{1} & x_{01∞}^{1} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{0∞}^{1} & (m^{1})_{0∞}^{−} \\ 0 & 1 \end{matrix} ] \\ [ & \begin{matrix}m_{1∞}^{1} & (m^{1})_{1∞}^{−} \\ 0 & 1 \end{matrix} & ] \end{aligned} ∈ N_{0}B_{2}^{2n} ,

the ((0, 1), 0) ← ((0, 1), 1) ← ((1, 1), 1) simplex to

\begin{aligned}[ \begin{matrix}m_{01}^{1} & x_{01∞}^{0} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{01}^{1} & x_{01∞}^{1}+ξ_{0∞} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{0∞}^{1} & (m^{1})_{0∞}^{−}+ξ_{0∞} \\ 0 & 1 \end{matrix} ] \\ & [ \begin{matrix}1 & ξ_{1∞} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{1∞}^{1} & (m^{1})_{1∞}^{−}+ξ_{1∞} \\ 0 & 1 \end{matrix} ] \\ [ & \begin{matrix}m_{1∞}^{1} & (m^{1})_{1∞}^{−} \\ 0 & 1 \end{matrix} & ] \end{aligned} ∈ N_{0}B_{2}^{2n} ,

(here the identity from Lemma 4.7 is used) and the (1, 1)-simplex \xymatrix{((0, 0), 0) & ((0, 1), 0) \ar[l] \\ ((0, 0), 1) \ar[u] & ((0, 1), 1) \ar[l] \ar[u]} to

[figure: \begin{matrix}[ \begin{matrix}m_{01}^{0} & x_{01∞}^{0} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{0∞}^{0} & (m^{0})_{0∞}^{−} \\ 0 & 1 \end{matrix} ] \\ [ & \begin{matrix}m_{1∞}^{0} & (m^{0})_{1∞}^{−} \\ 0 & 1 \end{matrix} & ] \end{matrix}; \begin{matrix}[ \begin{matrix}m_{01}^{1} & x_{01∞}^{0} \\ 0 & 1 \end{matrix} ] & [ \begin{matrix}m_{0∞}^{1} & (m^{1})_{0∞}^{−}+ξ_{0∞} \\ 0 & 1 \end{matrix} ] \\ [ & \begin{matrix}m_{1∞}^{1} & (m^{1})_{1∞}^{−}+ξ_{1∞} \\ 0 & 1 \end{matrix} & ] \end{matrix}; [ \begin{matrix}α_{01} & id \\ id & id \end{matrix} ] [ \begin{matrix}α_{0∞} & α_{0∞}^{−} \\ id & id \end{matrix} ] [ \begin{matrix}α_{1∞} & α_{1∞}^{−} \\ id & id \end{matrix} ]] ∈ N_{1}B_{1}^{2n} .

Here we have employed the formula x_{01∞}^{1} + ξ_{0∞} = m_{01}^{1} · ξ_{1∞} + x_{01∞}^{0} of Lemma 4.7.

4.6. General version of jnc. Consider

(1)  \xymatrix{m = (m^{0} & m^{1} \ar[l]_{α^{1}} & … \ar[l]_{α^{2}} & m^{p}) \ar[l]_{α^{p}}} ∈ N_{p}B_{q}^{n} .

Then (α^{u})_{i∞} is given by the tuple

((α^{u})_{i∞}^{±}: (m^{u})_{i∞}^{±} + (ξ^{u})_{i∞}^{0} → (m^{u−1})_{i∞}^{±} , {(ψ^{u})_{i∞}^{l}: (ξ^{u})_{i∞}^{l} → (ξ^{u})_{i∞}^{l−1}})

for u = 1, …, p, l = 1, …, ℓ, but we simplify notation by setting ξ_{i∞}^{u} = (ξ^{u})_{i∞}^{0}, x_{ij∞}^{u} = (x^{u})_{ij∞}^{0}, and ignoring the ψ’s. Then jnc sends an m as in (1) to the simplex jnc(m) ∈ z_{∗}(NB^{n})_{pq} with value at the ((a, b), c)-vertex in z^{∗}(Δ[p] × Δ[q]) given by

[\begin{matrix}(m^{b})_{c∞} & (m^{b})_{c∞}^{−} + ξ_{c∞}^{b} + ⋯ + ξ_{c∞}^{a+1} \\ 0 & 1 \end{matrix}] ∈ GL_{2n}(T_{ℓ}R)

with the convention that the ξ’s only occur if a + 1 ⩽ b. Higher simplices are given by the structural isomorphisms in m. Note that the elements in the off-diagonal blocks are actually all in R.

More precisely, a triple (ϕ, b, ψ) where ϕ: [r] → [p] and ψ: [r] → [q] are in Δ and ϕ(r) ⩽ b ⩽ p, determines a (0, r)-simplex in z^{∗}(Δ[p] × Δ[q]), because z^{∗}(Δ[p] × Δ[q])_{(0,r)} = Δ([r + 1], [p]) × Δ([r], [q]) and ϕ together with b determine an element in the first factor. We see that jnc(m)(ϕ, b, ψ) ∈ N_{0}B_{r}^{2n} is the element whose (0 ⩽ i < j ⩽ r)- and (0 ⩽ i ⩽ r < j = ∞)-entries are

[\begin{matrix}(m^{b})_{ψ(i)ψ(j)} & x_{ψ(i)ψ(j)∞}^{ϕ(j)} + ξ_{ψ(i)∞}^{ϕ(j)} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 0 & 1 \end{matrix}]

and

[\begin{matrix}(m^{b})_{ψ(i)∞} & (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 0 & 1 \end{matrix}] ,

respectively. (As before, the ξ’s only occur when ϕ(i) + 1 ⩽ ϕ(j) and ϕ(i) + 1 ⩽ b, respectively.)

Moving in the (nerve =) b-direction is easy because it amounts to connecting two values jnc(m)(ϕ, b, ψ) and jnc(m)(ϕ, b′, ψ) by morphisms. Since this is determined by the one-skeleton, it is enough to describe the case b′ = b − 1 < b. On the (0 ⩽ i < j ⩽ r)-entries it is induced by (α^{b})_{ψ(i)ψ(j)} : (m^{b})_{ψ(i)ψ(j)} → (m^{b−1})_{ψ(i)ψ(j)} (in the upper left hand corner, and otherwise the identity), and on the (0 ⩽ i ⩽ r < j = ∞)-entries it is given by (ξ_{ψ(i)∞}^{b}, (α^{b})_{ψ(i)∞}): (m^{b})_{ψ(i)∞} → (m^{b−1})_{ψ(i)∞} and (α^{b})_{ψ(i)∞}^{−}: (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{b} → (m^{b−1})_{ψ(i)∞}^{−} (in the upper row, and otherwise the identity).

Checking that this is well defined and simplicial amounts to the same kind of checking as we have already encountered, using the same identities. One should notice that at no time during the verifications is the symmetry of addition used. It is used, however, for the isomorphism that renders matrix multiplication associative up to isomorphism.

The simplicial homotopy from inc to jnc is gotten by multiplications (in the bar direction) by matrices of the form

[\begin{matrix}1 & x_{ψ(i)ψ(j)∞}^{ϕ(j)} + ξ_{ψ(i)∞}^{ϕ(j)} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 0 & 1 \end{matrix}]  and  [\begin{matrix}1 & (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 0 & 1 \end{matrix}] .

4.7. The homotopic maps knc and lnc. Consider the following variant knc of the map jnc: using the same notation as for jnc, when evaluated on (ϕ, b, ψ) where ϕ: [r] → [p] and ψ: [r] → [q] are in Δ and ϕ(r) ⩽ b ⩽ p, knc(m)(ϕ, b, ψ) ∈ N_{0}B_{r}^{2n} is the element whose (0 ⩽ i < j ⩽ r)- and (0 ⩽ i ⩽ r < j = ∞)- entries are

[\begin{matrix}(m^{b})_{ψ(i)ψ(j)} & x_{ψ(i)ψ(j)∞}^{ϕ(j)} + ξ_{ψ(i)∞}^{ϕ(j)} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 0 & 1 \end{matrix}]

and

[\begin{matrix}(m^{b})_{ψ(i)∞} + (ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1}, ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1}) & (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ (1, 1) & 1 \end{matrix}] ,

respectively. The (A, B)-notation is the “plus-minus” notation for objects in T_{ℓ}R. The entries for j = ∞

can be written more concisely as [\begin{matrix}(m^{b})_{ψ(i)∞} + (Ξ, Ξ) & (m^{b})_{ψ(i)∞}^{−} + Ξ \\ (1, 1) & 1 \end{matrix}] where Ξ = ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1}.

The entries for finite j are the same as for jnc.

There is a natural map (in the nerve direction) from jnc to knc (of the form (X, id): (A, B) → (A + X, B + X) ∈ T_{ℓ}R — induced by the identity), giving a homotopy.

Finally, let lnc: NB^{n} → z_{∗}NB^{2n} be induced by the constant map sending any matrix to [ \begin{matrix}1 & 0 \\ (0,1) & 1 \end{matrix} ]. Matrix multiplication yields

[\begin{matrix}(m^{b})_{ψ(i)∞}^{+} + ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} & (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ 1 & 1 \end{matrix}] · [\begin{matrix}1 & 0 \\ (0, 1) & 1 \end{matrix}]

= [\begin{matrix}(m^{b})_{ψ(i)∞} + (ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1}, ξ_{ψ(i)∞}^{b} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1}) & (m^{b})_{ψ(i)∞}^{−} + ξ_{ψ(i)∞}^{c} + ⋯ + ξ_{ψ(i)∞}^{ϕ(i)+1} \\ (1, 1) & 1 \end{matrix}] .

With the same abbreviation as above, this reads

[\begin{matrix}(m^{b})_{ψ(i)∞}^{+} + Ξ & (m^{b})_{ψ(i)∞}^{−} + Ξ \\ 1 & 1 \end{matrix}] · [\begin{matrix}1 & 0 \\ (0, 1) & 1 \end{matrix}] = [\begin{matrix}(m^{b})_{ψ(i)∞} + (Ξ, Ξ) & (m^{b})_{ψ(i)∞}^{−} + Ξ \\ (1, 1) & 1 \end{matrix}] .

We obtain a homotopy from lnc to knc.

Hence inc is connected by a chain of homotopies to a constant map. Since η_{∗}: id → z_{∗} is a monomorphism in the homotopy category, this means that the stabilization map in: B^{n} → B^{2n} is homotopically trivial, and so B(∗, GL(R), GL(T_{ℓ}R)) = colim_{n} B^{n} is contractible for each ℓ ⩾ 0. Hence B(∗, GL(R), GL(T R)) is also contractible. This concludes the proof of Theorem 1.1.

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Department of Mathematical Sciences, NTNU, 7491 Trondheim, Norway

   E-mail address: baas@math.ntnu.no

Department of Mathematics, University of Bergen, 5008 Bergen, Norway

   E-mail address: dundas@math.uib.no

Department Mathematik der Universität Hamburg, 20146 Hamburg, Germany

   E-mail address: richter@math.uni-hamburg.de

Department of Mathematics, University of Oslo, 0316 Oslo, Norway

   E-mail address: rognes@math.uio.no