Ring completion of rig categories

RING COMPLETION OF RIG CA TEGORIES NILS A. BAAS, BJØRN IAN DUNDAS, BIR GIT RICHTER AND JOHN ROGNES Abstract. W e offer a solution to the long-standing problem of group completing wi thin the con te xt of r ig categories (also kno wn as bimonoidal categories). Giv en a ri g category R we construct a natural additiv e group completion ¯ R that retains t he multiplicativ e structu re, hence has b ecome a ring c ategory . If w e start with a commut ativ e ri g cate gory R (also known as a symmetric bimonoidal category) the additiv e group completion ¯ R will b e a commut ativ e ring category . In an accompanying pap er [5] we sho w how to use this construction to prov e the conjecture fr om [3] that the algebraic K -theory of the connect ive top ological K -theory ring sp ectrum k u is equiv alen t to the algebraic K -theory of the rig category V of complex vec tor spaces. 1. Introduction and main resul t Multiplicative str uc tur e in algebraic K -theory is a delic a te matter. In 1980 Thomason [17] demon- strated that, after a dditiv e group completion, the most obvious approa ch es to multiplicativ e pa irings cease to make sense. F or instance, let us write ( −M ) M for the Grayson–Quillen [8] mo del fo r the algebraic K -theory o f a symmetric monoidal ca tegory M , wr itten additively . An o b ject in ( −M ) M is a pair ( a, b ) of o b jects of M , thought o f a s represent ing the difference “ a − b ”. The na ¨ ıve g uess for how to multiply elements is then dictated by the r ule that ( a − b )( c − d ) = ( ac + bd ) − ( ad + bc ). This, how ev er, do es not lea d to a dec e nt multiplicativ e structure: the resulting product is in most situatio ns not functor ia l. Several ways around this problem hav e bee n develop ed, but they all in v olv e first passing to sp ectra or infinite lo o p spaces by one of the equiv alent g roup completion machines, for instance the functor Spt from sy mmetr ic monoida l categor ie s to sp ectra defined in [18, App endix]. The or iginal problem has remained unanswered: ca n one a dditiv ely group complete and simultaneously keep the multiplicativ e structure, within the co n text of s ymmetric monoidal categorie s? W e ans w er this question affirmatively . Our motiv ation ca me from an outline o f pr o of in [3] of the conjecture that 2-vector bundles give r is e to a geometric cohomolog y theory of the same sort as elliptic cohomolog y , or mo r e precisely , to the algebraic K -theory of connec tiv e top olog ical K -theor y , which by work of Auso ni and the fo ur th author [1], [2] is a s pec tr um of tele s copic co mplexit y 2. The solution o f the ring completion problem g iven here enters as a step in our pr o of in [5] o f that conjecture. F or this application the alternatives provided in spe c tra were insufficient. Before stating our main result, let us fix some termino logy . Definition 1. 1. L et |C | denote the cla ssifying space of a small catego ry C , that is, the g eometric re- alization of its nerve N C . A functor F : C → D will b e called an u n stable e quivalenc e if it induces a homotopy equiv ale nc e of clas sifying spaces | F | : |C | → |D| , and will usually b e denoted C ∼ − → D . A lax s y mmetric monoida l functor F : M → N of symmetric monoidal categories, with or without zeros, is a stable e quivalenc e if it induces a stable eq uiv alence of sp ectra Spt F : Spt M → Spt N . An y lax symmetric monoidal functor whose underlying functor is an unsta ble equiv alenc e is a stable equiv alence . Unstable equiv alences are o ften called homotopy equiv alences, or weak equiv alences. W e use “un- stable” to emphasize the co n trast with stable equiv ale nces. These definitions rea dily extend to simplicial categorie s and functors b etw een them. By a rig (resp. comm utative rig) w e mean a r ing (resp. co mm uta tive ring) in the algebraic sense , except that nega tiv e elements are not assumed to exist. By a rig c ate gory (resp. c ommutative rig c ate gory ), Date : Octob er 23, 2018. 2000 Mathematics Subje ct Classific ation. Primary 19D23, 55R65; Secondary 19L41, 18D10. Key wor ds and phr ases. Algebraic K - theory , bimonoidal catego ry , bip ermuta tiv e category . The first autho r would l i k e to th ank the Institute for Adv anced Study , Princeton, for its hospitalit y and supp ort during his sta y in the spring of 2007. P art of the work w as done while the second author was on sabbatical at Stanford Unive rsity , whose hospitalit y and stimulating environmen t i s gratefully ackno wledged. The third author thanks the SFB 676 f or s upp ort and the top ology group in Sheffield for stimulating di scussions. 1 also k nown as a bimonoida l ca tegory (resp. symmetric bimono idal catego ry), we mea n a categ ory R with t wo binary op erations ⊕ and ⊗ , satisfying the axioms of a rig (resp. commutativ e rig) up to coherent natural isomo rphisms. By a (simplicial) ring c ate gory we mean a (s implicia l) rig categ o ry ¯ R such that π 0 | ¯ R| is a ring in the usual sense, with additiv e in verses. By a bip ermut ative c ate gory (resp. a strictly bimonoida l c ate gory ) we mean a commutativ e rig categor y (res p. a rig categor y) where as many of the coherence isomo r phisms as one c a n reaso nably demand are ident ities. See Definitions 2.1 and 2.4 b elow for precis e lists of axioms. Theorem 1.2. L et ( R , ⊕ , 0 R , ⊗ , 1 R ) b e a smal l simplicial rig c ate gory. Ther e ar e natur al morphisms R ∼ ← − Z R − → ¯ R of simplicial ri g c ate gories, such t hat (1) ¯ R is a simpl icial ring c ate gory, (2) R ∼ ← − Z R is an u nstable e quivalenc e, and (3) Z R − → ¯ R is a stable e qu ivalenc e. (4) If furthermor e (a) R is a gr oup oid, and (b) for every obje ct X in R the t r anslation functor X ⊕ ( − ) is faithful, then ther e is a natur al chain of unstable e quivalenc es of Z R -mo dules c onn e cting ¯ R to the Gr ayson–Quil len mo del ( − R ) R for the additive gr oup c ompletion of R . Addendum 1.3. L et R b e a smal l simplic ial c ommut ative rig c ate gory. The r e ar e natur al morphisms R ∼ ← − Z R − → ¯ R of simplicial c ommut ative rig c ate gories, such t hat al l four statements of the the or em ab ove hold . In particular , the induced ma ps Spt R ← Spt Z R → Spt ¯ R are stable equiv alences of r ing s pectr a, but the point is that ¯ R is r ing co mplete, before pas sing to sp ectra . Here are some ex amples of rig ca tegories that can be ring completed b y this metho d. • If R is a rig, then the discr ete categor y R with the elements of R as o b jects, and only iden tity morphisms, is a small rig catego r y . When R is comm utative, s o is R . T he sp ectrum Spt R is the Eilenberg– Mac Lane spe c trum of the a lgebraic ring co mpletio n of R . • There is a small commutativ e rig categor y E of finite s ets, with ob jects the finite sets n = { 1 , . . . , n } for n > 0 . In particular 0 is the empty set. There a r e no other mor phisms in E than the automor phisms, a nd the a utomorphism group of n is the s ymmetric g roup Σ n . Disjoint union and cartesian pr o duct o f se ts induce the op e rations ⊕ and ⊗ , and Spt E is equiv alent to the spher e sp ectrum. • F or ea ch commut ative ring A there is a small co mm utative rig categor y F ( A ) of finitely gener ated free A -mo dules. The o b jects o f F ( A ) are the free A -modules A n = L n i =1 A for n > 0. There are no other mo r phisms in F ( A ) than the automorphisms, and the a utomorphism group of A n is the genera l linear group GL n ( A ). Direct sum and tenso r pro duct of A -mo dules induce the op erations ⊕ a nd ⊗ , and Spt F ( A ) is the (free) algebra ic K -theory sp ectrum of the ring A . • Let V be the top ologic a l commutativ e rig ca tegory o f c o mplex (Hermitian) v ec to r s pa ces. It has one ob ject C n for eac h n > 0, with automorphism s pace eq ual to the unitar y group U ( n ). Ther e are no other morphisms. The sp ectrum Spt V is a mo del for the connective top olo gical K -theory sp ectrum ku . The case relev a n t to [3] and [5] is the 2-ca teg ory of 2-vector spaces of Kapra nov and V o evodsky [9], viewed as finitely generated fr e e V - mo dules. W e can functoria lly conv ert V to a simplicial commutativ e rig category b y replacing eac h mor phism space with its singular simplicial set. 1.1. Outline of proof. The proble m should b e appr oached with some trepidation, since the reasons for the failure of the obvious attempts at a solution to this long-s tanding pr oblem in alge br aic K -theory are fairly well hidden. The s tandard appro aches to additive gro up completion yie ld mo dels that are symmetric monoidal categorie s with resp ect to a n a dditiv e structure, but which have no meaningful m ultiplicative structure [17]. The failur e comes ab out essentially b ecause commutativit y fo r a dditio n only ho lds up to iso morphism. W e therefore need to mak e a mo del that provides eno ugh ro o m to circumv ent this difficult y . Our solution comes in the form of a graded construction, G R , related to iterations of the Grayson– Quillen mo del. It is a J -shap ed diag ram of symmetric monoidal categories , where the indexing ca tegory J = I ∫ Q is a certain p ermutativ e ca tegory ov er the catego ry I of finite sets n = { 1 , . . . , n } a nd injective 2 functions. Its definition can be motiv ated in a few steps. First, we use Thomaso n’s homoto p y colimit [18] of the diagr am 0 ← − R ∆ − → R × R in symmetric monoidal categor ies a s a model for the additive gro up completion of R . An ob ject ( a, b ) in the right hand categor y R × R r epresents the difference a − b , while an ob ject a in the middle categ ory R represents the r elation a − a = 0 , since a maps to ( a, a ) on the r ight hand side, and to zero in the left hand ca tegory . Group c ompletion is a homotopy idempo ten t pr o cess, meaning that we may rep eat it any p ositive nu m be r of times and a lw ays obtain unstably equiv alent results. F or each n > 0 we realize the n -fo ld iterated group completion of R a s the ho motopy colimit of a Q n - shap ed dia gram in symmetric monoidal categorie s, where Q 1 is the thr ee-ob ject catego ry indexing the dia gram displayed above, a nd in general Q n is isomo rphic to the pro duct of n copies of Q 1 . O ne distinguished entry in the Q n -shap ed dia gram is the pr o duct of 2 n copies of R . Its o b jects are given b y 2 n ob jects of R , which w e rega r d as b eing lo cated at the co rners o f a n n -dimensional cub e. These r epresent an alternating sum of terms in ¯ R , with s igns deter mined by the position in the n -cub e. The other en tries in the Q n -shap ed diag r am ar e diagonally embedded sub cub es of the n -cub e, or the zero ca tegory , and enco de cancella tion laws in the group completion. As regards the multiplicativ e structure, there is a natur a l pairing from the n - fold and the m -fold group completion to the ( n + m )-fold gr oup completion, with a ll p ossible ⊗ -pro ducts of the entries in the tw o original cub es being spr ead o ut o ver the big g er cube. F or instance, the pro duct of the tw o 1 -cub es ( a, b ) and ( c, d ) is a 2-cub e  ac ad bc bd  , wher e for brevity w e write ac for a ⊗ c , a nd so on. Rather than trying to turn an y single n - fold group completion in to a ring category , w e instead pass to the homotop y colimit ov er of all of them. T o allow the homotopy colimit to retain the multiplicativ e structure, we proceed as in [6] and index the itera ted gr o up completions b y the p ermutativ e category I , instead of the (non- symmetric) monoidal catego ry of finite sets a nd inclusions tha t index e s sequential colimits. F or each morphism m → n in I there is a preferred functor from Q m - shap ed to Q n -sha ped dia grams, involving extension by zero . F or instance, the unique morphism 0 → 1 takes a in R (for m = 0) to ( a, 0) in R × R in the display above (for n = 1 ). See section 3 for further examples a nd pictures in low dimensions. The res ulting homotopy co limit, modulo a technical p oint ab out zero ob jects, gives the desired r ing category ¯ R . As descr ib ed, this is the homo to p y colimit o f an I -s hap ed diagram, whose entry at n is the homoto p y colimit of a Q n - shap ed diagra m, for each n > 0. Suc h a double ho motopy colimit can b e condensed int o a single ho mo topy co limit ov er a la rger categ ory , namely the Gr othendieck construction J = I ∫ Q . In the end we therefore pr efer to present the ring categor y ¯ R as the one-step homotopy colimit o f a J -sha ped diagr am G R . The grade d multiplication G R ( x ) × G R ( y ) → G R ( x + y ) for x, y in J is defined as a bove, by multiplying tw o cub es together to get a bigger cub e, and makes G R a J -gr ade d rig c ate gory . The difficult y one usually encoun ters do es not appear , essen tially because we hav e spread the pro duct terms out ov er the vertices of the cub es, and no t attempted to add tog ether the “p ositive” a nd “neg ative” entries in some order or another. F rom a homo topy theo retic p oint o f view, the crucia l information lies in the fact that for each n > 0, the homotopy colimit of the sp ectra ass o ciated to the Q n - shap ed part o f the G R -diagram is stably equiv alent to the sp ectrum a sso ciated with R . F or ins ta nce, the homotopy colimit of the diagra m ∗ = Spt 0 ← − Spt R ∆ − → Spt( R × R ) (for n = 1) is the “ mapping co ne of the diag onal”, hence is ag ain a model for the s p ectr um asso cia ted with R . F rom a categor ic a l p oint of view, the p ossibility to interc ha nge the factors in R × R gives that the pass a ge to spectr a is unnecessa ry , since this flip induces the des ired “nega tiv e path comp onents”, without having to sta bilize. W e use Tho mason’s homotopy co limit in symmetric monoidal categ ories to transfo r m the J -gr aded rig category G R in to the rig ca tegory ¯ R , see Pr o po sition 3.2 and Lemma 5.2. The technical p oint alluded to ab ov e is that zero o b jects ar e tro ublesome (few symmetric monoidal categ o ries are “well p ointed”), and m ust b e handled with care. This gives rise to the intermediate simplicial rig c ategory Z R that app ear s in Theor em 1.2. 1.2. Plan. The structure of the pa pe r is as follo ws . After re pla cing the star ting commutativ e rig (resp. rig) c ategory R b y an equiv alent bip ermutativ e (r esp. strictly bimonoidal) categ ory , we discuss graded versions o f biper m utative and s trictly bimonoidal categorie s and their morphisms in section 2. 3 In section 3 we introduce the construction G R mentioned ab ov e, and show that it is a J -gr aded bip er- m utative (r e sp. stric tly bimonoidal) ca teg ory . Thomason’s ho motopy colimit of sy mmetr ic mo noidal ca tegories is defined in a non-unital (or zer o- less) setting. W e extend this to the unital setting by constructing a de r ived version of it in section 4, and in section 5 we s how that the homotopy colimit of a graded bip ermutativ e (resp. g raded strictly bi- monoidal) categ ory is almost a biper mu tative (resp. strictly bimonoida l) category —it o nly la cks a z ero. Section 6 des crib es how the results obtained so far com bine to lead to an a dditive gr oup co mpletion within the framework of (symmetric) bimonoida l categor ies. This ring completio n constr uction is given in Theor em 6.5. Most o f this pap er appea red e a rlier a s par t of a preprint [4] with the title “Two-v ecto r bundles define a for m of elliptic cohomology”. So me readers though t that title was hiding the result o n rig categories explained in the current pap er. W e therefore now o ffer the ring completion result separately , and a sk those readers interested in our main application to also turn to [5]. One should note that there was a mathematical error in the ear lier preprin t: the map T in the purp orted pro of of Lemma 3.7 (2 ) is not well defined, and so the v er sion o f the iterated Grayson–Quillen mo del used there might not hav e the right homotopy type. A piece o f no tation: if C is any small categ o ry , then the expres sion X ∈ C is shor t for “ X is an ob ject of C ” a nd likewise for mo rphisms a nd diagr ams. Displayed diagrams commute unless the contrary is stated ex plicitly . 2. Graded bipermut a tive ca tegories 2.1. P ermutativ e categori e s. A monoida l catego ry (r esp. symmetric monoidal category) is a cate- gory M with a bina r y op eration ⊕ satisfying the axioms o f a monoid (r esp. commutativ e monoid), i.e. , a group (resp. abelia n g roup) without negatives, up to coher en t natura l isomorphis ms . A p ermutativ e category is a sy mmetr ic monoida l category where the as s o ciativity and the left and right unitalit y iso- morphisms (but us ually not the commutativit y isomorphism), a re identities. F or the explicit definition of a p ermutativ e categor y see for instance [7, 3.1 ] or [12, § 4 ]; compare also [11, XI.1]. Since our p ermutativ e categorie s ar e t y pic a lly going to b e the under ly ing additive sy mmetr ic monoidal catego ries of categor ie s with some further multiplicative str ucture, we call the neutral element “zer o”, or simply 0. W e consider tw o kinds of functor s b etw een p ermutativ e categories ( M , ⊕ , 0 M , τ M ) and ( N , ⊕ , 0 N , τ N ), namely la x and strict symmetr ic monoidal functor s. A lax symmetric monoida l functor is a functor F in the sense of [11, XI.2], i.e. , there ar e morphisms f ( a, b ) : F ( a ) ⊕ F ( b ) → F ( a ⊕ b ) for all o b jects a, b ∈ M , which a re natura l in a and b , there is a mor phism n : 0 N → F (0 M ) , and these structure maps fulfill the coher ence conditions that are s pelled out in [11, XI.2]; in particular F ( a ) ⊕ F ( b ) f ( a,b ) / / τ N ( F ( a ) ,F ( b ))   F ( a ⊕ b ) F ( τ M ( a,b ))   F ( b ) ⊕ F ( a ) f ( b,a ) / / F ( b ⊕ a ) commutes for all a, b ∈ M . Let Perm b e the categor y o f small p ermutativ e categories and la x symmetric monoidal functors . W e mig h t say that f is a binatural tr ansformation, i.e. , a natura l transfor ma tion of functors M × M → N . Here “bi- ” r efers to the tw o v ariables, and sho uld not b e confused with the “bi-” in bipermutativ e, which refers to the tw o op era tions ⊕ a nd ⊗ . A strict symmetric monoidal fun ctor has furthermor e to satisfy that the morphis ms f ( a, b ) a nd n are ident ities, so that F ( a ⊕ b ) = F ( a ) ⊕ F ( b ) and F (0 M ) = 0 N [11, XI.2]. W e denote the category of small p ermutativ e catego ries and strict symmetric monoidal functors by Strict. A natura l transfor mation ν : F ⇒ G o f lax symmetric mo noidal functors, with comp onents ν a : F ( a ) → G ( a ), is requir ed to b e compatible with the structure mor phisms, so that ν a ⊕ b ◦ f ( a, b ) = g ( a, b ) ◦ ( ν a ⊕ ν b ) and ν 0 M ◦ n = n . Similarly for natura l transfor mations o f strict symmetric monoidal functors. 4 Since any sy mmetr ic monoidal ca tegory is natura lly eq uiv a len t to a p ermutative catego ry , we lo se no generality b y only considering permutativ e categories. W e mos tly consider the unital situation, except for the places in sections 4 and 5 where we explicitly state to b e in the ze r oless situation. 2.2. Bip ermutativ e categorie s. Roughly sp eaking, a rig ca tegory R co nsists of a symmetric monoidal category ( R , ⊕ , 0 R , τ R ) together with a functor R × R → R called “m ultiplication” and denoted by ⊗ or juxtap osition. Note that the multiplication is not a map of monoidal ca teg ories. The m ultiplication has a unit 1 R ∈ R , multiplying by 0 R is the zer o map, m ultiplying by 1 R is the identit y map, a nd the m ultiplication is (left and right) distributive o ver ⊕ up to appropria tely coherent na tural isomorphisms. If w e p ose the additional requir e men t that our rig ca tegories ar e commutative, then this co incides with what is often called a s ymmetric bimono ida l categ ory . Laplaza sp e lled o ut the coherence conditions in [10, pp. 31–3 5]. According to [13, VI, Pro p os ition 3.5 ] any c ommu tative rig categ ory is na turally equiv alent in the appropria te sens e to a bip ermutativ e categor y, and a similar rigidification r esult holds for r ig catego ries. Our main theorem (r esp. its addendum) is therefore equiv alent to the corres po nding statemen t where we assume that R is a strictly bimonoidal categor y (r e sp. a bip ermutativ e catego ry). W e will fo cus on the biper m utative case in the cour s e of this pa per , and indica te what has to b e adjusted in the str ictly bimonoidal ca s e. The reader can recov er the axioms for a biper m uta tiv e catego ry fro m Definition 2.1 below a s the sp ecial ca se of a “0-graded bip ermutativ e catego ry”, where 0 is the one- mo rphism catego r y . Other w is e one may for instance consult [7, 3 .6]. One word of warning: E lmendorf and Ma ndell’s left distributivity law is prec is ely what we (and [1 3, VI, Definition 3.3]) c all the rig h t distributivity law. Note that we demand strict right distributivity, and that this implies b oth cases of condition 3.3(b) in [7], in v iew of condition 3.3 (c). If R is a small rig ca tegory suc h that π 0 |R| is a ring (has additive in verses), then w e call R a ring c ate gory . Elmendorf and Ma ndell’s ring catego r ies are not r ing categories in our sense, but non- commutativ e rig ca tegories. In the course of this pap er we have to resolve rig ca tegories simplicially . If R is a small simplicial rig ca teg ory such that π 0 |R| is a ring, then we call R a s implicial ring c ate gory (even thoug h it is usually not a simplicial ob ject in the categor y o f ring categorie s ). If R is a strictly bimonoidal categ ory, a left R -mo dule is a per mu tative category M together with a multiplication R × M → M that is str ictly ass o ciativ e and coherently dis tr ibutiv e, as spelled out in [7, 9.1.1 ]. 2.3. J -graded bip ermutativ e categories and strictly bimonoidal categories. The following def- inition of a J -g raded bipermutative category is designed to axio matize the key prop er ties o f the functor G R de s crib ed in sectio n 3, and sim ultane o usly to gener alize the definition of a bip ermutativ e ca tegory (as the case J = 0). More g enerally , we could have int ro duced J -gra ded rig categor ies (resp. J -g raded com- m utative rig categ ories), generalizing rig catego ries (resp. commutativ e rig ca tegories), but this would hav e led to a n even more c umber some de finitio n. W e will there fo re a lw ays assume that the input R to our mac hiner y has b een transformed to an e q uiv alent bip ermutativ e or strictly bimonoidal catego ry befo re we start. Definition 2.1. Let ( J, + , 0 , χ ) b e a sma ll p ermutativ e catego ry . A J -gr ade d bip ermutative c ate gory is a functor C : J − → Strict from J to the categ ory Strict of small p er mutative categor ies and strict symmetric mo noidal functor s, together with data ( ⊗ , 1 , γ ⊗ ) as s pecified b elow, and sub ject to the follo wing conditions . The per mu tative structure of C ( x ) will b e denoted ( C ( x ) , ⊕ , 0 x , γ ⊕ ). (1) There a re comp osition functors ⊗ : C ( x ) × C ( y ) → C ( x + y ) for all x, y ∈ J , that are na tural in x and y . More explicitly , for each pa ir o f ob jects a ∈ C ( x ), b ∈ C ( y ) there is an ob ject a ⊗ b in C ( x + y ), and for each pair of morphisms f : a → a ′ , g : b → b ′ there is a morphism f ⊗ g : a ⊗ b → a ′ ⊗ b ′ , satisfying the usua l asso cia tivity and unitality 5 requirements. F o r each pair of morphisms k : x → z , ℓ : y → w in J the diagra m C ( x ) × C ( y ) ⊗ / / C ( k ) ×C ( ℓ )   C ( x + y ) C ( k + ℓ )   C ( z ) × C ( w ) ⊗ / / C ( z + w ) commutes. (2) There is a unit ob ject 1 ∈ C (0), such that 1 ⊗ ( − ) : C ( y ) → C ( y ) and ( − ) ⊗ 1 : C ( x ) → C ( x ) ar e the ident it y functors for all x, y ∈ J . Mo re precisely , the inclusion { 1 } × C ( y ) → C (0) × C ( y ) compo sed with ⊗ : C (0) × C ( y ) → C (0 + y ) = C ( y ) equals the pro jection isomorphism { 1 } × C ( y ) ∼ = C ( y ), and likewise for the functor from C ( x ) × { 1 } . (3) F or each pa ir of o b jects a ∈ C ( x ), b ∈ C ( y ) ther e is a twist isomorphism γ ⊗ = γ a,b ⊗ : a ⊗ b − → C ( χ y ,x )( b ⊗ a ) in C ( x + y ), where χ y ,x : y + x → x + y , such that a ⊗ b f ⊗ g   γ a,b ⊗ / / C ( χ y ,x )( b ⊗ a ) C ( χ y,x )( g ⊗ f )   a ′ ⊗ b ′ γ a ′ ,b ′ ⊗ / / C ( χ y ,x )( b ′ ⊗ a ′ ) commutes for any f , g as ab ov e, a nd C ( k + ℓ )( γ a,b ⊗ ) = γ C ( k )( a ) , C ( ℓ )( b ) ⊗ , for an y k , ℓ as above. W e r equire that C ( χ y ,x )( γ b,a ⊗ ) ◦ γ a,b ⊗ is equal to the identit y on a ⊗ b for all ob jects a and b : a ⊗ b id a ⊗ b / / γ a,b ⊗ & & M M M M M M M M M M M C ( χ y ,x ) C ( χ x,y )( a ⊗ b ) C ( χ y ,x )( b ⊗ a ) C ( χ y,x )( γ b,a ⊗ ) 5 5 k k k k k k k k k k k k k k and that γ a, 1 ⊗ and γ 1 ,a ⊗ are equa l to the ident it y o n a for all ob jects a . (4) The comp ositio n ⊗ is strictly a s so ciative, and the diagram a ⊗ b ⊗ c γ a ⊗ b,c ⊗ / / id ⊗ γ b,c   C ( χ z ,x + y )( c ⊗ a ⊗ b ) C ( χ z ,x + y )( γ c,a ⊗ id)   C (id + χ z ,y )( a ⊗ c ⊗ b ) C ( χ z ,x + y ) C ( χ x,z + id)( a ⊗ c ⊗ b ) commutes for all ob jects a , b a nd c (co mpare [11, p. 2 54, (7a)]). (5) Multiplication with the zero o b ject 0 x annihilates everything, for each x ∈ J . More pr ecisely , the inclusion { 0 x } × C ( y ) → C ( x ) × C ( y ) comp osed with ⊗ : C ( x ) × C ( y ) → C ( x + y ) is the c onstant functor to 0 x + y , and likewise for the co mpos ite functor from C ( x ) × { 0 y } . (6) Right distributivit y holds strictly , i.e. , ( C ( x ) × C ( x )) × C ( y ) ⊕× id / / ∆   C ( x ) × C ( y ) ⊗   ( C ( x ) × C ( y )) × ( C ( x ) × C ( y )) ⊗×⊗   C ( x + y ) × C ( x + y ) ⊕ / / C ( x + y ) commutes, wher e ⊕ is the mono idal structure and ∆ is the dia gonal on C ( y ) co mbined with the ident it y on C ( x ) × C ( x ), follow ed b y a twist. W e denote these instances of iden tities by d r , so d r = id : ⊕ ◦ ( ⊗ × ⊗ ) ◦ ∆ → ⊗ ◦ ( ⊕ × id). 6 (7) The left distributivity transformatio n, d ℓ , is given in terms of d r and γ ⊗ as d ℓ = γ ⊗ ◦ d r ◦ ( γ ⊗ ⊕ γ ⊗ ) . (Here we suppress the twist C ( χ ) fro m the notation.) More explicitly , for all x, y ∈ J and a ∈ C ( x ), b, b ′ ∈ C ( y ) the following diag ram defines d ℓ : a ⊗ b ⊕ a ⊗ b ′ d ℓ   γ a,b ⊗ ⊕ γ a,b ′ ⊗ / / C ( χ y ,x )( b ⊗ a ) ⊕ C ( χ y ,x )( b ′ ⊗ a ) a ⊗ ( b ⊕ b ′ ) C ( χ y ,x )( b ⊗ a ⊕ b ′ ⊗ a ) C ( χ y,x )( d r )=id   C ( χ y ,x ) C ( χ x,y )( a ⊗ ( b ⊕ b ′ )) C ( χ y ,x )(( b ⊕ b ′ ) ⊗ a ) C ( χ y,x )( γ b ⊕ b ′ ,a ⊗ ) o o (8) The diagr am ( a ⊗ b ) ⊕ ( a ⊗ b ′ ) d ℓ / / γ ⊕   a ⊗ ( b ⊕ b ′ ) id ⊗ γ ⊕   ( a ⊗ b ′ ) ⊕ ( a ⊗ b ) d ℓ / / a ⊗ ( b ′ ⊕ b ) commutes for all ob jects. The analogous diag ram for d r also co mm utes . Due to the definition of d ℓ in terms of γ ⊗ and the ident it y d r , it suffices to demand tha t γ ⊕ ◦ ( γ ⊗ ⊕ γ ⊗ ) = ( γ ⊗ ⊕ γ ⊗ ) ◦ γ ⊕ and ( γ ⊕ ⊗ id) ◦ γ ⊗ = γ ⊗ ◦ (id ⊗ γ ⊕ ). (9) The distributivity transfo rmations are ass o ciative, i.e. , the diagr am ( a ⊗ b ⊗ c ) ⊕ ( a ⊗ b ⊗ c ′ ) d ℓ   d ℓ * * T T T T T T T T T T T T T T T a ⊗ (( b ⊗ c ) ⊕ ( b ⊗ c ′ )) id ⊗ d ℓ / / a ⊗ b ⊗ ( c ⊕ c ′ ) commutes for all ob jects. (10) The following p e n tagon diag ram commutes ( a ⊗ ( b ⊕ b ′ )) ⊕ ( a ′ ⊗ ( b ⊕ b ′ )) d r   ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( a ⊗ b ) ⊕ ( a ⊗ b ′ ) ⊕ ( a ′ ⊗ b ) ⊕ ( a ′ ⊗ b ′ ) d ℓ ⊕ d ℓ 4 4 i i i i i i i i i i i i i i i i id ⊕ γ ⊕ ⊕ id   ( a ⊕ a ′ ) ⊗ ( b ⊕ b ′ ) ( a ⊗ b ) ⊕ ( a ′ ⊗ b ) ⊕ ( a ⊗ b ′ ) ⊕ ( a ′ ⊗ b ′ ) d r ⊕ d r * * U U U U U U U U U U U U U U U U (( a ⊕ a ′ ) ⊗ b ) ⊕ (( a ⊕ a ′ ) ⊗ b ′ ) d ℓ ? ?                    for all o b jects a, a ′ ∈ C ( x ) and b, b ′ ∈ C ( y ). Remark 2.2. In Definition 2.1, condition (1) says that we hav e a binatural transformatio n ⊗ : C × C ⇒ C ◦ (+) 7 of bifunctors J × J → Cat, where Cat de no tes the catego r y of small categ ories. Condition (3) demands that there is a mo dification [11, p. 2 78] C × C c Cat + 3 ⊗   γ ⊗ ⇛ ( C × C ) ◦ tw J ⊗   C ◦ (+) C ◦ (+) ◦ tw J C ( χ ) k s where c Cat is the symmetr ic structure on Cat (with resp ect to product) and tw J is the in terchange o f factors o n J × J . In the following w e will deno te a J - graded biper mut ative c a tegory C : J → Stric t b y C • if the catego ry J is clear fro m the context. F or the o ne-morphism categor y J = 0, a J - graded bip ermutativ e categ ory is the same as a bip e rmut ative category. Thus every J -grade d bip ermutativ e categ ory C • comes with a biper mu tative ca tegory C (0), a nd C • can b e v iewed as a functor J → C (0)-mo dules. Example 2.3. W e consider the small bip ermutativ e category of finite sets, whose ob jects a re the finite sets o f the form n = { 1 , . . . , n } for n > 0 , and whose morphisms m → n ar e all functions { 1 , . . . , m } → { 1 , . . . , n } . Disjoint union of sets gives rise to a p ermutativ e str ucture n ⊕ m := n ⊔ m and we identify n ⊔ m with n + m . F or functions f : n → n ′ and g : m → m ′ we define f ⊕ g as the map on the disjoint union f ⊔ g whic h we will denote by f + g . The additive twist c ⊕ is g iven by the shuffle maps χ ( n, m ) : n + m − → m + n with χ ( n, m )( i ) = ( m + i for i 6 n i − n for i > n. Multiplication o f sets is defined via n ⊗ m := nm . If we identif y the element ( i − 1) · m + j in nm with the pair ( i, j ) with i ∈ n and j ∈ m , then the function f ⊗ g is given by ( i, j ) 7→ ( f ( i ) , g ( j )) , and the multiplicative t w is t c ⊗ : n ⊗ m − → m ⊗ n sends ( i, j ) to ( j, i ). The empty set 0 is a strict zer o for the addition and the s ingleton s et 1 is a strict unit for the multiplication. Rig h t distributivity is the identit y and the left distributivity law is given by the res ulting p ermutation d ℓ = c ⊗ ◦ d r ◦ ( c ⊗ ⊕ c ⊗ ) . F or la ter reference we deno te this instanc e of d ℓ by ξ . Considering only the sub categ ory of bijections, ins tea d of a rbitrary functions, results in the bipermu- tative catego r y of finite sets E that we r e ferred to in the in tro duction. Later, we will ma ke use of the zeroless bip ermutativ e categor y of finite nonempty s e ts and s urjective functions. Definition 2.4. A J -gr ade d strictly bimonoid al c ate gory is a functor C : J → Stric t to the catego ry of per mut ative categories and strict symmetric mo noidal functors, satisfying the conditions of Definition 2 .1, except that w e do not require the existence of the natural isomorphism γ ⊗ , and the left distributivit y isomorphism d ℓ is not given in terms of d r . Axiom (7) of Definition 2.1 has to b e replaced b y the following condition: (7’) The diagr am a ⊗ b ⊗ c ⊕ a ⊗ b ′ ⊗ c d r / / d ℓ   ( a ⊗ b ⊕ a ⊗ b ′ ) ⊗ c d ℓ ⊗ id   a ⊗ ( b ⊗ c ⊕ b ′ ⊗ c ) id ⊗ d r / / a ⊗ ( b ⊕ b ′ ) ⊗ c commutes for all ob jects. 8 In the J -graded bip ermutativ e case condition (7’) follows fro m the other axioms. Definition 2.5. A lax morphism of bip ermut ative c ate gories , F : C → D , is a pair o f lax symmetric monoidal functors ( C , ⊕ , 0 C , c ⊕ ) → ( D , ⊕ , 0 D , c ⊕ ) and ( C , ⊗ , 1 C , c ⊗ ) → ( D , ⊗ , 1 D , c ⊗ ), with the same underlying functor F : C → D , that resp ect the le ft a nd right distributivit y laws. In other words, we hav e a binatural tr ansformation from ⊕ ◦ ( F × F ) to F ◦ ⊕ : η ⊕ = η ⊕ ( a, b ) : F ( a ) ⊕ F ( b ) → F ( a ⊕ b ) for a, b ∈ C , as well as a binatural transforma tio n from ⊗ ◦ ( F × F ) to F ◦ ⊗ : η ⊗ = η ⊗ ( a, b ) : F ( a ) ⊗ F ( b ) → F ( a ⊗ b ) for a, b ∈ C , plus morphisms 0 D → F (0 C ) and 1 D → F (1 C ). W e require that these commute with c ⊕ and c ⊗ , re s pec tively , and tha t the following diag ram (and the analo g ous one for d ℓ ) commutes F ( a ) ⊗ F ( b ) ⊕ F ( a ′ ) ⊗ F ( b ) η ⊗ ⊕ η ⊗   d r =id ( F ( a ) ⊕ F ( a ′ )) ⊗ F ( b ) η ⊕ ⊗ id / / F ( a ⊕ a ′ ) ⊗ F ( b ) η ⊗   F ( a ⊗ b ) ⊕ F ( a ′ ⊗ b ) η ⊕ / / F ( a ⊗ b ⊕ a ′ ⊗ b ) F ( d r )=id F (( a ⊕ a ′ ) ⊗ b ) for all o b jects a, a ′ , b ∈ C , i.e. , we hav e η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ ( η ⊕ ⊗ id) and F ( γ ⊗ ◦ ( γ ⊗ ⊕ γ ⊗ )) ◦ η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ (id ⊗ η ⊕ ) ◦ γ ⊗ ◦ ( γ ⊗ ⊕ γ ⊗ ) . F or a lax morphism of strictly bimonoida l c ate gories we demand that F is lax mo noidal with resp ect to ⊗ , lax symmetr ic monoidal with resp ect to ⊕ , and that F ( d ℓ ) ◦ η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ (id ⊗ η ⊕ ) ◦ d ℓ and F ( d r ) ◦ η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ ( η ⊕ ⊗ id) ◦ d r . Definition 2.6. A lax m orphism of J -gr ade d bip ermut ative c ate gories , F : C • → D • , consists o f a natural transformatio n F from C • to D • that is compatible with the bifunctors ⊕ , ⊗ and the units. Additively , we r e quire a tr ansformation η ⊕ from ⊕ ◦ ( F × F ) to F ◦ ⊕ : C ( x ) × C ( x ) ⊕ / / F x × F x   C ( x ) F x   D ( x ) × D ( x ) ⊕ / / η x ⊕ r r r r r r r r 4 < r r r r r r r r D ( x ) that commutes with γ ⊕ , is binatural with resp e c t to mo rphisms in C ( x ) × C ( x ), and is natural w ith resp ect to x . Multiplica tiv ely , we re q uire a tr a nsformation η ⊗ from ⊗ ◦ ( F × F ) to F ◦ ⊗ : C ( x ) × C ( y ) ⊗ / / F x × F y   C ( x + y ) F x + y   D ( x ) × D ( y ) ⊗ / / η x,y ⊗ p p p p p p p p 3 ; p p p p p p p p D ( x + y ) that c ommu tes with γ ⊗ , is binatural with resp ect to morphisms in C ( x ) × C ( y ), and is natural with resp ect to x and y . The functor F must resp ect the distributivity constra in ts in that it fulfills η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ ( η ⊕ ⊗ id) and F ( d ℓ ) ◦ η ⊕ ◦ ( η ⊗ ⊕ η ⊗ ) = η ⊗ ◦ (id ⊗ η ⊕ ) ◦ d ℓ . F or a lax morph ism of J -gr ade d strictly bimonoidal c ate gories ther e is no requir e men t o n ( F a nd) η ⊗ concerning the mu ltiplicative t wist γ ⊗ . 9 3. A cubical constr uction on (bi- )permut a tive ca tegories W e r emo del the Grayson–Q uillen cons truction [8] o f the group completion of a p ermutativ e category to suit o ur multiplicativ e needs. The na ¨ ıve pro duct ( ac ⊕ bd, ad ⊕ bc ) of tw o pairs ( a, b ) and ( c, d ) in their mo del will be replaced by the quadruple  ac ad bc bd  , where no order of a dding terms is c hosen. This av oids the “phonines s” of the multiplication [17], but r e q uires that we keep track o f n -cubical diag rams of ob jects, of v arying dimensions n > 0. W e star t by introducing the indexing catego ry I ∫ Q for all of these diagrams, and then describ e the I ∫ Q -sha ped diagra m G M asso ciated to a p ermutativ e category M . If we s ta rt with a bip e rmut ative category R , the result will b e an I ∫ Q - g raded bip ermutativ e categor y G R . 3.1. An indexing category. Let I b e the usual skeleton of the categor y of finite sets and injective functions, i.e. , its ob jects ar e the finite sets n = { 1 , . . . , n } for n > 0, and its morphisms are the injectiv e functions ϕ : m → n . W e define the sum o f tw o ob jects n and m to be n + m and use the twist ma ps χ ( n, m ) defined in Ex ample 2.3. Then ( I , + , 0 , χ ) is a per mu tative ca tegory . F or ea ch n > 0, let Q n b e the ca teg ory whose ob jects ar e subsets T of {± 1 , . . . , ± n } = {− n, . . . , − 1 , 1 , . . . , n } such that the abso lute v a lue function T → Z is injective. In other words, w e may have i ∈ T or − i ∈ T , but not b oth, for each 1 6 i 6 n . Mo rphisms in Q n ar e inclusions S ⊆ T of subsets. (The ob jects could equally well be des crib ed as pairs ( T , w ) where T ⊆ n and w is a function T → {± 1 } , and similarly for the morphisms.) Let P n ⊆ Q n b e the full sub categor y generated by the subsets of n = { 1 , . . . , n } , i.e. , the T with only p ositive elements. F or exa mple, the ca tegory Q 2 c an b e depicted as: {− 1 , 2 } { 2 } o o / / { 1 , 2 } {− 1 } O O   ∅ o o / / O O   { 1 } O O   {− 1 , − 2 } {− 2 } o o / / { 1 , − 2 } and P 2 is given by the upper r ight ha nd squa re. W e sha ll use P n and Q n to index n -dimensiona l cubical diagrams with 2 n and 3 n vertices, resp ectively . F or each morphism ϕ : m → n in I we define a functor Q ϕ : Q m → Q n as follows. First, let C ϕ = n \ ϕ ( m ) b e the complemen t of the image o f the injective function ϕ . Then extend ϕ to a n odd function {± 1 , . . . , ± m } → { ± 1 , . . . , ± n } , which we also call ϕ , and let ( Q ϕ )( S ) = ϕ ( S ) ⊔ C ϕ for each ob ject S ∈ Q m . F or exa mple, if ϕ : 1 → 2 is given by ϕ (1) = 2, then C ϕ = { 1 } a nd Q ϕ is the functor {− 1 } _   ∅ _   o o / / { 1 } _   { 1 , − 2 } { 1 } o o / / { 1 , 2 } embedding Q 1 into the rig h t hand column of Q 2 . Similar ly , the function ϕ : 1 → 2 with ϕ (1) = 1 embeds Q 1 into the upp er row of Q 2 . If S ⊆ T then clearly ( Q ϕ )( S ) ⊆ ( Q ϕ )( T ). If ψ : k → m is a second morphism in I , w e see tha t Q ϕ ◦ Q ψ = Q ( ϕψ ), and so n 7→ Q n defines a functor Q : I → Cat. Restricting to sets with only pos itiv e ent ries, we g et a subfunctor P ⊆ Q that may b e easie r to gras p: if ϕ : m → n ∈ I , then P ϕ : P m → P n is the functor sending S ⊆ m to ϕ ( S ) ⊔ C ϕ , wher e C ϕ = n \ ϕ ( m ) is the co mplemen t of the image of ϕ . Our main indexing category will b e the Grothendieck construction J = I ∫ Q . This is the category with ob jects pairs x = ( m , S ) with m ∈ I and S ∈ Q m , and with morphisms x = ( m , S ) → ( n , T ) = y consisting of pairs ( ϕ, ι ) with ϕ : m → n a morphism in I and ι : ( Q ϕ )( S ) ⊆ T a n inclusion. T o giv e a functor C from I ∫ Q to any catego r y is e q uiv alent to giv ing a functor C n from Q n for each n > 0, together with natura l transformations C ϕ : C m ⇒ C n ◦ Q ϕ for all ϕ : m → n in I , which must be compa tible with ident ities a nd comp osition in I . 10 Consider the functor + : Q n × Q m → Q ( n + m ) defined as follows. The injective functions in 1 : n → n + m and in 2 : m → n + m ar e given b y in 1 ( i ) = i and in 2 ( j ) = n + j . Extending to o dd functions we define T + S to be the dis jo in t union o f images in 1 ( T ) ⊔ in 2 ( S ) ⊆ {± 1 , . . . , ± ( n + m ) } . F or example, if T = {− 1 , 2 } ⊆ {± 1 , ± 2 , ± 3 } and S = { 1 , − 2 } ⊆ { ± 1 , ± 2 } , then T + S = { − 1 , 2 , 4 , − 5 } ⊆ {± 1 , . . . , ± 5 } . These functors, for v arying n, m > 0, combine to an addition functor o n I ∫ Q . F or each pair of ob jects ( n , T ) , ( m , S ) ∈ I ∫ Q we define ( n , T ) + ( m , S ) = ( n + m , T + S ), and likewise o n morphisms. Lemma 3.1. A ddition makes I ∫ Q and I ∫ P into p ermutative c ate gories. Pr o of. The zero ob ject is ( 0 , 0 ), and the isomorphis m ( χ ( n, m ) , id) : ( n + m , T + S ) → ( m + n , S + T ) provides the symmetric structure.  3.2. The cub e construc tion. Let M b e a p ermutativ e categor y (with zero). Define a functor M n : P n → Strict for eac h n > 0, b y sending a subset T ⊆ n to M n ( T ) = M P T , the permutative categor y of functions from the set P T of subsets of T to M , i.e. , the pro duct of one copy of M for ea ch subset o f T . If ι : S ⊆ T we get a s trict symmetric monoidal functor M n ( ι ) : M P S → M P T by sending the ob ject a = ( a U | U ⊆ S ) ∈ M P S to ( a V ∩ S | V ⊆ T ), and likewise with morphisms. Thes e a r e diago nal functors, since each a U gets rep eated once fo r each V with V ∩ S = U . F or n = 0 , 1 , 2 the diag rams M n hav e the following shapes : M , M → M × M and M × M / / M × 4 M O O / / M × M O O where the mo rphisms a re the appro priate dia gonals. In pa rticular, M n ( n ) is the pro duct of 2 n copies of M , viewed a s sprea d out ov e r the cor ne r s of an n -dimensional cub e. F or ϕ : m → n we define a natur al tr ansformation M ϕ : M m ⇒ M n ◦ P ϕ : fo r S ∈ P m we let M ϕ ( S ) be the comp osite M m ( S ) = M P S ∼ = M P ( ϕ ( S )) → M P ( ϕ ( S ) ⊔ C ϕ ) = M n (( P ϕ )( S )) where the isomor phism is just the reindexa tion induced by ϕ , and the functor M P ( ϕ ( S )) → M P ( ϕ ( S ) ⊔ C ϕ ) is the identit y on fa c tors indexed by subsets of ϕ ( S ) and zer o on the fa c to rs that are no t hit by ϕ . Explicitly , M ϕ ( S )( f ) V = ( f ϕ − 1 ( V ) if V ⊆ ϕ ( S ) 0 otherwise for any morphism f : a → b ∈ M P S and V ⊆ ϕ ( S ) ⊔ C ϕ . These a re extension by zero functor s, not diagonals . E ach f U gets rep eated exactly once, as M ϕ ( S )( f ) V for V = ϕ ( U ). F or instance, if ϕ : 1 → 2 is given by ϕ (1) = 2 , then M 1 ( ∅ ) = M → M × M = M 2 ( { 1 } ) and M 1 ( 1 ) = M × M → M × 4 ∼ = M 2 ( 2 ) are given by appro priate inclusions onto factor s in pr o ducts. F or bo th morphisms ϕ : 1 → 2 the asso ciated functors M → M × M are the inclusion onto the ∅ -factor, whereas the tw o functors M × M → M × 4 include onto either the ∅ and { 1 } factor s, or the ∅ and { 2 } factors, dep ending on ϕ . W e se e that for all S ⊆ T ⊆ m and ϕ : m → n , the diagra m M P S / / M ϕ ( S )   M P T M ϕ ( T )   M P ( ϕ ( S ) ⊔ C ϕ ) / / M P ( ϕ ( T ) ⊔ C ϕ ) commutes, s e nding a ∈ M P S bo th wa ys to W 7→ a ϕ − 1 ( W ) ∩ S if W ⊆ ϕ ( T ) and 0 other w is e. 11 If ψ : k → m ∈ I , then we hav e a n eq uality M ϕψ = M ϕ M ψ of natura l transfo rmations M k ⇒ M n ◦ P ( ϕψ ) in: P k P ψ   P ( ϕψ ) ' ' M k # # H H H H H H H H H P m P ϕ   M m / / Strict P n M n ; ; v v v v v v v v v (This diagram is not strictly commutativ e. The t wo right hand triangles only co mm ute up to the natural transformations M ψ and M ϕ , res pectively .) Both natural trans formations are r epresented by the functors M P S → M P ( ϕψ ( S ) ⊔ C ( ϕψ )) sending a to V 7→ a ( ϕψ ) − 1 V if V ⊆ ϕψ ( S ) and 0 otherwise. Thus M ca n b e viewed as a left lax tra nsformation from the functor P : I → Ca t to the constant functor at Strict. (W e recall the definition of a left la x transfor mation in subsectio n 4.1 b elow.) This left lax tra nsformation M : P ⇒ Strict extends to a left la x tra ns formation M : Q ⇒ Strict by declaring that M n ( T ) = 0 if T contains negative elemen ts. The first three diagr ams now loo k like: M , 0 ← M → M × M and 0 M × M o o / / M × 4 0 O O   M O O   / / o o M × M O O   0 0 o o / / 0 Another wa y of saying that w e have a left lax tra nsformation Q ⇒ Strict is to say tha t w e ha ve a functor I ∫ M : I ∫ Q → I ∫ Strict ∼ = I × Strict (o ver I ). Pro jecting to the second factor, I ∫ M gives rise to a functor G M : I ∫ Q → Strict . Explicitly , G M ( n , T ) = M n ( T ), whic h is M P T if T contains no negatives and 0 other wis e. If ϕ : m → n ∈ I a nd ι : ( Q ϕ )( S ) ⊆ T ∈ Q n , then G M ( ϕ, ι ) : M m ( S ) → M n ( T ) is the comp osite of G M ( ϕ, id) = M ϕ S : M m ( S ) → M n (( Q ϕ )( S )) and G M (id , ι ) = M n ( ι ) : M n (( Q ϕ )( S )) → M n ( T ). 3.3. Multipl icativ e s tructure. Since the diagr am G M : I ∫ Q → Strict is so simple, only consisting o f diagonals and inclusions o nto factors in pro ducts, algebraic structur e on M is ea sily transfer red to G M . Prop ositio n 3.2. If R is a strictly bimonoidal c ate gory, then G R is an I ∫ Q - gr ade d strictly bimonoida l c ate gory. If R is a bip ermutative c ate gory, then G R is an I ∫ Q -gr ade d bip ermutative c at e gory. Pr o of. W e m ust sp ecify comp osition functor s ⊗ : G R ( n , T ) × G R ( m , S ) → G R ( n + m , T + S ) for all ( n , T ) , ( m , S ) ∈ I ∫ Q . Let a ∈ G R ( n , T ) and b ∈ G R ( m , S ). If S and T only contain p ositive elements, then a ⊗ b ∈ G R ( n + m , T + S ) is defined by ( a ⊗ b ) V + U = a V ⊗ b U , where the ⊗ -pr o duct on the rig ht is formed in R . As V and U rang e ov er all the subsets o f T and S , resp ectively , V + U ranges o ver all the subsets o f T + S . If T or S contain negativ e elements, we set a ⊗ b = 0. Likewise for morphisms in G R ( n , T ) and G R ( m , S ). These comp ositio n functors are clearly natural in ( n , T ) and ( m , S ). The unit ob ject 1 o f G R ( 0 , 0 ) ∼ = R corresp onds to the unit o b ject 1 R of R . I n the bip ermutativ e case, the twist iso morphism γ ⊗ : a ⊗ b − → G R ( χ ( m, n ) , id)( b ⊗ a ) has comp onents ( a ⊗ b ) V + U = a V ⊗ b U γ R ⊗ − → b U ⊗ a V = ( b ⊗ a ) U + V for all V ⊆ T and U ⊆ S , wher e γ R ⊗ is the twist isomorphism in R . Since everything is defined p oint wise, the multiplicativ e structure o n R forces all the axio ms of an I ∫ Q -g raded strictly bimonoidal categor y (or I ∫ Q - g raded biper mutative ca tegory) to hold for G R .  12 4. Hocolim-lemma t a W e r ecall Thomason’s homotopy colimit constructio n in the case of a J - shap ed dia gram of zerole s s per - m utative categories , and then construct a derived version of this co ns truction for p er mutative c a tegories with zero . 4.1. The case without zeros. Let P erm nz be the category of p er m uta tiv e categor ies without zer o ob jects, and lax symmetric monoidal functors. Let Strict nz be the subca tegory with the sa me ob jects, but w ith strict symmetric monoidal functors a s mor phisms. There a re for getful functors V : Str ict nz → Perm nz and U : Perm nz → Ca t, with comp osite W = U V : Strict nz → Cat. F or any sma ll categor y J , let Cat J be the categor y o f functors J → Cat a nd left lax tr ansformations . Recall that for functors C , D : J → Cat, a left lax transformation F : C → D as s igns to each o b ject x ∈ J a functor F x : C ( x ) → D ( x ), and to eac h morphism k : x → y in J a natural tra nsformation ν k : D ( k ) ◦ F x ⇒ F y ◦ C ( k ) of functor s C ( x ) → D ( y ): C ( x ) C ( k ) / / F x   C ( y ) F y   D ( x ) D ( k ) / / ν k 7 ? x x x x x x x x x x x x x x x x D ( y ) These must b e compatible with co mpo sition in J , so that ν id = id and ν ℓk = ν ℓ C ( k ) ◦ D ( ℓ ) ν k for ℓ : y → z in J . If e ach ν k = id, we hav e a natur al transfor mation in the usual sense . Similarly , let Perm nz J be the category of functors J → P erm nz and left lax trans formations. In this case, the ca tegories C ( x ) , C ( y ) , D ( x ) , D ( y ) etc. ar e symmetr ic mono ida l without zero, the functors C ( k ) , D ( k ) , F x , F y etc. are lax symmetric monoidal, and ν k is a natural transforma tion of la x symmetric monoidal functors. Finally , let Strict nz J be the categ ory of functors J → Strict nz and left lax transfor- mations. In this case, all of the symmetric mo noidal functors are str ict. Let ∆ : Cat → Cat J be the constant J -shap ed dia gram functor. Given a functor C : J → Ca t, the Grothendieck construction J ∫ C is a model for the homotopy colimit in Cat [16]. W e recall tha t a n ob ject in J ∫ C is a pair ( x, X ) wher e x ∈ J and X ∈ C ( x ) are ob jects, while a morphism ( x, X ) → ( y , Y ) is a pair ( k , f ) where k : x → y ∈ J and f : C ( k )( X ) → Y ∈ C ( y ) are morphisms. This construction defines a fun ctor J ∫ ( − ) : Cat J → Cat, which is left adjoint to ∆ : Cat → Ca t J . Here it is, of course, impo rtant that w e are allowing left lax transformations as morphisms in Cat J , since otherwise the left adjoint would b e the categor ical colimit. Thomason’s homo topy colimit o f p ermutativ e ca tegories [18] is co ns tructed to have a simila r universal prop erty with resp ect to the comp osite ∆ V : Strict nz → Perm nz J , wher e V is as ab ov e and ∆ : P er m nz → Perm nz J is the c o nstant J -shap ed diagra m functor. W e briefly recall the explicit description. Definition 4.1. Let C : J → Perm nz be a functor. An ob ject in ho co lim J C is an expression n [( x 1 , X 1 ) , . . . , ( x n , X n )] where n > 1 is a na tur al nu mber, each x i is an ob ject of J , a nd each X i is an ob ject of C ( x i ). A morphism from n [( x 1 , X 1 ) , . . . , ( x n , X n )] to m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] consists of thre e pa rts: a surjective function ψ : n → m , mor phisms ℓ i : x i → y ψ ( i ) in J for each 1 6 i 6 n , and mor phisms  j : L i ∈ ψ − 1 ( j ) C ( ℓ i )( X i ) → Y j in C ( y j ) for each 1 6 j 6 m . W e will write ( ψ , ℓ i ,  j ) to signify this morphism. See [18, 3.22] for the definition of comp osition in the category ho c o lim J C . This category is p ermu- tative, without a z e ro, if one defines addition to b e giv e n b y concatenation [18, p. 163 2]. Each le ft lax tr ansformation F : C → D induces a strict s y mmetric monoidal functor hocolim J F : ho colim J C → ho colim J D , so this cons truction defines a functor ho colim J : Perm nz J → Strict nz . The universal prop er ty in [18, 3.21] says that ho colim J : Perm nz J → Strict nz is left adjoint to ∆ V : Strict nz → Perm nz J . Again, it is critical that we ar e allowing left lax transformations as mor- phisms in Perm nz J . Recall Definition 1 .1 o f unstable equiv alences in Cat and stable equiv alences in Perm nz and Strict nz . W e use the cor r esp onding po in t wise notio ns in diagram ca tegories like Cat J and Perm nz J , so a left lax transformatio n F : C → D b etw ee n functors C , D : J → Perm nz is a stable (re sp. unstable) equiv alence if every o ne of its comp onents F x : C ( x ) → D ( x ) is a s ta ble (resp. unstable) equiv alence, for x ∈ J . Lemma 4.2. L et F : C → D b e a stable (r esp. un stable) e quivalenc e in P er m nz J . Then ho colim J F : ho colim J C → ho colim J D 13 is a st able (r esp. unstable) e quivalenc e in Strict nz . If C : J → Perm nz is a c onstant functor and J is c ont r actible, then C ( x ) ∼ − → ho co lim J C is an u nstable e quivalenc e for e ach x ∈ J . Pr o of. The stable case follows from [18, 4 .1], since homotopy colimits of sp ectra pr eserve sta ble equiv a- lences. The unstable case follows b y the sa me line o f ar gument , see [18, p. 163 5] for an overview. First co nsider the strict case, when F : C ∼ − → D is a left lax tra nsformation and unstable e q uiv alence of functors J → Strict nz . The doubly forgetful functor W : Strict nz → Cat has a left adjoin t, the free functor P : Cat → Strict nz , with P C = ` n > 1 f Σ n × Σ n C × n , where f Σ n is the translation categ ory of the symmetric gr oup Σ n . The free–forgetful adjunction ( P , W ) gener ates a simplicial resolution ( P W ) • +1 C = { [ q ] 7→ ( P W ) q +1 C } of C by free ze r oless permutativ e categories. See [18, 1.2] for details. By Thoma s on’s argumen t [18, p. 1641 –1644 ], the augmentation ( P W ) • +1 C → C induces an unstable equiv alence ho colim J V ( P W ) • +1 C ∼ − → ho co lim J V C , and simila rly for D and F . Hence it suffices to prove that ho colim J V ( P W ) q +1 F is an unsta ble equiv a- lence, for each q > 0. Let C ′ = W ( P W ) q C , D ′ = W ( P W ) q D be functors J → Cat, and F ′ = W ( P W ) q F the resulting left lax transforma tion C ′ → D ′ . T hen F ′ is an unstable equiv alence, by q applications of Lemma 4.3 b elow. W e m us t pr ov e that V P F ′ : V P C ′ → V P D ′ induces an unstable equiv ale nc e of homotopy colimits. This follows fr om Lemma 4 .4 be low, the fact that the Gro thendieck constr uction J ∫ F ′ : J ∫ C ′ → J ∫ D ′ resp ects unstable equiv alences, and one mo re applicatio n of Lemma 4.3. Finally c o nsider the lax case, when F : C ∼ − → D is a left lax transformatio n and unstable equiv alence of functor s J → Perm nz . F or each x ∈ J let ˆ C ( x ) = ho colim 0 C ( x ) b e the homotopy colimit o f the functor 0 → Perm nz taking the unique o b ject o f 0 to C ( x ). By the universal prop erty of ho c olim 0 this defines a functor ˆ C : J → Strict nz , and a natural transformatio n C → V ˆ C . It is an unstable equiv alence by [14, 4.3]. Summation in the p ermutative catego r ies C ( x ) induces a left la x natural tra nsformation V ˆ C → C , s uch that the comp osite C → V ˆ C → C equals the identit y tr ansformation. By natura lit y of these constructio ns with r esp ect to F , we see that F : C → D is a r etract of V ˆ F : V ˆ C → V ˆ D as a morphism in Perm nz J , wher e ˆ F : ˆ C ∼ − → ˆ D is a left lax transformatio n and unstable e q uiv alence of functors J → Strict nz . B y functoriality , ho colim J F is a retract of ho c olim J V ˆ F , which is an unstable equiv alence by the first case of the pro of applied to ˆ F : ˆ C → ˆ D . It follows that ho colim J F is also an unstable eq uiv alence. The claim in the case of a consta n t diagra m follows by the same a rgument.  Lemma 4.3. The fr e e functor P : Ca t → Strict nz sends unst able e qu ivalenc es to u nstable e quivalenc es. Pr o of. This follows from the natural homeomor phism of cla ssifying spaces | P C | ∼ = ` n > 1 E Σ n × Σ n |C | × n and the fa ct that | f Σ n | = E Σ n is a free Σ n -space.  Lemma 4.4. L et C ′ : J → Cat b e any funct or. Ther e is a natu r al unstable e qu ivalenc e P ( J ∫ C ′ ) ∼ − → ho colim J V P C ′ . Pr o of. Thomaso n prov e d this in [1 8]. There the statement appe a rs in the second parag raph on page 1639, in ra ther different –lo oking notation, and the pro of s tarts with the las t paragr aph o n page 1 6 37.  Lemma 4.5. L et I b e the c ate gory of fi nite set s and inje ctive functions, and let m ∈ I . If C : I → Perm nz is a functor that sends e ach ϕ : m → n ∈ I to a st able (r esp. unstable) e quivalenc e C ( ϕ ) : C ( m ) → C ( n ) , then the c anonic al functor C ( m ) → ho colim I C is a st able (r esp. unstable) e quivalenc e. Pr o of. This is a weak v ersion of B¨ okstedt’s lemma [6, 9.1 ], which holds for homo to p y colimits in Cat since it holds for homotop y co limits in simplicial sets. By the a rgument ab ov e, using the r esolution by free p ermutativ e categorie s , it also holds in Perm nz .  4.2. The case with zero. W e shall nee d a version o f the homoto p y colimit for diagr ams of p ermutativ e categorie s with zero. Thoma s on comments that suc h a homotopy colimit with zer o is not a homotopy functor, unless the category is “ well based”. Hence we m ust derive our functor to get a homotop y inv ariant version. W e do this by means of another simplicial resolution, this time generated b y the free–forg etful adjunction b et ween per m utative ca tegories with and without zeros. 14 The functor R : Strict → Strict nz that forgets the sp ecia l ro le of the zer o ob ject ha s a left adjoint L : Strict nz → Strict, given b y adding a disjoint zer o ob ject: L N = N + for N ∈ Strict nz . Lik ewise, the fo r getful functor R ′ : Perm J → Perm nz J has a left adjoint L ′ : Perm nz J → Perm J , given by adding disjoint zeros p oint wise: L ′ C : x 7→ C ( x ) + for C : J → Perm nz and a ll x ∈ J . Let Strict iz ⊂ Strict b e the full sub category ge nerated by ob jects of the form L N = N + , i.e. , the p er- m utative catego ries with an isolate d zer o ob ject. Similarly , let Perm iz J ⊂ Perm J be the full sub categor y generated by ob jects of the form L ′ C = C + . In the statement and pro of of following lemma w e omit the fo rgetful functors R and R ′ from the notation, and wr ite N + and C + for L N and L ′ C , resp e c tiv ely . Note that ∆ V ( N + ) = ∆ V ( N ) + , where ∆ V : Strict nz → Perm nz J is a s in s ubs ection 4.1. Lemma 4.6. The functor (∆ V ) iz : Strict iz → Perm iz J , taking N + to the c onstant diagr am x 7→ N + for x ∈ J , has a left adjoint hocolim iz J : Perm iz J → Strict iz , satisfying ho colim iz J ( C + ) = (ho colim J C ) + for al l zer oless di agr ams C : J → Perm nz . Pr o of. T o define the functor ho c o lim iz J , we m ust sp ecify a str ict symmetric monoidal functor ho colim iz J F : (ho colim J C ) + − → (ho co lim J D ) + for each left lax transforma tion F : C + → D + . T he morphis m η + : D + → (∆ V (hoco lim J D )) + = ∆ V ((hoco lim J D ) + ) , where η : id → ∆ V ◦ ho colim J is the a djunction unit, induces a function Perm J ( C + , D + ) ∼ = Perm nz J ( C , D + ) → Perm nz J ( C , ∆ V ((hocolim J D ) + )) ∼ = Strict nz (ho colim J C , (hoco lim J D ) + ) ∼ = Strict((ho c olim J C ) + , (hoco lim J D ) + ) . W e dec la re the image of F : C + → D + to b e ho colim iz J F . A diagra m c hase shows that ho colim iz J ( GF ) = ho colim iz J G ◦ ho colim iz J F for ea ch G : D + → E + , so ho co lim iz J is a functor . The adjunction prop erty follows from the chain of natura l bijections Strict(ho colim iz J ( C + ) , N + ) = Strict((ho colim J C ) + , N + ) ∼ = Strict nz (ho colim J C , N + ) ∼ = Perm nz J ( C , ∆ V ( N + )) = Perm nz J ( C , (∆ V ) iz ( N + )) ∼ = Perm J ( C + , (∆ V ) iz ( N + )) .  Definition 4.7. F o r e a ch per m utative catego r y with zero M ∈ Perm let Z M ∈ Perm iz ∆ op ⊂ Perm ∆ op be the simplicial ob ject in p ermutativ e categor ies with isola ted zero es g iven by [ q ] 7→ Z q M = ( L R ) q +1 ( M ) . The face and degenera cy maps are induced by the adjunction counit LR → id and unit id → RL , a s usual. The c ounit also induces a natural augmentation ma p ǫ : Z M → M of simplicial p er m utative categorie s with zero, wher e M is viewed as a co nstant simplicial ob ject. Lemma 4. 8. L et M b e a p ermutative c ate gory. The augmen t ation map ǫ : Z M ∼ − → M is an unstable e quivalenc e. Pr o of. The map Rǫ : RZ M → R M of simplicial zero less p ermutativ e categor ies admits a simplicial homotopy in verse, induced by the adjunction unit. Hence the map of cla ssifying spa ces | ǫ | : | Z M| → |M| admits a homotopy inverse, since the classifying space only dep ends on the underlying categor y .  W e extend Z point wise to define a simplicial res olution ǫ : Z C ∼ − → C for any C : J → P erm, with Z C : x 7→ Z C ( x ) fo r all x ∈ J . T his allows us to define a derived homotopy co limit fo r per m utative categorie s with zero. Definition 4.9. The derive d h omotopy c olimit Dhoc olim J : Perm J → Strict iz ∆ op ⊂ Strict ∆ op is defined b y Dhoc olim J C = ho colim iz J ( Z C ) = { [ q ] 7→ ho colim iz J ( LR ) q +1 C } . The construction deserves its name: 15 Lemma 4.10. L et C → D b e a stable (r esp. u nstable) e quivalenc e in Perm J . Then Z q C → Z q D is a stable (r esp. unstable) e qu ivalenc e for e ach q > 0 , so t he induc e d functor Dhoc olim J C − → Dho colim J D is a st able (r esp. unstable) e quivalenc e, to o. Pr o of. The functor LR adds a disjoin t base p oint to the classifying space, and the counit LR → id induces a sta ble equiv a lence of spec tr a [18, 2.1]. Hence LR preserves b oth stable and unstable equiv alences. Iterating ( q + 1) times y ields the a ssertion for Z q .  Lemma 4.11. L et I b e the c ate gory of finite sets and inje ctive functions, and let m ∈ I . If C : I → Perm is a functor that sends e ach ϕ : m → n ∈ I to a st able (r esp. unstable) e quivalenc e C ( ϕ ) : C ( m ) → C ( n ) , then the c anonic al functors C ( m ) ∼ ← − Z C ( m ) − → Dho colim I C ar e stable (re sp. un stable) e quivalenc es. Pr o of. This follows fro m Lemmas 4.5 and 4 .8.  5. The homotopy col imit o f a graded bipermut a tive ca tegor y W e a r e now r e ady for a key prop osition. Prop ositio n 5.1. L et J b e a p ermutative c ate gory, and let C • b e a J -gr ade d bip ermu t ative c ate gory. Then Dho colim J C • is a simplicial bip ermu tative c ate gory, and C 0 ∼ ← − Z C 0 − → Dho colim J C • ar e lax morphisms of simplicial bip ermutative c ate gories. The same statement s hold when r eplacing “bip ermutative” by “strictly bimonoidal”. F urt hermor e, for e ach x ∈ J , the c anonic al functors C x ∼ ← − Z C x − → Dho colim J C • ar e maps of Z C 0 -mo dules. Pr o of. Recall the adjoin t pair ( L ′ , R ′ ) from subse c tion 4.2. If C • is a J -g raded bip ermutativ e categ ory, then so is L ′ R ′ C • = C • + , and Z C • bec omes a simplicial J -graded biper m utative ca tegory. By Lemma 5.2, which we will pr ov e b elow, we get that ho colim J R ′ ( L ′ R ′ ) q C • bec omes a ze r oless bip ermutativ e catego r y for each q > 0. Hence ho colim iz J Z q C • = L hocolim J R ′ ( L ′ R ′ ) q C • is a bip ermutativ e categor y, and all the simplicial structure maps are lax morphisms of bip ermutativ e categor ies. Therefore Dho colim J C • bec omes a s implicial bip ermutativ e categor y. Likewise, for each q > 0, Lemma 5 .2 be low g uarantees that Z q C 0 → ho co lim iz J Z q C • is a lax mor phism of bip ermutativ e catego ries and that each Z q C x → ho co lim iz J Z q C • is a map of Z q C 0 -mo dules, so w e are done by functoriality .  W e omit the forge tful functors R and R ′ in the statemen t and pr o of of the following lemma, which contains the most detailed diagr am chasing re quired in this pap er. Lemma 5. 2. L et J b e a p ermutative c ate gory. If C • is a J -gr ade d bip ermutative c ate gory, then Thoma- son ’s ho m otopy c olimit of p ermut ative c ate gories hoco lim J C • is a zer oless bip ermutative c ate gory. The c anonic al functor C 0 → ho co lim J C • is a lax morphism of zer oless bip ermutative c ate gories. F u rthermor e, for e ach x ∈ J , t he c anonic al f unctor C x − → ho co lim J C • is a map of zer oless C 0 -mo dules. If C • is a J -gr ade d st rictly bimonoi dal c ate gory, then ho colim J C • is a zer oless strictly bimonoidal c ate gory with a lax morphism of zer oless strictly bimonoidal c ate gories C 0 → ho colim J C • , and zer oless C 0 -mo dule maps C x → ho co lim J C • . 16 Pr o of. Thomaso n show ed that the homotopy colimit is a per mu tative category without zero. The additive t wist iso morphism τ ⊕ : n [( x 1 , X 1 ) , . . . , ( x n , X n )] ⊕ m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] ∼ = − → m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] ⊕ n [( x 1 , X 1 ) , . . . , ( x n , X n )] is g iven by ( χ ( n, m ) , id , id), wher e χ ( n, m ) ∈ Σ n + m is a s in Example 2.3. Let [ X ] and [ Y ] b e shorthand notations for the ob jects n [( x 1 , X 1 ) , . . . , ( x n , X n )] and m [( y 1 , Y 1 ) , . . . , ( y m , Y m )], resp ectively . The twist isomorphism for ⊕ then app ears as τ ⊕ : [ X ] ⊕ [ Y ] ∼ = − → [ Y ] ⊕ [ X ] . In order to distinguish the m ultiplica tive structure of C • from the o ne on the homotopy colimit, we shall simply denote the comp osition functor ⊗ on C • by juxtapo s ition of ob jects, o r by · . The multiplicativ e bifunctor ⊗ on the homotopy colimit is then defined at the ob ject level by n [( x 1 , X 1 ) , . . . , ( x n , X n )] ⊗ m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] := nm [( x 1 + y 1 , X 1 Y 1 ) , . . . , ( x 1 + y m , X 1 Y m ) , . . . , ( x n + y 1 , X n Y 1 ) , . . . , ( x n + y m , X n Y m )] . W e will use the shorthand notation [ X ] ⊗ [ Y ] for this o b ject. The ob ject 1 := 1 [(0 , 1)] is a unit for ⊗ . With these definitions , as ex tended b elow to the mor phism level, (ho co lim J C • , ⊗ , 1) is a strict mo no idal catego ry . W e will define the m ultiplicative twist map τ ⊗ : [ X ] ⊗ [ Y ] ∼ = − → [ Y ] ⊗ [ X ] a s a comp osite of t wo morphisms. First, we apply the twist map γ ⊗ for the multiplication in C • to every en try of the form X i Y j . T he triple (id nm , χ x i ,y j , γ ⊗ ) defines a mor phism nm [( x 1 + y 1 , X 1 Y 1 ) , . . . , ( x 1 + y m , X 1 Y m ) , . . . , ( x n + y 1 , X n Y 1 ) , . . . , ( x n + y m , X n Y m )] → nm [( y 1 + x 1 , Y 1 X 1 ) , . . . , ( y m + x 1 , Y m X 1 ) , . . . , ( y 1 + x n , Y 1 X n ) , . . . , ( y m + x n , Y m X n )] . (Here χ is the t wist map in J . T o b e precis e, the third co ordinate of the mor phism is really C ( χ x i ,y j )( γ ⊗ ), but we omit C ( χ x i ,y j ) from the notation.) Seco nd, we use the p ermutation σ n,m ∈ Σ nm that induces matrix tra ns po sition. The triple ( σ n,m , id y j + x i , id) defines a morphism nm [( y 1 + x 1 , Y 1 X 1 ) , . . . , ( y m + x 1 , Y m X 1 ) , . . . , ( y 1 + x n , Y 1 X n ) , . . . , ( y m + x n , Y m X n )] → mn [( y 1 + x 1 , Y 1 X 1 ) , . . . , ( y 1 + x n , Y 1 X n ) , . . . , ( y m + x 1 , Y m X 1 ) , . . . , ( y m + x n , Y m X n )] . Let the twist ma p for ⊗ b e the comp osite mo rphism τ ⊗ = ( σ n,m , id y j + x i , id) ◦ (id nm , χ x i ,y j , γ ⊗ ). As matrix transp os ition squares to the identit y , χ y j ,x i ◦ χ x i ,y j = id and γ 2 ⊗ = id, we obtain that τ 2 ⊗ = id. If [ X ] = 1 is the mu ltiplicative unit, then we hav e that σ 1 ,m is the iden tity in Σ m and χ 0 ,y j is the identit y as w ell, so τ ⊗ : 1 ⊗ [ Y ] → [ Y ] ⊗ 1 is the iden tit y . Similarly one s hows tha t τ ⊗ gives the ident it y mor phism if [ Y ] = 1 is the multiplicative unit. W e hav e now verified prop erties (1), (2) and (3) of Definition 2.1, a t the level of o b jects. W e leave to the rea der to check prop erty (4). Prop er ty (5) is disrega rded in the zeroless situa tio n. W riting out ([ X ] ⊗ [ Y ]) ⊕ ([ X ′ ] ⊗ [ Y ]) and ([ X ] ⊕ [ X ′ ]) ⊗ Y we get the same ob ject, and we define the right distributivity d r to b e the iden tity morphism b etw een these tw o expr essions. The left distributivity d ℓ inv olves a reorder ing o f elements. It is a mo rphism d ℓ : ([ X ] ⊗ [ Y ]) ⊕ ([ X ] ⊗ [ Y ′ ]) − → [ X ] ⊗ ([ Y ] ⊕ [ Y ′ ]) . The so urce is ( nm + nm ′ )[( x 1 + y 1 , X 1 Y 1 ) , . . . , ( x n + y m , X n Y m ) , ( x 1 + y ′ 1 , X 1 Y ′ 1 ) , . . . , ( x n + y ′ m ′ , X n Y ′ m ′ )] , while the target is n ( m + m ′ )[( x 1 + y 1 , X 1 Y 1 ) , . . . , ( x 1 + y ′ m ′ , X 1 Y ′ m ′ ) , . . . , ( x n + y 1 , X n Y 1 ) , . . . , ( x n + y ′ m ′ , X n Y ′ m ′ )] . The same terms ( x i + y j , X i Y j ) and ( x i + y ′ j , X i Y ′ j ) o cc ur in bo th the source and ta rget, but their ordering differs by a suitable p er mutation ξ ∈ Σ nm + nm ′ . Thus we define the mor phism d ℓ by the tr iple ( ξ , id , id). Note that ξ is the left distributivity is omorphism in the bip e r mut ative catego ry of finite sets and functions, as defined in E xample 2.3. W e have to chec k that the so defined distributivity tra nsformation d ℓ coincides with τ ⊗ ◦ ( τ ⊗ ⊕ τ ⊗ ). The t wis t terms γ ⊗ and χ occur twice in the comp osition, so they reduce to the identit y . What is left is a p ermutation that is caused by τ ⊗ ◦ ( τ ⊗ ⊕ τ ⊗ ), and this is precisely ξ . 17 W e hav e now verified pr op erties (6) and (7) o f Definition 2.1. Since the isomor phisms τ ⊕ , d r and d ℓ are all of the form ( σ, id , id) for suitable p er mutations σ , pr o pe r ties (8), (9) a nd (10) all fo llow fro m the corres p onding o nes in the biper m utative category of finite sets a nd functions. This finishes the pro o f that the zer o less bipermutativ e categ ory structure works fine on ob jects. It remains to establish that ⊕ and ⊗ a re bifunctors on ho c olim J C • , that the v a r ious a sso ciativity and distributivity laws ar e natural, a nd that the additive and multiplicativ e twists are natural. F or ⊕ this is straightforward and can be found in [18]: supp ose given tw o morphisms ( ψ , ℓ i ,  j ) : n [( x 1 , X 1 ) , . . . , ( x n , X n )] → n ′ [( x ′ 1 , X ′ 1 ) , . . . , ( x ′ n ′ , X ′ n ′ )] and ( ϕ, k i , π j ) : m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] → m ′ [( y ′ 1 , Y ′ 1 ) , . . . , ( y ′ m ′ , Y ′ m ′ )] in the homotopy co limit, with ψ : n → n ′ , ℓ i : x i → x ′ ψ ( i ) and  j : L ψ ( i )= j C ( ℓ i )( X i ) → X ′ j , and ϕ : m → m ′ with corresp onding k i and π j . T hen there is a surjection ψ + ϕ from n + m to n ′ + m ′ , and we can recycle the morphisms ℓ i and k i to give corr esp onding morphisms in J . In the third co or dinate we can use the mor phisms  j and π j to g et new o nes, b ecause the preimages of n ′ and m ′ under ψ + ϕ are disjoin t. T aken tog ether, this results in a morphism from the sum ( n + m )[( x 1 , X 1 ) , . . . , ( y m , Y m )] to the sum ( n ′ + m ′ )[( x ′ 1 , X ′ 1 ) , . . . , ( y ′ m ′ , Y ′ m ′ )]. It is straig h tforward to see tha t ⊕ defines a bifunctor, that the ass o ciativity law for ⊕ is natur al, and that the a dditiv e twist τ ⊕ is natural. F or the remainder of this pro of let us denote the elements in the set nm = { 1 , . . . , nm } as pair s ( i, j ) with 1 6 i 6 n and 1 6 j 6 m . The tens o r pro duct of the morphisms ( ψ , ℓ i ,  j ) and ( ϕ, k i , π j ) has three co ordinates. O n the first we take the pro duct of the surjections, i.e. , nm ∋ ( i, j ) 7→ ( ψ ( i ) , ϕ ( j )) ∈ n ′ m ′ , and on the second w e take the sum ℓ i + k j : x i + y j → x ′ ψ ( i ) + y ′ ϕ ( j ) of the mor phisms ℓ i , k j ∈ J . The third co o rdinate of the morphism ( ψ , ℓ i ,  j ) ⊗ ( ϕ, k i , π j ) has to b e a mor phism M ( ψ ( i ) ,ϕ ( j ))=( r,s ) C ( ℓ i + k j )( X i · Y j ) = M ( ψ ( i ) ,ϕ ( j ))=( r,s ) C ( ℓ i )( X i ) · C ( k j )( Y j ) − → X ′ r · Y ′ s in C ( x ′ r + y ′ s ), for each 1 6 r 6 n ′ and 1 6 s 6 m ′ . Here, the sum is taken with resp ect to the lexicogra phical o rdering of the indices ( i, j ). Cons ider the fo llowing diagr am: L ( ψ ( i ) ,ϕ ( j ))=( r,s ) C ( ℓ i )( X i ) · C ( k j )( Y j ) id t t i i i i i i i i i i i i i i i i σ * * U U U U U U U U U U U U U U U U L ψ ( i )= r L ϕ ( j )= s C ( ℓ i )( X i ) · C ( k j )( Y j ) L ψ ( i )= r d ℓ   L ϕ ( j )= s L ψ ( i )= r C ( ℓ i )( X i ) · C ( k j )( Y j ) L ϕ ( j )= s d r   L ψ ( i )= r C ( ℓ i )( X i ) ·  L ϕ ( j )= s C ( k j )( Y j )  L ψ ( i )= r id · π s   L ϕ ( j )= s  L ψ ( i )= r C ( ℓ i )( X i )  · C ( k j )( Y j ) L ϕ ( j )= s  r · id   L ψ ( i )= r C ( ℓ i )( X i ) · Y ′ s d r   L ϕ ( j )= s X ′ r · C ( k j )( Y j ) d ℓ    L ψ ( i )= r C ( ℓ i )( X i )  · Y ′ s  r · id * * T T T T T T T T T T T T T T T T T X ′ r ·  L ϕ ( j )= s C ( k j )( Y j )  id · π s t t j j j j j j j j j j j j j j j j j X ′ r · Y ′ s 18 The isomo rphism σ is an a ppropriate p ermutation of the summands. The distributivity laws in C • are natural with resp ect to morphisms in C • , and therefore we hav e the identities: d r ◦  M ψ ( i )= r id C ( ℓ i )( X i ) · π s  =  id L ψ ( i )= r C ( ℓ i )( X i )  · π s  ◦ d r d ℓ ◦  M ϕ ( j )= s  r · id C ( k j )( Y j )  =   r ·  id L ϕ ( j )= s C ( k j )( Y j )  ◦ d ℓ Combining thes e with the ge ne r alized p en tagon equa tion d r ◦ M ψ ( i )= r d ℓ = d ℓ ◦ M ϕ ( j )= s d r ◦ σ we see that the diagram co mm utes. W e define the thir d co ordina te in the tensor product morphis m to be the comp osition given by either of the tw o branches. Note that fo r ( ψ , ℓ i ,  j ) ⊗ id the definition reduce s to (  j · id) ◦ d r , and similarly the thir d co or dinate of id ⊗ ( ϕ, k i , π j ) is (id · π j ) ◦ d ℓ . In pa rticular, the tensor pro duct of iden tity morphisms is an ide ntit y morphism. Comp ositions of morphisms in the ho motopy colimit inv olve an additive twist [18, p. 1 631]. F o r ( ψ ′ , ℓ ′ i ,  ′ j ) : n ′ [( x ′ 1 , X ′ 1 ) , . . . , ( x ′ n ′ , X ′ n ′ )] → n ′′ [( x ′′ 1 , X ′′ 1 ) , . . . , ( x ′′ n ′′ , X ′′ n ′′ )] the morphism L ψ ′ ψ ( i )= r C ( ℓ ′ ψ ( i ) ℓ i )( X i ) → X ′′ r is given as a co mpo sition. First, one has to per mu te the summands σ : M ψ ′ ψ ( i )= r C ( ℓ ′ ψ ( i ) ℓ i )( X i ) → M ψ ′ ( k )= r M ψ ( i )= k C ( ℓ ′ k ℓ i )( X i ) . Then, as we a ssumed that C is a functor to Strict, we know that M ψ ′ ( k )= r M ψ ( i )= k C ( ℓ ′ k ℓ i )( X i ) = M ψ ′ ( k )= r M ψ ( i )= k C ( ℓ ′ k ) C ( ℓ i )( X i ) = M ψ ′ ( k )= r C ( ℓ ′ k )  M ψ ( i )= k C ( ℓ i )( X i )  . Finally , we a pply the mor phism M ψ ′ ( k )= r C ( ℓ ′ k )(  k ) : M ψ ′ ( k )= r C ( ℓ ′ k )  M ψ ( i )= k C ( ℓ i )( X i )  − → M ψ ′ ( k )= r C ( ℓ ′ k )( X ′ k ) and contin ue with  ′ r to end up in X ′′ r . In order to prove that the tensor pro duct actually defines a bifunctor, we will show tha t ( ψ , ℓ i ,  j ) ⊗ ( ϕ, k i , π j ) = (( ψ , ℓ i ,  j ) ⊗ id) ◦ (id ⊗ ( ϕ, k i , π j )) = (id ⊗ ( ϕ, k i , π j )) ◦ (( ψ , ℓ i ,  j ) ⊗ id) and (( ψ ′ , ℓ ′ i ,  ′ j ) ⊗ id) ◦ (( ψ , ℓ i ,  j ) ⊗ id) = (( ψ ′ , ℓ ′ i ,  ′ j ) ◦ ( ψ , ℓ i ,  j )) ⊗ id , and leav e the check of the rema ining identit y to the r eader. The first equation is straig h tforward to see, beca use (( ψ , ℓ i ,  j ) ⊗ id) ◦ (id ⊗ ( ϕ, k i , π j )) corres p onds to the left branch of the diag r am ab ov e and the other comp osition is given b y the right branch. F or the second equation we hav e to chec k that (((  ′ ◦  ) · id ) ◦ d r ) s = ((  ′ · id) ◦ d r ◦ (  · id ) ◦ d r ) s . Both morphisms have sourc e M ψ ′ ψ ( i )= s C ( ℓ ′ ψ ( i ) ℓ i + id)( X i · Y j ) = M ψ ′ ψ ( i )= s C ( ℓ ′ ψ ( i ) ℓ i )( X i ) · Y j 19 and the left hand side corresp onds to the left br anch o f the follo wing diag ram and the right hand side to the r ight br a nch. L ψ ′ ψ ( i )= s C ( ℓ ′ ψ ( i ) ℓ i )( X i ) · Y j d r   L ψ ′ ψ ( i )= s C ( ℓ ′ ψ ( i ) ℓ i + id)( X i · Y j ) σ    L ψ ′ ψ ( i )= s C ( ℓ ′ ψ ( i ) ℓ i )( X i )  · Y j σ · id   L ψ ′ ( k )= s L ψ ( i )= k C ( ℓ ′ ψ ( i ) + id) C ( ℓ i + id)( X i · Y j )  L ψ ′ ( k )= s L ψ ( i )= k C ( ℓ ′ ψ ( i ) ) C ( ℓ i )( X i )  · Y j L ψ ′ ( k )= s C ( ℓ ′ k + id)  L ψ ( i )= k C ( ℓ i + id)( X i · Y j )  L ψ ′ ( k )= s C ( ℓ ′ k +id)((  k · id) ◦ d r )    L ψ ′ ( k )= s C ( ℓ ′ k )  L ψ ( i )= k C ( ℓ i )( X i )  · Y j ( L ψ ′ ( k )= s C ( ℓ ′ k )(  k ) ) · id   L ψ ′ ( k )= s C ( ℓ ′ k )( X ′ k ) · Y j d r    L ψ ′ ( k )= s C ( ℓ ′ k )( X ′ k )  · Y j  ′ s · id ( ( Q Q Q Q Q Q Q Q Q Q Q Q  L ψ ′ ( k )= s C ( ℓ ′ k )( X ′ k )  · Y j  ′ s · id u u k k k k k k k k k k k k k k X ′′ s · Y j Naturality of d r in C • ensures that d r can ch ange place with L ψ ′ ( k )= s C ( ℓ ′ k + id)(  k · id) on the right branch. That d r ◦ σ = ( σ · id) ◦ d r holds be c ause C • satisfies prop erty (8) from Definition 2.1, and hence the diag ram commutes. In order to show that the asso cia tivit y identification is natural, we hav e to prov e that (( ψ 1 , ℓ 1 i ,  1 j ) ⊗ ( ψ 2 , ℓ 2 i ,  2 j )) ⊗ ( ψ 3 , ℓ 3 i ,  3 j ) = ( ψ 1 , ℓ 1 i ,  1 j ) ⊗ (( ψ 2 , ℓ 2 i ,  2 j ) ⊗ ( ψ 3 , ℓ 3 i ,  3 j )) for morphisms in the homo to p y co limit. The claim is obvious on the co ordinates of the sur jections a nd the mor phis ms in J . F or proving the ident it y in the thir d coo rdinate of morphis ms , note that the natur ality of ⊗ implies that we can write (( ψ 1 , ℓ 1 i ,  1 j ) ⊗ ( ψ 2 , ℓ 2 i ,  2 j )) ⊗ ( ψ 3 , ℓ 3 i ,  3 j ) =((( ψ 1 , ℓ 1 i ,  1 j ) ⊗ id) ⊗ id) ◦ ((id ⊗ ( ψ 2 , ℓ 2 i ,  2 j )) ⊗ id) ◦ ((id ⊗ id) ⊗ ( ψ 3 , ℓ 3 i ,  3 j )) . Therefore, it suffices to prov e the claim for each of the factors. W e will show it for the middle one and lea ve the other ones to the curious reader. Recall that id ⊗ ( ψ 2 , ℓ 2 i ,  2 j ) ha s as third coor dinate the comp osition (id ·  2 j ) ◦ d ℓ and therefore (id ⊗ ( ψ 2 , ℓ 2 i ,  2 j )) ⊗ id has third c o ordinate (((id ·  2 j ) ◦ d ℓ ) · id) ◦ d r = (id ·  2 j · id) ◦ (id · d ℓ ) ◦ d r . But (id · d ℓ ) ◦ d r = ( d r · id) ◦ d ℓ (equation (7’) o f Definition 2 .4) holds in C • , and therefo r e the third co ordinate equals (id ·  2 j · id) ◦ (id · d ℓ ) ◦ d r = (id ·  2 j · id) ◦ ( d r · id) ◦ d ℓ which is the third co ordina te of id ⊗ (( ψ 2 , ℓ 2 i ,  2 j ) ⊗ id). Naturality of the multiplicative twist map can b e seen as follows. W e have to show that τ ⊗ ◦ (( ψ , ℓ i ,  j ) ⊗ ( ϕ, k i , π j )) = (( ψ , ℓ i ,  j ) ⊗ ( ϕ, k i , π j )) ◦ τ ⊗ . On the first co o rdinate of the morphisms this reduces to the equality σ n ′ ,m ′ ◦ ( ψ , ϕ )( i, j ) = ( ϕ ( j ) , ψ ( i )) = ( ϕ, ψ ) ◦ σ n,m ( i, j ) , and on the sec o nd co ordinate we hav e the equation χ ◦ ( ℓ i + k j ) = ( k j + ℓ i ) ◦ χ 20 bec ause χ is natural. Thus, it r emains to prov e that the a b ove equation holds in the third co or dinate, which amounts to showing that the fo llowing diag ram commutes. L ψ ( i )= r L ϕ ( j )= s C ( ℓ i )( X i ) · C ( k j )( Y j ) ( L L γ ⊗ ) ◦ σ / / L ψ ( i )= r d ℓ   L ϕ ( j )= s L ψ ( i )= r C ( k j )( Y j ) · C ( ℓ i )( X i ) σ − 1   L ψ ( i )= r L ϕ ( j )= s C ( k j )( Y j ) · C ( ℓ i )( X i ) L ψ ( i )= r d r   L ψ ( i )= r C ( ℓ i )( X i ) ·  L ϕ ( j )= s C ( k j )( Y j )  L ψ ( i )= r id · π s   L ψ ( i )= r γ ⊗ / / L ψ ( i )= r  L ϕ ( j )= s C ( k j )( Y j )  · C ( ℓ i )( X i ) L ψ ( i )= r π s · id   L ψ ( i )= r C ( ℓ i )( X i ) · Y ′ s d r   L ψ ( i )= r Y ′ s · C ( ℓ i )( X i ) d ℓ    L ψ ( i )= r C ( ℓ i )( X i )  · Y ′ s  r · id   Y ′ s ·  L ψ ( i )= r C ( ℓ i )( X i )  id ·  r   X ′ r · Y ′ s γ ⊗ / / Y ′ s · X ′ r The top diagram commutes b ecause d ℓ is defined in terms of d r and γ ⊗ . F or the b ottom diagr am w e apply the same ar gument toge ther with the naturality of γ ⊗ . W e ha ve to chec k that r ight distributivity is the iden tity o n morphisms. Consider three morphisms as above. When w e fo cus on the surjections ψ 1 : n → n ′ , ψ 2 : m → m ′ , and ψ 3 : l → l ′ , we see that a condition like ( ψ 1 + ψ 2 ) ψ 3 ( i, j ) = ( r , s ) o nly affects either the preimage of n ′ l ′ or the preimage of m ′ l ′ in ( n + m ) l , but never b oth. Therefore, the third co o rdinate of the morphism (( ψ 1 , ℓ 1 i ,  1 j ) ⊕ ( ψ 2 , ℓ 2 i ,  2 j )) ⊗ ( ψ 3 , ℓ 3 i ,  3 j ) is either a third co ordinate of ( ψ 1 , ℓ 1 i ,  1 j ) ⊗ ( ψ 3 , ℓ 3 i ,  3 j ) o r of ( ψ 2 , ℓ 2 i ,  2 j ) ⊗ ( ψ 3 , ℓ 3 i ,  3 j ), and th us right distributivity is the identit y on morphisms. In the J -gr aded biper m utative case the naturality of the left distributivity isomo rphism follows from the o ne of d r and the multiplicativ e t wist. In bo th the biper m utative a nd the strictly bimo noidal ca se left distributivity is g iven by ( ξ , id , id). Therefor e naturalit y of d ℓ in the bip e rmut ative s e tting prov es naturality in the strictly bimonoida l setting as well. This finishes the pro of that ho colim J C • is a bip ermutativ e ca tegory without zero. W e now prove the remaining sta temen ts of the lemma. There is a natur al functor G : C 0 → ho co lim J C • which sends X ∈ C 0 to G ( X ) = 1[(0 , X )]. Note that the functor G is strict (symmetric) monoida l with res pec t to ⊗ , be cause G (1) = 1[(0 , 1)] and G ( X ) ⊗ G ( Y ) = 1[(0 , X )] ⊗ 1[(0 , Y )] = 1 [(0 + 0 , X ⊗ Y )] = 1 [(0 , X ⊗ Y )] = G ( X ⊗ Y ) . How ever, G is only lax symmetric monoidal with res pec t to ⊕ : there is a binatur al tra nsformation η ⊕ = ( ψ , id , id) fro m G ( X ) ⊕ G ( X ′ ) = 1[(0 , X )] ⊕ 1[(0 , X ′ )] = 2[(0 , X ) , (0 , X ′ )] to G ( X ⊕ X ′ ) = 1[(0 , X ⊕ X ′ )], given by the canonical s ur jection ψ : 2 → 1 and identit y mor phisms in the other tw o comp onents. This morphism is of cour se not an isomorphism. W e hav e to show that the functor G resp ects the distr ibutivit y constraints d r = id and d ℓ . In our situation we hav e that η ⊗ = id, so we have to chec k that η ⊕ = η ⊕ ⊗ id and G ( τ ⊗ ◦ ( τ ⊗ ⊕ τ ⊗ )) ◦ η ⊕ = (id ⊗ η ⊕ ) ◦ τ ⊗ ◦ ( τ ⊗ ⊕ τ ⊗ ) . 21 The firs t equation is just s tating the fact that 2[(0 , X ) , (0 , X ′ )] ⊗ 1[(0 , Y )] η ⊕ ⊗ id / / 1[(0 , X ⊕ X ′ )] ⊗ 1[(0 , Y )] 2[(0 , X ⊗ Y ) , (0 , X ′ ⊗ Y )] η ⊕ / / 1[(0 , ( X ⊕ X ′ ) ⊗ Y )] commutes, in view of the ident it y d r : ( X ⊗ Y ) ⊕ ( X ′ ⊗ Y ) = ( X ⊕ X ′ ) ⊗ Y . F or the left distributivity law w e should observe that the m ultiplicative t wist τ ⊗ on the homotopy colimit r educes to the mor phism (id , χ, γ ⊗ ) in the case of elements of length 1 in the ho motopy c olimit, and that χ 0 , 0 = id. F urthermore, id ⊗ ( ψ , id , id) = ( ψ , id , id) holds. Therefore (id ⊗ η ⊕ ) ◦ d ℓ = (id ⊗ ( ψ , id , id)) ◦ τ ⊗ ◦ ( τ ⊗ ⊕ τ ⊗ ) = ( ψ , id , id) ◦ (id , id , γ ⊗ ◦ ( γ ⊗ ⊕ γ ⊗ )) = (id , id , γ ⊗ ◦ ( γ ⊗ ⊕ γ ⊗ )) ◦ ( ψ , id , id) = G ( d ℓ ) ◦ η ⊕ . The cla im ab out the mo dule structure is o b vious. As the le ft dis tr ibutivit y o n the ho mo topy c o limit is of the form ( ξ , id , id), the ab ove pr o of ca rries ov er to the s trictly bimono idal case.  Lemma 5.3. If F : C • → D • is a lax morphism of J -gr ade d bip erm u tative c ate gories (r esp. J -gr ade d strictly bimonoidal c ate gories) then it induc es a lax morphism F ∗ : ho colim J C • → ho colim J D • of zer o- less bip ermutative c ate gories ( re sp. zer oless strictly bimonoida l c ate gories). Pr o of. Of course, we define F ∗ : ho colim J C • → ho co lim J D • on o b jects b y F ∗ ( n [( x 1 , X 1 ) , . . . , ( x n , X n )]) := n [( x 1 , F ( X 1 )) , . . . , ( x n , F ( X n ))] . Note that with this definition F ∗ is strict symmetric monoidal with resp ect to ⊕ even if F was only lax symmetric monoida l. Given a mor phis m ( ψ , ℓ i ,  j ) from n [( x 1 , X 1 ) , . . . , ( x n , X n )] to m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] we define the induced mor phism ( ψ , ℓ i ,  F j ) : F ∗ ( n [( x 1 , X 1 ) , . . . , ( x n , X n )]) → F ∗ ( m [( y 1 , Y 1 ) , . . . , ( y m , Y m )]) as fo llows: w e keep the surjection ψ and the morphisms ℓ i , and for the third co ordinate we take the comp osition  F j : M ψ ( i )= j D ( ℓ i )( F ( X i )) = M ψ ( i )= j F ( C ( ℓ i )( X i )) η ⊕ − → F ( M ψ ( i )= j C ( ℓ i )( X i )) F (  j ) − → F ( Y j ) . The natur ality of η ⊕ ensures that comp osition of morphisms is well-defined. Let n [( x 1 , X 1 ) , . . . , ( x n , X n )] and m [( y 1 , Y 1 ) , . . . , ( y m , Y m )] be tw o ob jects in ho colim J C • . Applying ⊗ ◦ ( F ∗ , F ∗ ) yields nm [( x 1 + y 1 , F ( X 1 ) ⊗ F ( Y 1 )) , . . . , ( x n + y m , F ( X n ) ⊗ F ( Y m ))] whereas the comp osition F ∗ ◦ ⊗ g ives nm [( x 1 + y 1 , F ( X 1 ⊗ Y 1 )) , . . . , ( x n + y m , F ( X n ⊗ Y m ))] . Thu s, we c a n use (id , id , η ⊗ ) to obtain a natura l tr ansformation η ∗ ⊗ from ⊗ ◦ ( F ∗ , F ∗ ) to F ∗ ◦ ⊗ . This transformatio n inherits all pro per ties from η ⊗ . In particular, η ∗ ⊗ is la x symmetric monoidal if η ⊗ was so . It remains to check the pr o pe r ties co ncerning the distributivity laws. As d r is the identit y on the J -gra ded bipermutative categor y a nd on the homoto p y colimit, and η ⊕ is the identit y on the homo topy colimit, the equalities reduce to η ∗ ⊗ ⊕ η ∗ ⊗ = η ∗ ⊗ and F ∗ ( d ℓ ) ◦ ( η ∗ ⊗ ⊕ η ∗ ⊗ ) = η ∗ ⊗ ◦ d ℓ . The firs t equation is straightforward to check. The left distributivit y law in the homoto p y colimit is given by d ℓ = ( ξ , id , id) and η ∗ ⊗ ⊕ η ∗ ⊗ is equal to η ∗ ⊗ ⊕ η ∗ ⊗ = (id nm , id , η ⊗ ) ⊕ (id nm ′ , id , η ⊗ ) . 22 As addition in the ho mo topy colimit is given by co ncatenation, we c a n simplify the ab ov e expression to (id nm + nm ′ , id , η ⊗ ). As d ℓ differs fr o m the identit y o nly in the firs t co or dinate, and η ∗ ⊗ ⊕ η ∗ ⊗ only in the third co o rdinate, these mo r phisms commute.  6. A ring completion device Recall from s ubsection 3.2 the cons truction G M : I ∫ Q → Strict. Lemma 6.1. L et M b e a p ermut ative c ate gory. Then (1) the c anonic al functor M → ho colim I ∫ Q G M is a st able e quivalenc e, (2) ho colim I ∫ Q G M is gro u p c omplete, and (3) the c anonic al functor ho c o lim T ∈Q 1 G M ( 1 , T ) ∼ − → ho co lim I ∫ Q G M is an unstable e quivalenc e. Pr o of. Recall that sp ectrificatio n co mmutes with ho mo topy c o limits [18, 4.1], i.e. , hoco lim J Spt is e quiv- alent to Spt ho colim J . Given n ∈ I , the homotopy colimit ho co lim T ∈Q n Spt G M ( n , T ) can b e calculated by taking the homotopy colimit in ea ch of the n directions of Q n success ively . Since a ll nontrivial maps inv olved are dia gonal maps, we see that the homotopy colimit in the n -th direction can be iden tified with ho colim S ∈Q ( n − 1 ) Spt G M ( n − 1 , S ), through the inclusion n − 1 → n tha t skips n . By induction it fol- lows that each morphism in the I -sha ped diagram n 7→ ho colim T ∈Q n G M ( n , T ) is a stable equiv alence. Lemma 4 .5 then says that the functor M → ho colim n ∈ I ho colim T ∈Q n G M ( n , T ) is a sta ble equiv alence . The claim that the functor M → ho colim I ∫ Q G M is a stable equiv alence follows, since by extending Thomason’s pr o of [16 ] of hoco lim I |Q| ≃ | I ∫ Q| = ho colim I ∫ Q ∗ (for the trivial functor ∗ ) to allow for arbitrar y functors from I ∫ Q , we hav e a n equiv alence ho colim I ∫ Q Spt G M ≃ ho colim n ∈ I ho colim T ∈Q n Spt G M ( n , T ) . See a ls o [15, 2.3 ] for a write-up in the dual situation. That π 0 of ho co lim I ∫ Q G M is a gro up can b e se e n as follows. I t is enough to s how that ele ments of the form 1[(( n , S ) , a )] hav e negatives, for n > 1. If S 6 = n then there is an inclusion S ⊆ T ∈ Q n with T containing a negative n umber, so there is a path 1[(( n , S ) , a )] → 1[(( n , T ) , 0)] ← 1[(( 0 , 0 ) , 0)] in the homotopy co limit, and the element represents zero. If S = n , so that a ∈ M P n , let b ∈ M P n be given b y b U = a V , where V = U ∪ { n } if n / ∈ U and V = U \ { n } if n ∈ U , for all U ⊆ n . Then a ⊕ b is isomorphic to M n ( ι )( c ) for s ome c ∈ M P S , where S = n − 1 a nd ι : S ⊂ n is the inclusion. Hence there is a path 1[( n , n ) , a ] ⊕ 1[( n , n ) , b ] → 1[( n , n ) , a ⊕ b ] ↔ 1 [( n , n ) , M n ( ι )( c )] ← 1 [( n , S ) , c ] in the homotopy co limit, and, as we saw a bove, the right ha nd element repres ent s zero. Now, since stable equiv alences betw een group complete symmetr ic monoidal ca tegories a re unsta ble equiv alences, the third claim also follows.  Lemma 6.2. If M is a p erm u tative c ate gory with z er o, such that al l morphisms ar e isomorphisms and e ach additive tr anslation is faithf ul, t hen ther e is an unstable e qu ivalenc e ho colim S ∈Q 1 G M ( 1 , S ) ∼ − → ( −M ) M . Pr o of. This is en tirely due to Thomason. Theorem 5.2 in [18] asserts that there is an unstable eq uiv alence from hoco lim S ∈Q 1 G M ( 1 , S ) to the “simplified double mapping cylinder” , and his arg ument on pp. 1 657– 1658 exhibits an unstable equiv alence from the simplified double mapping cylinder to ( − M ) M .  Remark 6.3. The unstable equiv alence ho colim S ∈Q 1 G M ( 1 , S ) ∼ − → ( −M ) M is the additive extension of the as signment that sends 1 [ {− 1 } , 0] and 1[ ∅ , a ] to (0 , 0) ∈ ( −M ) M , a nd 1[ { 1 } , ( a, b )] to ( a, b ). The map on morphisms is straig h tforward, once one declares that the morphism 1[ ∅ , a ] → 1[ { 1 } , ( a, a )] is sent to [id a , a ] : (0 , 0) → ( a, a ) ∈ ( −M ) M . Collecting Pr op osition 3.2, Lemma 5.2 a nd Lemma 6.1, we o btain our zer oless ring completion. Corollary 6.4. L et R b e a bip ermutative c ate gory (r esp. a strictly bimonoidal c ate gory). The c anonic al lax morphism R − → ho colim I ∫ Q G R is a st able e quivalenc e of zer oless bip ermutative c ate gories (r esp. zer oless strictly bimonoidal c ate gories), and ho colim S ∈Q 1 G R ( 1 , S ) ∼ − → ho co lim I ∫ Q G R 23 is an u nstable e quivalenc e of R -mo dules. Using Pro po sition 5.1 to a dd zer os, and tracing the ac tio n of Z R , we hav e the main result: Theorem 6.5. If R is a c ommu tative rig c ate gory (r esp. a rig c ate gory), then ¯ R = Dho colim I ∫ Q G R is a simplicial c ommutative ring c ate gory (r esp. a simplicial ring c ate gory). Her e G R is the I ∫ Q -gr ade d bip ermutative c ate gory (r esp. I ∫ Q -gr ade d strictly bimonoidal c ate gory) of Pr op osition 3.2 applie d to the bip ermutative c ate gory (r esp. strictly bimonoidal c ate gory) asso ciate d with R . The simplicial rig maps of Pr op osition 5.1 R ∼ ← − Z R − → ¯ R ar e stable e quivalenc es of Z R -mo dules. F urt hermor e, if R is a gr oup oid with faithful add itive tr anslation, then the maps ( −R ) R ∼ ← − Z ( −R ) R ∼ ← − Z ho colim S ∈Q 1 G R ( 1 , S ) ∼ − → ¯ R form a ch ain of u nstable e quivalenc es of Z R -mo dules. References [1] Christian Ausoni, On the algebr aic K -the ory of t he c omplex K -the ory sp e ctrum , In ven t. Math. 180 (2010), no. 3, 611–668. [2] Christian Ausoni and John Rognes, Al gebr aic K -the ory of top olo g ic al K - t he ory , Acta M ath. 188 (2002), no. 1, 1–39. [3] Nils A. Baas, Bjørn Ian Dundas, and John Rognes, Two-ve ctor bund les and forms of el liptic c ohomolo gy , T op ology , geometry and quan tum field theory, London Math. Soc. 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Dep ar tment of Ma them a tical S ciences, NTNU, 7491 Trondheim, Nor w a y E-mail addr ess : baas@math.ntnu .no Dep ar tment of Ma them a tics, University of Bergen, 5008 Bergen, Nor w ay E-mail addr ess : dundas@math.ui b.no Dep ar tment M a thema tik der Universit ¨ at Ham burg, 20 146 Hamb urg, Germ any E-mail addr ess : richter@math.u ni-hambu rg.de Dep ar tment of Ma them a tics, University of Oslo, 03 16 Oslo, Nor w ay E-mail addr ess : rognes@math.ui o.no 24