Stable bundles over rig categories

ST ABLE BUNDLES O VER RIG CA TEGORIES NILS A. BAAS, BJØRN IAN DUNDAS, BIR GIT RICHTER AND JOHN ROGNES Abstract. The point of t his paper i s to p rov e the con jecture that virtual 2 -vect or bundles are classi fied b y K ( ku ), the algebraic K -theory o f topological K -theory . Hence, b y the w ork of Ausoni and the fourth author, virtual 2-v ector bundles giv e us a ge ometric cohomology theory of the same telescopic complexity as elliptic cohomology . The main tec hnical step i s showing that for w ell-b ehav ed small rig categories R (also known as bimonoidal categories) the algebraic K -theory space , K ( H R ), of the ring sp ectrum H R associated to R is equiv alen t to K ( R ) ≃ Z × | BGL ( R ) | + , where GL ( R ) is the mono idal category of weak ly inv ertible matrices ov er R . The title refers to the sharp er result tha t B GL ( R ) is equiv alen t to B GL ( H R ). If π 0 R i s a ring this is almost f ormal, and our approach is to r eplace R by a ring completed version, ¯ R , provided b y [BDRR1] with H R ≃ H ¯ R and π 0 ¯ R the ri ng completion of π 0 R . The r emaining step is then to show that “stable R -bund les” and “stable ¯ R -bundles” are the same, which is done by a h ands-on con traction of a custom-buil t m odel f or the difference b etw een B GL ( R ) and B GL ( ¯ R ). 1. Introduction and main resul t In telesc o pic c o mplexity 0, 1 and ∞ there ar e cohomo lo gy theories that p ossess a geometric definition: de Rham cohomolo gy of manifolds is given in terms o f differential forms, co homology classes in rea l and c omplex K - theory a re clas ses of virtual vector bundles, and complex cob or dism has a geo metr ic definition p er se . In order to unders ta nd phenomena of in termediate teles c opic c omplexity , it is desir a ble to hav e geometric in terpretations fo r such cohomology theories as w ell. In [BDR] it w as co njectur ed that virtual 2- vector bundles provide a geometric in terpretation of a cohomolog y theory of telescopic complexity 2 which q ua lifies as a form of e lliptic cohomology . More precisely , it w a s conjectured that the algebraic K -theory o f a commutativ e rig ca tegory R is equiv a lent to the algebraic K -theory of the ring spectrum asso ciated with R . The case of virtual 2-vector bundles arises when R is the categ ory of finite dimensional complex v ector spaces, with ⊕ a nd ⊗ C as sum and m ultiplication. This, together with the analysis of the K -theory of complex topo lo gical K -theory due to Ausoni and the fourth author [AR, A], and the Quillen–L icht enbaum c o njecture for the integers, gives the desired relation to elliptic co homology . In this pap er we pr ov e the conjecture fro m [BDR ]. This work was motiv ated by the study of extended top ological qua n tum field theories and the sear ch for a geometric definition of elliptic cohomolog y , see [BDR], [F], [K ] and [L]. Let R b e a rig ca tegory (also known as a bimo no idal category ), i.e. , a categ ory w ith t wo op eratio ns ⊕ and ⊗ satisfying the axioms of a rig (ring wit hout negativ e elements) up to coherent natural isomorphisms. In a na logy with Quille n’s definition of the algebra ic K -theory s pace K ( A ) = Ω B ( ` n B GL n ( A )) of a ring A , the algebraic K -theo r y of R w as defined in [BDR] as K ( R ) = Ω B ( ` n | B GL n ( R ) | ) ≃ Z × | B GL ( R ) | + where B and GL n are versions of the bar constr uc tio n and the genera l linear g roup appr opriate for rig categ ories. On the other hand, fo rgetting the multiplicativ e structure, R has a n underlying symmetr ic monoidal category , a nd so it makes sense to sp eak a b out its K -theory sp ectrum H R with re spe ct to ⊕ . The K - theory sp ectrum cons truction H R is a dir ect ex tension of the usual Eilenberg– Mac Lane construction, and can, since R is a rig category, be endowed with the str ucture of a strict ring sp ectrum, for instance through the mo del given b y Elmendor f and Mandell in [EM]. Hence, w e may spea k about its algebraic Date : N o ve mber 9, 2018. 2000 Mathematics Subje ct Classific ation. Primary 19D23, 55R65; Secondary 19L41, 18D10. Key wo r ds and phr ases. Al gebraic K -theory , top ological K - theory , 2-v ector bundles, elliptic cohomology , bimonoidal categories, bi p ermutativ e categ ories. The firs t author w ould like to thank the Institu te for Adv anced Study , Princeton, for their hospitalit y and supp ort during his sta y in the spring of 2007. Part of the work was done while the second author was on s abbatical at Stanford Unive rsity , whose hospitality and stimulat ing env ironment i s gratefully ack nowledg ed. The third author thanks the SFB 676 f or s upport and the topology group i n Sheffield for stimulating dis cussions on the sub ject. 1 K - theory space K ( H R ). W e prov e that, under certa in mild restr ic tions on R , ther e is a n equiv alence K ( R ) ≃ K ( H R ) . In the sp ecia l situation where R is a ring ( i.e. , R is discr ete as a category and has negative elements), this is the sta ndard assertion that the K -theory of a r ing is equiv alent to the K -theo ry of its asso cia ted Eilenberg-Ma c Lane sp ectrum. The key difficulty in establishing the equiv alence ab ov e lies in proving that the lac k of negative ele ments makes no difference for alg ebraic K -theory , even for rig categories. More precisely , w e prove the following result: Theorem 1.1. L et ( R , ⊕ , 0 R , ⊗ , 1 R ) b e a s m al l top olo gic al rig c ate gory satisfying the fol lowing c ondi- tions: (1) Al l morphisms in R ar e isomorphisms, i.e. , R is a gr oup oid. (2) F or every obje ct X ∈ R the tr anslation funct or X ⊕ ( − ) is fa ithful. Then | B GL ( R ) | and B GL ( H R ) ar e we akly e quivalent. Henc e, the algebr aic K -the ory sp ac e of R as a rig c ate gory, K ( R ) = Ω B  a n > 0 | B GL n ( R ) |  ≃ Z × | B GL ( R ) | + , is we akly e quivalent to the algebr aic K -t he ory sp ac e of the strict ring sp e ctrum asso ciate d to R , K ( H R ) = Ω B  a n > 0 B GL n ( H R )  ≃ Z × B GL ( H R ) + . Addendum 1.2. In p articular, if R is t he c ate gory of finite dimensional c omplex ve ctor sp ac es, the the or em states that stable 2 -ve ctor bund les, in the sense of [BDR] ar e classifie d by B GL ( k u ) , wher e k u = H R is the c onne ctive c omplex K - t he ory sp e ctrum with π ∗ k u = Z [ u ] , | u | = 2 . In co n trast to K ( H R ), which is built in a tw o -stage pro c ess, the K -theory o f the (strictly ) bimonoida l category R is built using b oth monoidal struc tur es at once, s o in this sense K ( R ) is a mo del that is easier to understand and handle than K ( H R ). The conditions (1) a nd (2) on R are not restr ictive for the applications we hav e in mind, and a r e asso ciated with the fact tha t in [BDRR1] w e chose to work with v ariants o f the Grayson–Quillen mo del for K -theory . Pr obably , the restrictions can b e remo ved if o ne uses ano ther tec hno logical platform. Among those rig ca tegories that satisfy the r equirements of Theo rem 1.1 are the following ‘standard’ ones, usually c o nsidered in the context o f K -theor y constructions. • If R is the dis crete categor y (having only identit y morphis ms ) with ob jects the e le men ts of a ring with unit, R , then H R is the Eilenberg-Ma c Lane sp ectrum H R . • The sphere spe ctrum S is the algebra ic K - theory sp ectrum o f the small r ig catego r y of finite sets E . The ob jects of E ar e the finite sets n = { 1 , . . . , n } for n > 0, with the conv ention that 0 is the empt y set. Morphisms from n to m only exist for n = m , and in this ca se they constitute the symmetric group on n letters. The algebra ic K -theor y of S is e quiv alent to W a ldha usen’s A - theory of a point A ( ∗ ) [W], and so gives info r mation about diffeomorphisms of hig h dimensional disks. Thus we obtain that A ( ∗ ) ≃ K ( S ) ≃ K ( E ) ≃ Z × | B GL ( E ) | + . • F or a comm utative ring A w e consider the fo llowing small r ig catego ry of finitely gene r ated free A -mo dules, F ( A ). Ob jects of F ( A ) are the finitely generated free A -mo dules A n for n > 0. The set of mo r phisms from A n to A m is empty unless n = m , and the morphisms from A n to itself are the A -mo dule automo rphisms of A n , i.e. , GL n ( A ). O ur result allows us to identify the tw o-fo ld iterated algebr aic K - theory of A , K ( K ( A )), with Z × | B GL ( F ( A ) ) | + . • The case that star ted the pro ject is the catego ry of 2- vector spaces of Kapranov and V o evo dsky [KV], view ed as modules ov er the rig catego ry V of co mplex (Hermitian) vector spaces. Here V has ob jects C n for n > 0, and the automorphism space of C n is the unitary g roup U ( n ). This ident ifies K ( H V ) = K ( k u ) with K ( V ) ≃ Z × | B GL ( V ) | + , whic h was called the K -theory of the 2-catego ry of complex 2-vector spaces in [B DR ]. Auso ni’s calculatio ns [A] s how that K ( k u p ) has telescopic complexity 2 for every prime > 5, and th us qualifies as a form of elliptic co homology . • Replacing the complex num b ers by the reals yields an identification o f K ( k o ) with the K -theory of the 2- category of r e al 2-vector spaces. 2 • Considering o ther subgroups of GL n ( C ) or GL n ( R ) as morphisms in a categor y with ob jects n = { 1 , . . . , n } with n > 0 giv es a lar ge v ariety of K -theory sp ectra that are in the range of our result. F or a sample of suc h sp ecies ha ve a lo ok at [M2, pp. 161– 167]. 1.1. The s pi ne of the argument giving a pro of of Theorem 1.1. Although the pro of contains some length y technical lemmas, it is p ossible to give the main flo w of the argument in a few paragr aphs, referring aw ay the hard parts. Remem b er that the group-like mo noid GL n ( H R ) is defined b y the pullback GL n ( H R ) / /   ho colim m ∈ I Ω m M n ( H R ( S m ))   GL n ( π 0 H R ) / / M n ( π 0 H R ) . where I is a strictly mo no idal skeleton of the ca tegory of finite sets and injective functions, ma king GL n ( H R ) a group-like monoid (here we hav e written H R in the form of a simplicial functor with asso ciated symmetric spectrum m 7→ H R ( S m )). Let M n ( ¯ R ) b e the monoida l ca tegory of n × n -matrices ov er ¯ R (see Section 2). The s e t of co mpo nent s π 0 M n ( ¯ R ) can b e identified with M n ( π 0 ¯ R ), a nd w e let GL n ( ¯ R ) b e the s ubmonoidal catego ry of M n ( ¯ R ) consisting of the components that are inv er tible as matrices over the additive gro up completion of π 0 ¯ R (see Definitions 2.3 and 2.4). Since the effect of stabilization in H ¯ R is exactly g roup completion, the natural isomor phism | ¯ R| → H ¯ R ( S 0 ) together with stabilization induces a na tural map | GL n ( ¯ R ) | → GL n ( H ¯ R ). Lemma 1 .3. If ¯ R is a ring c ate gory, i.e. , a rig c ate gory with π 0 ¯ R a ring, t hen | GL n ( ¯ R ) | ∼ − → GL n ( H ¯ R ) is a homotopy e quivalenc e. Pr o of. By assumption π 0 ¯ R is isomorphic to its gr oup completion Gr ( π 0 ¯ R ) = π 0 H ¯ R , so it is enough to show that | M n ( ¯ R ) | a nd ho colim m ∈ I Ω m M n ( H ¯ R ( S m )) are equiv alent. Both are n 2 -fold pr o ducts, so it suffices to s how that | ¯ R| and hocolim m ∈ I Ω m H ¯ R ( S m ) are equiv alen t. All the structure maps Ω m H ¯ R ( S m ) → Ω m ′ H ¯ R ( S m ′ ) are equiv a lences for m → m ′ an injection o f nonempty finite sets, and since ¯ R is already group-like | ¯ R| ≃ H ¯ R ( S 0 ) maps by an equiv a lence to Ω H ¯ R ( S 1 ).  W e kno w from [BDRR1] that there is a chain of simplicia l rig categ ories R ∼ ← − − − − Z R − − − − → ¯ R such that • R ← Z R b ecomes a weak equiv alence upon realization, • H Z R → H ¯ R is a s table equiv alence and • π 0 ¯ R is a ring. Consider the comm utative diagr am | B GL ( R ) |   | B GL ( Z R ) | / / ∼ o o   | B GL ( ¯ R ) | ∼   B GL ( H R ) B GL ( H Z R ) ∼ o o ∼ / / B GL ( H ¯ R ) , where the definition of GL of r ig ca teg ories is given in Section 2 and the ba r constructio n is r ecalled in Section 3. Both constructions preserve weak equiv alences. • The left ward pointing horiz o nt al maps are weak equiv alences s ince R ← Z R is, • the bottom right ward p ointing arrow is a w e a k equiv alence since H Z R → H ¯ R is a stable equiv alence and • the right hand vertical map is a weak equiv alence by Lemma 1.3. T o show that the left hand vertical a rrow is a weak equiv alence, it ther efore suffices to pr ov e that the upper rig ht hand horizo nt al map is a weak e quiv alence. By P rop osition 3.8, the homotopy fib er of B GL ( Z R ) → B GL ( ¯ R ) is given by the one-sided bar cons truction B ( ∗ , GL ( Z R ) , GL ( ¯ R )), and s o we hav e reduced the problem to giv ing a contraction of the asso ciated space. 3 Under the as sumptions of Theorem 1.1 w e k now from [BDRR1] that ther e is a c ha in o f weak equiv a - lences ( −R ) R ∼ ← − − − − Z ( −R ) R ∼ − − − − → ¯ R of Z R -mo dules. Here, ( −R ) R is the Grayson-Q uillen model [G], also discussed in Section 4.1 , and so by Remar k 2.5 we get weak equiv alences B ( ∗ , GL ( R ) , GL (( −R ) R )) ∼ ← − − − − B ( ∗ , GL ( Z R ) , GL ( Z ( −R ) R )) ∼ − − − − → B ( ∗ , GL ( Z R ) , GL ( ¯ R )) . This mea ns that the somewha t complicated construction of ¯ R from [BDRR1] may b e safely forgotten once we kno w it ex is ts. Just one simplification r emains: in Lemma 4.2 we display a weak equiv alenc e T R → ( −R ) R of R - mo dules. The R -mo dules T R and ( −R ) R are gener alizations of how to cons truct the integers from the natural num b er s b y c onsidering equiv alence classes of pairs o f natura l n umber s. W e are left with showing that B ( ∗ , GL ( R ) , GL ( T R )) is contractible. This is done through a concr ete contraction. It is an elab ora tion of the following path  a − b 0 0 1  →  a − b b 0 1  ←  1 0 − 1 1  , a, b ∈ N in B ( ∗ , GL 2 ( N ) , GL 2 ( Z )) = { [ q ] 7→ GL 2 ( N ) × q × GL 2 ( Z ) } , with 1-simplices given b y  1 b 0 1  ,  a − b 0 0 1  and  a b 1 1  ,  1 0 − 1 1  ∈ GL 2 ( N ) × GL 2 ( Z ) resp ectively , showing tha t the inclusion of B ( ∗ , GL 1 ( N ) , GL 1 ( Z )) in B ( ∗ , GL 2 ( N ) , GL 2 ( Z )) is n ull-ho mo- topic. 1.2. Plan. The structure o f the pap er is as follows. In Section 2 we discuss the mo noidal category GL n ( R ) o f (w eakly) invertible matric e s ov er a s tr ictly bimono idal catego ry R . Section 3 recalls the definition of the bar constr uction of monoidal categories as in [BDR] a nd in tro duces a version with co efficients in a mo dule. In Section 4 w e construct the pro mised co ntraction of B ( ∗ , GL ( R ) , GL ( T R )), th us co mpleting the pro o f of the main theorem. This pap er has circulated in preprint form under the title “Two-v ector bundles define a for m of elliptic cohomolog y”, which, while not highlighting the nuts and b olts of the pap er p erhaps better repre s ent ed the reason for writing (o r rea ding) it. The old preprint [BDRR] also contained the main result of the pap er [BDRR1]. Graeme Segal cons tructed a ring completion o f the rig categ o ry of complex vector spaces [S2, p. 3 00] (see also the App endix of [S1]). His mo del is a topo lo gical categor y consis ting of certain spaces of bo unded chain co mplexes and spaces of quasi-isomo rphisms. One ca n probably build a v ariant of his mo del that could replace the co nstruction ¯ R in o ur pro of o f Theorem 1.1, in the specia l case R = V . A piece o f notation: if C is any small categor y , then the expres sion X ∈ C is short for “ X is an ob ject of C ” and likewise for mo rphisms and diagra ms. D isplay ed diagr ams commute unles s the contrary is stated explicitly . F or basics on bipermutativ e and rig categor ies w e refer to [BDRR1] Section 2 . 2. Weakl y inver tible ma trices Let R b e a strictly bimonoidal category. Definition 2.1. The c ate gory of n × n - m atric es over R , M n ( R ), is defined as fo llows. The ob jects of M n ( R ) are matrices X = ( X i,j ) n i,j =1 of ob jects of R and morphisms from X = ( X i,j ) n i,j =1 to Y = ( Y i,j ) n i,j =1 are matrices F = ( F i,j ) n i,j =1 where each F i,j is a morphism in R fro m X i,j to Y i,j . Lemma 2.2. F or a strictly bimonoida l c ate gory ( R , ⊕ , 0 R , c ⊕ , ⊗ , 1 R ) the c ate gory M n ( R ) is a monoidal c ate gory with r esp e ct to t he matrix multiplic ation bifunctor M n ( R ) × M n ( R ) · − → M n ( R ) ( X i,j ) n i,j =1 · ( Y i,j ) n i,j =1 = ( Z i,j ) n i,j =1 with Z i,j = n M k =1 X i,k ⊗ Y k,j . The unit of this structu r e is given by the unit matrix obje ct E n which has 1 R ∈ R as diagonal entries and 0 R ∈ R in the other plac es. 4 The prop erty of R being bimonoida l giv es π 0 R the structure of a r ig, and its (additiv e) group com- pletion Gr ( π 0 R ) = ( − π 0 R ) π 0 R is a r ing. Definition 2 .3. W e define the we akly invertible n × n -matric es over π 0 R , GL n ( π 0 R ), to b e the n × n - matrices over π 0 R that a re inv e r tible as matrice s over Gr ( π 0 R ). Note that we can define GL n ( π 0 R ) by the pullback square 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 c ate gory of we akly invertible n × n -matric es over R , GL n ( R ), is the full sub- category o f M n ( R ) with ob jects all matrices X = ( X i,j ) n i,j =1 ∈ M n ( R ) who se matrix of π 0 -classes [ X ] = ([ X i,j ]) n i,j =1 is contained in GL n ( π 0 R ). Matrix m ultiplica tio n is of c ourse compatible with the pro pe r ty of being weakly in vertible. Thus, the category GL n ( R ) inherits a monoidal structure fro m M n ( R ). How ever, even if our base ca tegory is no t bimono idal it still makes sense to talk a bo ut ma trices and even weakly inv ertible matrices, as lo ng as π 0 of that category is a rig . Remark 2.5. If M is an R -mo dule, matrix multiplication mak e s the categor y M n ( M ) in to a module ov er the mono idal categor y GL n ( R ). F or our applicatio ns the following situa tion will b e pa rticularly impo rtant: let M → N be a map of R -mo dules, where the map o f π 0 R -mo dules π 0 M → π 0 N comes from a rig map under π 0 R . Then we ge t a map GL n ( M ) → GL n ( N ) of GL n ( R )-modules which induces a weak equiv alence up on realization if M → N do es. There is a canonica l stabilization functor GL n ( R ) → GL n +1 ( R ) which is induced b y taking the blo ck 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-kno wn facts ab out the o ne-sided bar constructio n of monoida l categorie s. Definition 3.1. Let ( M , · , 1 ) b e a monoidal ca teg ory and T a left M -mo dule. The one-side d b ar c onstruction B ( ∗ , M , T ) is the simplicial catego ry whose category o f q - simplices B q ( ∗ , M , T ) is as follows: consider the or dered set [ q ] + = [ q ] ⊔ { ∞} , i.e. , in a dditio n to the n umber s 0 < 1 < · · · < q there is a greatest element ∞ . An ob ject a in B q ( ∗ , M , T ) consists of the follo wing data. (1) F or eac h 0 6 i < j 6 q there is an ob ject a ij ∈ M , and for each 0 6 i 6 q a n ob ject a i ∞ ∈ T . (2) F or eac h 0 6 i < j < k 6 ∞ there is an is o morphism a ij k : a ij · a j k → a ik (in M if k < ∞ and in T if k = ∞ ) such tha t if 0 6 i < j < k < l 6 ∞ , the following diagr am commutes ( a ij · a j k ) · a kl a ijk · id   struct. iso. / / a ij · ( a j k · a kl ) id · a jk l   a ik · a kl a ikl / / a il a ij · a j l . a ijl o o A morphis m f : a → b consis ts of morphisms f ij : a ij → b ij (in M if j < ∞ a nd in T if j = ∞ ) s uc h that if 0 6 i < j < k 6 ∞ f ik a ij k = b ij k ( f ij · f j k ) : a ij · a j k → b ik . The simplicia l structure is g otten as follows: if φ : [ q ] → [ p ] ∈ ∆ the functor φ ∗ : B p ( ∗ , M , T ) → B q ( ∗ , M , T ) is obtained by preco mpo s ing with φ + = φ ⊔ { ∞} . So fo r instance d 1 ( a ) is gotten by deleting all entries with indices co nt aining 1 from the data giving a . In o rder to allow for degenera cy maps s i , we use the conv ention that all ob jects of the form a ii are the unit of the mo noidal structure, and all isomorphisms of the form a iik and a ikk are identities. 5 Remark 3.2. A g o o d wa y to think abo ut this comes from the dis crete case when M is a monoid and T is an M -set. Then an ob ject a ∈ B q ( ∗ , M , T ) is uniquely given b y the “sup erdiag o nal” ( a 01 , a 12 , . . . , a q − 1 q , a q ∞ ), and B ( ∗ , M , T ) is isomorphic to the nerve o f the category with ob jects T and morphisms a 1 ∞ → a 01 · a 1 ∞ = a 0 ∞ corres p o nding to ( a 01 , a 1 ∞ ). The reason w e have to include all of the “ upp er tria ng ular” ele men ts is that asso ciativity may not be strict. F or instance, it is no t strict in our main example: matrix multiplication o ver a r ig categor y is in general not strictly asso ciative. Hence, a 01 · ( a 12 · a 23 ) may be differen t from ( a 01 · a 12 ) · a 23 , and so the sup e rdiagona l elements do not car ry enoug h information to turn the obvious choice of face maps into a simplicial structure. W e remedy this by adding choices for all faces in our simplices. This just adds more elements in each isomorphism class in every simplicial deg ree, and is a standard trick used in many places, for ins tance by W aldhause n in his S • -construction. Example 3. 3. (1) If T is the one- po int catego ry ∗ , then B ( ∗ , M , ∗ ) is is omorphic to the bar con- struction B M of [BDR]. (2) If F : M → M ′ is a lax monoidal functor, then M ′ may b e co nsidered a s an M -mo dule, and we wr ite without fur ther ado B ( ∗ , M , M ′ ) for the corre s po nding bar construction (with F sup- pressed). In case F is a n isomorphism, B ( ∗ , M , M ′ ) is contractible. W e think of elemen ts of B q ( ∗ , M , T ) in terms of strictly upper tr iangular arrays of ob jects, suppr essing the isomo r phisms, so that a t ypica l element in B 2 ( ∗ , M , T ) is written a 01 a 02 a 0 ∞ a 12 a 1 ∞ a 2 ∞ with d 1 given b y a 02 a 0 ∞ a 2 ∞ . The one-sided bar constr uction is functorial in “natura l modules”. A natura l module is a pair ( M , T ) where M is a mo no idal category and T is an M -mo dule. A morphism ( M , T ) → ( M ′ , T ′ ) consists of a pair ( F, G ) where F : M → M ′ is a la x monoidal functor a nd G : T → F ∗ T ′ is a map o f M -mo dules, where F ∗ T ′ is T ′ endow ed with the M -mo dule structure g iven by restricting a long F . Lemma 3.4. F or e ach q ther e is an e quivalenc e of c ate gories b etwe en B q ( ∗ , M , T ) and the pr o duct c ate gory M × q × T . Pr o of. The equiv alence is given b y the forgetful functor F : B q ( ∗ , M , T ) → M × q × T sending a to the “super diagonal” F ( a ) = ( a 01 , . . . , a q − 1 q , a q ∞ ). The inv erse 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 ij k given b y the structur a l is o- morphisms.  Corollary 3.5. L et ( F , G ) : ( M , T ) → ( M ′ , T ′ ) b e a map of n atur al mo dules such that F and G ar e e quivalenc es of c ate gories. Then the induc e d map B ( ∗ , F , G ) : B ( ∗ , M , T ) → B ( ∗ , M ′ , T ′ ) is a de gr e ewise e qu ivalenc e of simpli cial c ate gories. Remark 3 . 6. The same res ult holds, if instea d of equiv alences of categor ies we consider w eak equiv a- lences. Usually M × q × T is no t functoria l in [ q ], but if ( M , T ) is s trict, the monoidal str uc tur e gives a simplicial catego ry B strict ( ∗ , M , T ) = { [ q ] 7→ M × q × T } . In this situation Lemma 3.4 rea ds: Corollary 3.7. L et M b e a strict monoidal c ate gory and T a s t rict M -m o dule. Then ther e is a de gr e e- wise e quivalenc e b etwe en the simplici al c ate gories B ( ∗ , M , T ) and B strict ( ∗ , M , T ) . 6 Prop ositio n 3.8. L et F : M → G b e a str ong monoidal functor such t hat the monoidal structu r e on G induc es a gr oup structur e on π 0 G . Then B ( ∗ , M , G ) − − − − → B M   y   y B ( ∗ , G , G ) − − − − → B G is homotopy c artesian, me aning that it induc es a homotopy c artesian diagr am up on applying the nerve functor in every de gr e e. The (nerve of the) lower lef t hand c orner is c ontr actible. Pr o of. By [JS] there is a diagram of monoidal categ ories st M ∼ − − − − → M st F   y F   y st G ∼ − − − − → G such that the horizontal maps are mo noidal eq uiv alences, and st F is a stric t monoidal functor betw een strict monoidal catego ries. T ogether with Corolla ries 3.5 a nd 3.7 this tells us that we ma y just as well consider the str ict situation, and use the strict bar construction. How ever, note that the nerve o f the strict monoida l catego ry st M is a simplicial monoid, a nd that re versal of prio rities gives a natural isomorphism B ( ∗ , N st M , N st G ) ∼ = N B strict ( ∗ , st M , st G ) , so that o ur statement reduces to the statemen t that B ( ∗ , N st M , N st G ) → B ( N s t M ) → B ( N st G ) is a fiber sequence up to homoto p y , which is a classical result [M1] given that N st G is gr oup-like.  4. Contracting the one-sided bar construction 4.1. A mo de l for K -theory o f R as an R -mo dule. In order to construct concrete homotopies, w e offer a sligh t v ariant of the Gra yson–Q uillen mo del where morphisms are not entire equiv alence clas ses. The price is as usua l that the resulting o b ject is a 2-catego ry . Since there was some confusion ab out this point while the paper was still at a preprint stag e , we emphasize that this is not the co ns truction o f Thomason [Th1, 4.3.2] and Jar dine [J]. Definition 4.1. Let ( M , ⊕ , 0 M , τ M ) b e a p ermutativ e categ o ry wr itten additively . Let T M b e the following 2- c ategory . The ob jects of T M are pa ir s ( A + , A − ) =: A o f ob jects in M , thought of as plus and minus ob jects in M . Giv en tw o ob jects A, B ∈ T M , the categor y o f morphisms T M ( A, B ) has ob jects the pairs ( X , α ) where X is an ob ject 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 isomo rphism φ : X → Y such that β ± (1 ⊕ φ ) = α ± . Co mpo s ition T M ( B , C ) × T M ( A, B ) → T M ( A, C ) is given by sending (( Y , β ) , ( X , α )) to the pair co nsisting of X ⊕ Y and the co mp os ite maps A ± ⊕ ( X ⊕ Y ) = ( A ± ⊕ X ) ⊕ Y α ± ⊕ id − − − − → B ± ⊕ Y β ± − − − − → C ± . Comp osition o n mor phisms is simply given b y addition. Comp osition is stric tly as so ciative because M is per mut ative; if M is merely symmetric monoidal, standard modifications are necessary . Symmetry allows for a symmetric mo noidal structur e on T M : if w e 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 mo del for the K -theo ry of M is the categor y ( −M ) M with the same ob jects as T M a nd with morphism sets the pa th components ( −M ) M ( A, B ) = π 0 T M ( A, B ). If a ll morphisms in M a r e is omorphisms and if additive translation in M is faithful, ( − M ) M is s hown in [G] to be a group c ompletion of M . Under these hypothes e s, there is at most one morphism b etw ee n tw o given ob jects ( X , α ) a nd ( Y , β ) in T M ( A, B ). Co nsequently the morphism spac es are homotopy discrete: the pro jection T M ( A, B ) → π 0 T M ( A, B ) is a weak equiv alence. In the case of a top ologica l categ ory , we interpret π 0 as the co eq ua lizer of the sour ce a nd tar get maps fr o m the mo r phism s pace to the ob ject space. The assignment that is the identit y on ob jects and otherwis e is induced by the pro jection T M ( A, B ) → π 0 T M ( A, B ) gives a 2-functor T M → ( −M ) M (consider ing ( − M ) M as a 2-ca tegory with only identit y 2- morphism). 7 Lemma 4.2. L et M b e a p ermutative c ate gory with al l morphisms in M isomorphi sms and faithful additive tr anslation. The 2 -functor T M → ( −M ) M is a we ak e quivalenc e and the standar d inclusion M → T M is a gr oup c ompletion. Note that if R is a rig category, T R will not be a rig ca tegory (essen tially beca use of the non-strict symmetry in quadratic terms, as in [Th2, p. 57 2]), but it will still be an R -mo dule: Lemma 4 .3. L et ( R , ⊕ , 0 R , c ⊕ , ⊗ , 1 R ) b e a strictly bimonoidal c ate gory. The map R × T R → T R given on obje cts by ( A, ( B + , B − )) 7→ ( A ⊗ B + , A ⊗ B − ) , and on morphisms by sending φ : A → B ∈ R and ( X, α ) ∈ T R ( C, D ) to the p air c onsisting of A ⊗ X and t he map A ⊗ C ± ⊕ A ⊗ X − − − − → A ⊗ ( C ± ⊕ X ) φ ⊗ α ± − − − − → B ⊗ D ± (wher e the first map is the left distributivity isomorphi sm) induc es an R -m o dule structu r e on T R . W e consider T R as a simplicial category by taking the nerve o f each categor y of mor phisms; th us in simplicial degree ℓ , the ob jects of T ℓ R are the ob jects of T R . The mor phisms in T ℓ R from ( A + , A − ) to ( B + , B − ) co nsist o f ob jects X 0 , . . . , X ℓ , a 1-morphism α ± : A ± ⊕ X 0 → B ± , and isomorphisms φ r : X r → X r − 1 for r = 1 , . . . , ℓ . The simplicial structur e is given by co mp os ing and forgetting φ r ’s and inserting identit y maps. 4.2. Sub divisi ons. W e will use the following v aria n t o f edg ewise sub division to make r o om for an explicit simplicial co n traction, who se co nstruction b egins in Subse c tion 4.4. Consider the shea r functor z : ∆ × ∆ → ∆ × ∆ given by sending ( S, T ) to ( T ⊔ S, T ) where T ⊔ S is the disjoint unio n with the ordering obtained from T and S w ith the extra decla ration that ev ery ob ject in S is g reater than every ob ject in T . If B is a bisimplicia l ob ject, we let z ∗ B = B ◦ z . The standa rd inclusion S → T ⊔ S gives a natural transformation η in ∆ × ∆ from the iden tity to z , and hence a natural transforma tion in bisimplicial se ts η ∗ : z ∗ → id. Let E ns denote the categor y of sets and functions . Lemma 4.4. F or any bisimplici al set X t he map η ∗ : z ∗ X → X b e c omes a we ak e quivalenc e up on r e alization. Pr o of. The diagonal of z ∗ X is equa l to the ev aluation of X o n the o ppo site of the co mpo site ∆ S 7→ ( S,S ) − − − − − − → ∆ × ∆ ( S,T ) 7→ ( S ⊔ S,T ) − − − − − − − − − − → ∆ × ∆ , so since a map o f bisimplicial sets is an equiv alence if it is one in every (vertical) degre e , it is eno ugh to k now that for each fix ed T ∈ ∆ the natural map { S 7→ X ( S ⊔ S, T ) } → { S 7→ X ( S, T ) } is a w eak equiv alence. But this is a s ta ndard weak equiv alence from the (seco nd) edgewise sub division, which is known to b e homoto pic to a homeomorphism after rea liz ation. See [BHM, Lemma 1.1] and the pro of of [BHM, Pr o p o sition 2.5].  V ertices in z ∗ (∆[ p ] × ∆[ q ]) (where pro ducts of simplicial sets are viewed as bisimplicial sets, and vertices are (0 , 0)-simplices) are for instance indexed by tuples (( a, b ) , c ) wher e 0 6 a 6 b 6 p and 0 6 c 6 q . Here are pictures of z ∗ (∆[2] × ∆[0]) and z ∗ (∆[2] × ∆[1]): ((2 , 2) , 0) / / & & M M M M M M M M M M ((1 , 2) , 0)   / / ((1 , 1) , 0)   ((0 , 2) , 0) / / & & M M M M M M M M M M ((0 , 1) , 0)   ((0 , 0) , 0) 8 ((2 , 2) , 1) / /   + + W W W W W W W W W W W W W W W W W W W W W W W W W W " " E E E E E E E E E E E E E E E E E E E E E   ((1 , 2) , 1)   & & M M M M M M M M M M / / 2 2 2 2   2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ((1 , 1) , 1)   & & M M M M M M M M M M 2 2 2 2 2 2 2 2 2 2 2   2 2 2 2 2 2 2 2 2 2 2 2 ((0 , 2) , 1) , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y / /   ((0 , 1) , 1) & & M M M M M M M M M M   ((2 , 2) , 0) / / + + W W W W W W W W W W W W W W W W W W W W W W W W W W ((1 , 2) , 0) & & M M M M M M M M M M / / ((1 , 1) , 0) & & M M M M M M M M M M ((0 , 0) , 1)   ((0 , 2) , 0) , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y / / ((0 , 1) , 0) & & M M M M M M M M M M ((0 , 0) , 0) . Note that for any bisimplicial set X X ( p,q ) ∼ = Hom bisimp. sets (∆[ p ] × ∆[ q ] , X ) = Z ([ s ] , [ t ]) Ens(∆([ s ] , [ p ]) × ∆([ t ] , [ q ]) , X ( s,t ) ) as a ca tegorical end. Therefore, the r ight adjoint of z ∗ , z ∗ , is g iven by ( z ∗ X ) ( p,q ) = Hom bisimp. sets (∆[ p ] × ∆[ q ] , z ∗ X ) ∼ = Hom bisimp. sets ( z ∗ (∆[ p ] × ∆[ q ]) , X ) and thus z ∗ X = { [ p ] , [ q ] 7→ { bisimp. maps z ∗ (∆[ p ] × ∆[ q ]) → X }} = { [ p ] , [ q ] 7→ Z ([ s ] , [ t ]) Ens(∆([ t ] ⊔ [ s ] , [ p ]) × ∆([ t ] , [ q ]) , X ( s,t ) ) } . Let η ∗ : X → z ∗ X be the natural tra nsformation ass o ciated with η . Notice that η ∗ maps X (0 ,q ) isomorphica lly to ( z ∗ X ) (0 ,q ) for all q > 0, so ( z ∗ X ) (0 ,q ) ∼ = X (0 ,q ) . Lemma 4 .5. In the homotopy c ate gory (with r esp e ct to m aps that b e c ome we ak e quivalenc es up on r e al- ization), η ∗ : X → z ∗ X is a split m onomorphism. Pr o of. By for mal considera tions the diagra m z ∗ X z ∗ η ∗ / / η ∗ $ $ H H H H H H H H H H z ∗ z ∗ X counit   X commutes, and η ∗ is a w eak equiv alence after realization. Hence z ∗ η ∗ (and so η ∗ ) is a split monomorphism in the homotop y category .  4.3. The bar construction on matrices. Let R b e a strictly bimono idal catego ry s uch that all mor- phisms ar e isomorphisms and e a ch tra nslation functor is faithful. Consider the one-s ide d bar co nstruction B ( ∗ , GL n ( R ) , GL n ( T R )). Viewing T R as a simplicial cate- gory we get that B ( ∗ , GL n ( R ) , GL n ( T R )) is a bisimplicial categor y . W e are go ing 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 co nt ractible for eac h ℓ . T o ease readability , we will abandon the c umber some ⊕ and ⊗ in favor o f the mo re reada ble + and · — reminding us of the matrix nature of our efforts. Fix ℓ once and for all, and le t B n = B ( ∗ , GL n ( R ) , GL n ( T ℓ R )). An ob ject in B n q is a colle c tio n m ij of n × n matrices in R for 0 6 i < j 6 q , and for each 0 6 i 6 q a matrix m i ∞ in T ℓ R , tog e ther with suitably compatible structural is omorphisms m ij k : m ij · m j k → m ik . The matric es are drawn from the “weakly inv ertible co mpo nen ts”. The matrice s m i ∞ and the structural isomor phis ms relating these need sp ecial attent ion. Each e ntry is in T ℓ R , so m i ∞ can b e viewed as a pair m ± i ∞ of ma tr ices, and the str uctural isomorphism m ij ∞ : m ij · m j ∞ → m i ∞ is a tuple ( m ± ij ∞ , φ 1 ij ∞ , . . . , φ ℓ ij ∞ ), where the φ r ij ∞ : x r ij ∞ → 9 x r − 1 ij ∞ ∈ M n ( R ) for r = 1 , . . . , ℓ ar e matrices of isomor phisms, a nd the m ± ij ∞ : m ij · m ± j ∞ + x 0 ij ∞ → m ± i ∞ are iso mo rphisms. The assumed commutativit y of ( m ij · m j k ) · m k ∞ ∼ = / / m ijk · id   m ij · ( m j k · m k ∞ ) id · m jk ∞   m ik · m k ∞ m ik ∞ / / m i ∞ m ij · m j ∞ m ij ∞ o o says that tw o morphisms from ( m ij · m j k ) · m k ∞ agree: one is an isomo rphism with source ( m ij · m j k ) · m k ∞ + x r ik ∞ , the other one is an iso morphism with sour ce ( m ij · m j k ) · m k ∞ + m ij · x r j k ∞ + x r ij ∞ . Ther efore we obta in the fo llowing equality . Lemma 4 .6. In t he situation ab ove one has the identity x r ik ∞ = m ij · x r j k ∞ + x r ij ∞ for r = 0 , 1 , . . . , ℓ , and the diagr am m ij · ( m j k · m ± k ∞ ) + x 0 ik ∞ m ij · ( m j k · m ± k ∞ ) + m ij · x 0 j k ∞ + x 0 ij ∞ id · m ± jk ∞ +id   ( m ij · m j k ) · m ± k ∞ + x 0 ik ∞ ∼ = O O m ijk · id+id   m ij · m ± j ∞ + x 0 ij ∞ m ± ij ∞   m ik · m ± k ∞ + x 0 ik ∞ m ± ik ∞ / / m ± i ∞ c ommutes. Here the map id · m ± j k ∞ already incorp or a tes the distributivit y isomo rphism, as sp ecified in Lemma 4.3. A morphism α : m → ˜ m in B n q consists of an n × n matrix of maps α ij : m ij → ˜ m ij in R for 0 6 i < j 6 q , and of morphisms ( α ± i ∞ , ψ 1 i ∞ , . . . , ψ ℓ i ∞ ) : m ± i ∞ → ˜ m ± i ∞ in T ℓ R for 0 6 i 6 q , all co mpatible with the structure maps of m and ˜ m . Th us there are matrices of ob jects ξ r i ∞ of R for 0 6 r 6 ℓ , each α ± i ∞ is a map m ± i ∞ + ξ 0 i ∞ → ˜ m ± i ∞ , and the ψ r i ∞ for r = 1 , . . . , ℓ a re maps ξ r i ∞ → ξ r − 1 i ∞ of matric e s in R . The compatibility condition α i ∞ m ij ∞ = ˜ m ij ∞ ( α ij · α j ∞ ) allows us to draw the follo wing conclusio n. Lemma 4 .7. In t he situation ab ove one has the identity x r ij ∞ + ξ r i ∞ = m ij · ξ r j ∞ + ˜ x r ij ∞ for e ach r = 0 , . . . , ℓ . 4.4. Start of the pro of that B ( ∗ , GL ( R ) , GL ( T ℓ R )) is contrac tible. In the following, 0 and 1 ar e short for z ero resp. unit matrices ov er R of v ar ying size. W e will s how that colim n B n = B ( ∗ , GL ( R ) , GL ( T ℓ R )) is contractible by sho wing that each matrix stabilization functor in : B n → B 2 n is trivial in the homo topy category . Her e in( m ) = [ m 0 0 1 ]. W e regar d the simplicial categories B n and B 2 n as bisimplicial sets, b y way of their r esp ective nerves N B n and N B 2 n . T o be precis e, the ( p, q )-simplices of N B 2 n are N p B 2 n q . By Lemma 4 .5 it then suffices to show that the comp osite map inc = η ∗ ◦ in : N B n → z ∗ N B 2 n is trivial in the homotopy ca tegory . As remarked ab ov e, z ∗ ( N B 2 n ) (0 ,q ) ∼ = ( N B 2 n ) (0 ,q ) = N 0 B 2 n q , so the sub division op erato r z ∗ do es not make any differ ence b efore w e sta rt to consider positive-dimensional simplices ( p > 0) in the nerve direction. Seeing that the imag e lies in a s ing le path comp onent is easy: if m ∈ N 0 B n 0 = ob GL n ( T ℓ R ) then there is a path  m 0 0 1  →  m m − 0 1  →  m m − (1 , 1) 1  ←  1 0 (0 , 1) 1  . The first a rrow repr esents the o ne-simplex in the bar direction g iven by the matrix multiplication  1 m − 0 1  ·  m 0 0 1  =  m m − 0 1  ∈ GL 2 n ( T ℓ R ) . 10 The second a rrow represe nts the one-simplex in the ner ve direction induced by the mo rphism 0 = (0 , 0) → (1 , 1) ∈ T 0 R ⊂ T ℓ R . The third map repr esents the one-simplex in the bar direction g iven by m ultiplication b y  m + m − 1 1  ∈ GL 2 n ( R ) . The rest of this section ex tends this path to a full homotopy , fr om inc via maps jnc and knc to a constant map lnc. 4.5. The homotopi c maps inc and jnc. Recall that ℓ > 0 is fixed, B n = B ( ∗ , GL n ( R ) , GL n ( T ℓ R )) is the s implicial ca tegory given by the o ne-sided bar cons truction, and N B n : [ p ] , [ q ] 7→ N p B n q is the bisimplicial se t given by its deg r eewise nerve. W e already let inc : N B n → z ∗ N B 2 n be the comp osite of the matrix s tabilization map in : N B n → N B 2 n and the natur a l map η ∗ : N B 2 n → z ∗ N B 2 n . There is another map jnc : N B n → z ∗ N B 2 n which is homotopic to inc. On N 0 B n q it is easy to describe: if m ∈ N 0 B n q , we declare that X ( m ) is giv en by X ( m ) ij =                " 1 m − i ∞ 0 1 # if i < j = ∞ " 1 x 0 ij ∞ 0 1 # if i < j < ∞ and let jnc( m ) = X ( m ) · inc( m ) ∈ N 0 B 2 n q = z ∗ ( N B 2 n ) (0 ,q ) . Here jnc( m ) ij =                " m i ∞ m − i ∞ 0 1 # if i < j = ∞ " m ij x 0 ij ∞ 0 1 # if i < j < ∞ with jnc( m ) ij k : jnc( m ) ij · jnc ( m ) j k → jnc( m ) ik being the is omorphisms induced b y m ij k as fo llows: for k < ∞ we use the iden tity x 0 ik ∞ = m ij · x 0 j k ∞ + x 0 ij ∞ from Lemma 4.6 and obtain  m ij k id id id  :  m ij x 0 ij ∞ 0 1  ·  m j k x 0 j k ∞ 0 1  =  m ij · m j k m ij · x 0 j k ∞ + x 0 ij ∞ 0 1  →  m ik x 0 ik ∞ 0 1  and for k = ∞ we use the string of isomorphisms  x ℓ ij ∞ 0 0 0  → . . . →  x 0 ij ∞ 0 0 0  together with the isomorphism  m ij ∞ m − ij ∞ id id  :  m ij x 0 ij ∞ 0 1  ·  m j ∞ m − j ∞ 0 1  +  x 0 ij ∞ 0 0 0  =  m ij · m j ∞ + x 0 ij ∞ m ij · m − j ∞ + x 0 ij ∞ 0 1  →  m i ∞ m − i ∞ 0 1  . W e no tice that the T ℓ -direction do e s no t add any complications o ther tha n notational. This contin ues to b e true in general, so w e s implify notation by considering only the case ℓ = 0. The rele v ant complications a rise when one sta rts moving in the nerve direc tion. As the cons truction of the map jnc is quite in volved, we will give some examples first. The impatient reader can skip this part and restart reading in Subsection 4.6 where the form ula in the g eneral case is given. As an illustr ation, 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 n 0 = N 2 GL n ( T 0 R ) . 11 Then jnc( m ) is captured by the picture h m 2 ( m 2 ) − 0 1 i h 1 ξ 2 0 1 i / / h 1 ξ 2 + ξ 1 0 1 i ' ' O O O O O O O O O O O h m 2 ( m 2 ) − + ξ 2 0 1 i h 1 ξ 1 0 1 i   / / h m 1 ( m 1 ) − 0 1 i h 1 ξ 1 0 1 i   h m 2 ( m 2 ) − + ξ 2 + ξ 1 0 1 i / / ' ' P P P P P P P P P P P P h m 1 ( m 1 ) − + ξ 1 0 1 i   h m 0 ( m 0 ) − 0 1 i where the bar direction is written in the “ g m − − − − → mg ” for m, and the unlab eled ar rows corr esp ond to the nerve direction, with en tries consisting of the a ppropriate α ’s. An ev en more complicated example, essen tially displaying all the complexity of the gener a l c ase: ℓ = 0, p = q = 1, a nd α : m 1 → m 0 ∈ N 1 B n 1 with ( m u ) ± 01 ∞ : ( m u ) 01 · ( m u ) ± 1 ∞ + ( x u ) 0 01 ∞ → ( m u ) ± 0 ∞ (for u = 0 , 1 running in the nerve dir ection), α 01 : m 1 01 → m 0 01 and α ± i ∞ : ( m 1 ) ± i ∞ + ξ i ∞ → ( m 0 ) ± i ∞ . Then jnc( m ) is the map from z ∗ (∆[1] × ∆[1]) = ((0 , 0) , 0) ( (0 , 1) , 0) o o ((1 , 1) , 0) o o ((0 , 0) , 1) O O ((0 , 1) , 1) o o O O ((1 , 1) , 1) o o O O f f M M M M M M M M M M sending the ((0 , 1) , 0) ← ((1 , 1) , 0) ← ((1 , 1) , 1) simplex to  1 ξ 0 ∞ 0 1  h m 1 01 x 1 01 ∞ + ξ 0 ∞ 0 1 i h m 1 0 ∞ ( m 1 ) − 0 ∞ + ξ 0 ∞ 0 1 i h m 1 01 x 1 01 ∞ 0 1 i h m 1 0 ∞ ( m 1 ) − 0 ∞ 0 1 i h m 1 1 ∞ ( m 1 ) − 1 ∞ 0 1 i ∈ N 0 B 2 n 2 , the ((0 , 1) , 0) ← ((0 , 1) , 1) ← ((1 , 1) , 1) simplex to h m 1 01 x 0 01 ∞ 0 1 i h m 1 01 x 1 01 ∞ + ξ 0 ∞ 0 1 i h m 1 0 ∞ ( m 1 ) − 0 ∞ + ξ 0 ∞ 0 1 i  1 ξ 1 ∞ 0 1  h m 1 1 ∞ ( m 1 ) − 1 ∞ + ξ 1 ∞ 0 1 i h m 1 1 ∞ ( m 1 ) − 1 ∞ 0 1 i ∈ N 0 B 2 n 2 , (here the identit y from L emma 4.7 is used) and the (1 , 1)-simplex ( (0 , 0) , 0) ((0 , 1) , 0) o o ((0 , 0) , 1) O O ((0 , 1) , 1) o o O O to h m 0 01 x 0 01 ∞ 0 1 i h m 0 0 ∞ ( m 0 ) − 0 ∞ 0 1 i h m 0 1 ∞ ( m 0 ) − 1 ∞ 0 1 i h m 1 01 x 0 01 ∞ 0 1 i h m 1 0 ∞ ( m 1 ) − 0 ∞ + ξ 0 ∞ 0 1 i h m 1 1 ∞ ( m 1 ) − 1 ∞ + ξ 1 ∞ 0 1 i  α 01 id id i d  h α 0 ∞ α − 0 ∞ id id i h α 1 ∞ α − 1 ∞ id id i o o ∈ N 1 B 2 n 1 . Here we ha ve e mployed the fo r mula x 1 01 ∞ + ξ 0 ∞ = m 1 01 · ξ 1 ∞ + x 0 01 ∞ of Lemma 4 .7. 4.6. General v ersion of jnc. Consider (1) m = ( m 0 m 1 α 1 o o . . . α 2 o o m p ) α p o o ∈ N p B n q . Then ( α u ) i ∞ is given by the tuple (( α u ) ± i ∞ : ( m u ) ± i ∞ + ( ξ u ) 0 i ∞ → ( m u − 1 ) ± i ∞ , { ( ψ u ) r i ∞ : ( ξ u ) r i ∞ → ( ξ u ) r − 1 i ∞ } ) 12 for u = 1 , . . . , p , r = 1 , . . . , ℓ , but we simplify notation by se tting ξ u i ∞ = ( ξ u ) 0 i ∞ , x u ij ∞ = ( x u ) 0 ij ∞ , and ignoring the ψ ’s. Then jnc se nds a n m as in (1) to the simplex jnc( m ) ∈ z ∗ ( N B n ) pq with v alue at the (( a, b ) , c )-vertex in z ∗ (∆[ p ] × ∆[ q ]) giv en by  ( m b ) c ∞ ( m b ) − c ∞ + ξ b c ∞ + · · · + ξ a +1 c ∞ 0 1  ∈ GL 2 n ( T ℓ R ) with the conv ention that the ξ ’s only o ccur if a + 1 6 b . Higher simplices a re given by the structural isomorphisms in m . Note that the elemen ts in the o ff-diagonal blo cks a re actually all in R . More precisely , a tr iple ( φ, b, ψ ) where φ : [ r ] → [ p ] and ψ : [ r ] → [ q ] a re in ∆ and φ ( r ) 6 b 6 p , determines a (0 , r )-simplex in z ∗ (∆[ p ] × ∆[ q ]), b e c ause z ∗ (∆[ p ] × ∆[ q ]) (0 ,r ) = ∆([ r + 1] , [ p ]) × ∆([ r ] , [ q ]) and φ tog ether with b determine an element in the first factor. W e see that jnc( m )( φ, b, ψ ) ∈ N 0 B 2 n r is the element whose (0 6 i < j 6 r )- and (0 6 i 6 r < j = ∞ )-entries are " ( m b ) ψ ( i ) ψ ( j ) x φ ( j ) ψ ( i ) ψ ( j ) ∞ + ξ φ ( j ) ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 0 1 # and " ( m b ) ψ ( i ) ∞ ( m b ) − ψ ( i ) ∞ + ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 0 1 # , resp ectively . (As b efor e, the ξ ’s o nly o ccur when φ ( i ) + 1 6 φ ( j ) and φ ( i ) + 1 6 b , resp ectively .) Moving in the (nerve =) b -direction is easy because it amounts to connecting t wo v a lue s jnc( m )( φ, b, ψ ) and jnc( m )( φ, b ′ , ψ ) by mo rphisms. Since this is determined b y the one - skeleton, it is enough to describe the case b ′ = b − 1 < b . On the (0 6 i < j 6 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 upp er left hand corner , and otherwise the identit y), and on the (0 6 i 6 r < j = ∞ )-entries it is given by ( ξ b ψ ( i ) ∞ , ( α b ) ψ ( i ) ∞ ) : ( m b ) ψ ( i ) ∞ → ( m b − 1 ) ψ ( i ) ∞ and ( α b ) − ψ ( i ) ∞ : ( m b ) − ψ ( i ) ∞ + ξ b ψ ( i ) ∞ → ( m b − 1 ) − ψ ( i ) ∞ (in the upper r ow, a nd otherwise the iden tity). Checking that this is well defined and simplicial a mounts to the sa me kind of chec king as we ha ve already enco untered, using the same identities. One sho uld notice that at no time during the verifica- tions is the symmetry of addition used. It is used, how ever, for the iso morphism that renders matrix m ultiplication asso ciative up to isomorphism. The s implicia l homotopy from inc to jnc is gotten b y mult iplications (in the bar direction) by matrices of the form " 1 x φ ( j ) ψ ( i ) ψ ( j ) ∞ + ξ φ ( j ) ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 0 1 # and " 1 ( m b ) − ψ ( i ) ∞ + ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 0 1 # . 4.7. The hom otopic maps knc and lnc. Consider the following v ar iant knc of the ma p jnc: using the sa me no ta tion a s for jnc, when ev aluated on ( φ, b, ψ ) where φ : [ r ] → [ p ] and ψ : [ r ] → [ q ] are in ∆ and φ ( r ) 6 b 6 p , knc( m )( φ, b, ψ ) ∈ N 0 B 2 n r is the elemen t whose (0 6 i < j 6 r )- and (0 6 i 6 r < j = ∞ )- ent ries a re " ( m b ) ψ ( i ) ψ ( j ) x φ ( j ) ψ ( i ) ψ ( j ) ∞ + ξ φ ( j ) ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 0 1 # and " ( m b ) ψ ( i ) ∞ + ( ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ , ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ ) ( m b ) − ψ ( i ) ∞ + ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ (1 , 1) 1 # , resp ectively . The entries for j = ∞ can b e written co ncisely as  ( m b ) ψ ( i ) ∞ + (Ξ , Ξ) ( m b ) − ψ ( i ) ∞ + Ξ (1 , 1) 1  where Ξ = ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ . The entries for finite j ar e the sa me 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 b y the identit y), g iving a homo to p y . 13 Finally , let lnc : N B n → z ∗ N B 2 n be induced by the constant ma p sending a n y matrix to  1 0 (0 , 1) 1  . Matrix multiplication yields " ( m b ) + ψ ( i ) ∞ + ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ ( m b ) − ψ ( i ) ∞ + ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ 1 1 # ·  1 0 (0 , 1) 1  = " ( m b ) ψ ( i ) ∞ + ( ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ , ξ b ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ ) ( m b ) − ψ ( i ) ∞ + ξ c ψ ( i ) ∞ + · · · + ξ φ ( i )+1 ψ ( i ) ∞ (1 , 1) 1 # . With the sa me abbreviatio n as ab ov e, this reads  ( m b ) + ψ ( i ) ∞ + Ξ ( m b ) − ψ ( i ) ∞ + Ξ 1 1  ·  1 0 (0 , 1) 1  =  ( m b ) ψ ( i ) ∞ + (Ξ , Ξ) ( m b ) − ψ ( i ) ∞ + Ξ (1 , 1) 1  . W e obtain a homotopy from lnc to k nc. Hence inc is connected by a c hain of homotopies to a constant map. Since η ∗ : id → z ∗ is a monomorphism in the homotopy categor y , this means that the stabilization map in : B n → B 2 n is homotopically trivial, and so B ( ∗ , GL ( R ) , GL ( T ℓ R )) = colim n B n is con tractible for each ℓ > 0 . Hence B ( ∗ , GL ( R ) , GL ( T R )) is also contractible. This concludes the pro of of Theor em 1.1. References [A] Christian Ausoni, On the algebr aic K - the ory of the c omplex K -t he ory sp e ctrum , In ven t. Math. 180 (2010) 611– 668. [AR] Christian Ausoni and John Rognes, Algebr aic K -the ory of top olo gica l K -the ory , Acta Math. 188 (2002) 1–39. [BDR] Nils A. Baas, Bj ør n Ian Dundas and John Rognes, Two-ve ctor bund les and forms of el liptic co homolo gy , T opo- logy , geometry and quan tum field theory . Proceedings of the 2002 Oxford symp osium in honour of the 60th birthda y of Graeme Sega l, Oxford, UK, June 24-29, 2002. Cam br i dge Univ ersity Pr ess, London Mathematical Society Lecture Note Series 308, (2004) 18–45. [BDRR1] Nils A. Baas, Bj ør n Ian Dundas, Birgit Rich ter and John Rognes, Ring c ompletion of rig c ate gories , to appear in Journal f ¨ ur die reine und angewandt e Mathemat ik (Crelle’s Journal). [BDRR] N ils A. Baas, Bjørn Ian D undas, Bi rgit Rich ter and John Rognes, Two-ve ct or bund les define a form of el liptic c ohomolo gy , Preprint, [BHM] Marcel B¨ okstedt, W u Ch ung Hsiang and Ib M adsen, The cy clotomic tr ac e and algebr aic K - the ory of sp ac es In ve nt . Math. 11 1 (1993) 865–940. [EM] An thon y D. Elmendorf and Michael A. Mandell, R ings, mo dules, and algebr as in infinite lo op sp ac e the ory , Adv. Math. 205 , (2006) 163–228. [F] Daniel S. F reed, Hig her algebr aic structur es and quantization . Comm. M ath. Phys. 1 59 (1994) 343–398. [G] Daniel R. Gra yson, Higher algebr aic K -the ory: II , Algebr. K -Theory , Pr oc. Conf. Ev anston 1976, Lect. Notes Math. 551 (1976) 217–240. [J] John F rederick Jardine, Sup er co her enc e , J. Pur e Appl. Algebra 7 5 , No. 2 (1991) 103–194. [JS] Andr ´ e Jo ya l and Ross Street, Br aide d tensor c ate gories , Adv. M ath. 102 (1993) 20–78. [K] Mikhail M. Kaprano v, Analo gies b e twe en t he L anglands c orr esp ondenc e and top olo gic al quantum field the ory . F unct ional analysis on the eve of the 21st cen tury , V ol. 1 (New Br unswick , NJ, 1993), 119–151, Progr. Math., 131 , Birkh¨ auser Boston, Boston, MA (1995). [KV] Mikhail M. Kapranov and Vladimi r A. V oevodsky , 2 -c ate g ories and Zamolo dchikov t e tr ahe dr a e quations . Alge- braic groups and their generalizations: quant um and infinite-dimensional methods (Universit y Park, P A, 1991), Pro c. Sympos. Pur e Math., 56 , Part 2, Amer . M ath. So c., Providence, RI (1994) 177–259. [L] R uth J. Lawrence, An intr o duction to top olo gic al field the ory . The int erface of knots and ph ysics (San F rancisco, CA, 1995), Pro c. Symp os. A ppl. M ath., 5 1 , Amer. M ath. So c., Pr ov idence, RI (1996) 89–128. [M1] J. P eter May , Classifying sp ac es and fibr ations , Mem. Amer. Math. So c. 1 (1975), 1, no. 155. [M2] J. Pet er May , E ∞ ring sp ac es and E ∞ ring sp e ctr a , With contributions by F rank Quinn, Ni gel Ray , and Jørgen T orneha ve, Lecture Notes in Mathematics 577 , Springer (1977). [S1] Graeme Segal, Equivariant K -the ory , Inst. Hautes ´ Etudes Sci. Publ. Math. 34 (1968) 129–151. [S2] Graeme Segal, Cate g ories and c ohomolo gy the ories , T op ology 1 3 (1974) 293–312. [Th1] Robert W. Thomason, Homotopy c olimits in the c ate gory of smal l cate gories , Math. Pro c. Cambridge Philos. Soc. 8 5 (1979) 91–109 . [Th2] Robert W. Thomason, Be war e the phony multiplic ation on Quil len ’s A − 1 A , Pr oc. A m . Math. So c. 80 (1980) 569–573. [W] F riedhelm W aldhausen, Algebr aic K - the ory of sp ac es , Algebraic and geometric topology , Pro c. Conf., New Brunswick/USA 1983, Lect. Notes Math. 112 6 (1985) 318–419. Dep ar tm ent of M a thema tical S ciences, NTNU, 7491 Trondheim, Nor w a y E-mail addr ess : baas@mat h.ntnu.no Dep ar tm ent of M a thema tics, University of Bergen, 5 008 Bergen, Nor w a y E-mail addr ess : dundas@m ath.uib.no 14 F achbereich Ma them a tik der Un iversit ¨ at Hamburg, 20 146 Ham burg, Germ any E-mail addr ess : richter@ math.uni-h amburg.de Dep ar tm ent of M a thema tics, University of Oslo, 03 16 Oslo, Nor w a y E-mail addr ess : rognes@m ath.uio.no 15