An involution on the K-theory of bimonoidal categories with anti-involution

AN INV OLUTION ON THE K -THEOR Y OF BIMONOID AL CA TEGORIES WITH ANTI-IN V OLUTION BIRGIT RICHTER Abstra ct. W e construct a combinatoria lly defined inv olution on the algebraic K - theory of t he ring sp ectru m associated to a bimonoidal category with anti-inv olution. Pa rticular examples of such are braided bimonoidal categories. W e inves tigate examples such as K ( k u ), K ( k o ), and W aldhausen’s A -theory of spaces of the form B B G , for abelian groups G . W e sho w that the in volution agrees with the classic al one for a bimonoidal category associated to a ri ng and prov e that it is not t riv ial in the abov e men tioned examples. 1. Introduction Sev eral multiplica tiv e cohomolog y theories p ossess a sp ectrum mo del that is the ring sp ectrum asso ciated to a b im on oidal category . The passage from bimonoidal categories to sp ectra u s es the additive structure of the bimonoidal cate gory; its multiplicat ion is then u sed to obtain the ring stru cture. F or instance, in the case of sin gu lar cohomology with co efficien ts in a rin g R , H ∗ ( − ; R ), w e can view the ring R as a discrete bimonoidal catego ry . The asso ciated sp ectrum is the Eilen b erg-Mac Lane sp ectrum of the ring R , H R . In general, w e d en ote the sp ectrum asso ciated to a bimonoidal categ ory R by H R . The main result of [BDRR] identifies the algebraic K -theory of H R w ith an algebraic K - theory construction d efi ned in [BDR], K ( R ), whic h uses the ring-lik e features of R , n amely uses addition and multiplicatio n in R to build K -theory . W e w ill recall the construction of K ( R ) in Section 2. In some examples, one can therefore read off some extra structure on K ( R ) us in g this equiv- alence. F or instance, if R is a ring with anti-in vo lution, then there is an inv olution on t he K -theory of the rin g R and th is yields an in volutio n on K ( R R ) = K ( H R ) = K ( R ) where R R denotes the d iscr ete categ ory asso ciated to t he ring R . F or the bip erm utativ e category of complex vecto r spaces, V C , we obtain that K ( V C ) ∼ K ( H V C ) = K ( k u ) where k u denotes the co nnectiv e sp ectrum a sso ciated to complex top ological K -theory . As complex conjugation give s rise to an action of the group of order tw o on k u we obtain an induced action of Z / 2 Z on K ( k u ) and h ence on K ( V C ). The aim of this pap er is to place these tw o examples in a broader con text and to inv estigate further examples. On the one han d w e will construct an inv olution on K ( R ) for ev ery strictly bimonoidal category w ith an ti-in volution. P articular examples of suc h categories are braided bimonoidal catego ries. Hence in the sp ecial case w h ere the br aiding is symmetric we obtain bip ermutativ e categories as a class of examples. W e prov e that in the classica l case of K -theory of a rin g with an ti-inv olution our inv olution coincides with the classical one. F ur th ermore, w e 2000 Mathematics Subje ct Classific ation. P rimary 55S25; Secondary 19D10. Key wor ds and phr ases. A lgebraic K - theory , top ological K -theory , W aldhausen A -theory , inv olution. The author would like t o thank Kob e Universit y for the hospitality during her stay in Marc h 2008 which wa s partially supp orted by Gran t- in-Aid for Scientific R esearch (C) 1954012 7 of the Japan Society for the Pro motion of Science. She thanks Christian Ausoni for asking a question that led to an important correction and Hann ah K¨ onig f or spottin g some anno ying t yp os. Nov emb er 9, 2021. 1 will consider bimonoidal catego ries w ith group actions and inv estigate h o w these relate to the constructed inv olution. W e close with the example of the inv olution on W aldhausen’s A -theory of a space X for sp aces of the form X = B B G for an ab elian grou p G . W e sho w that in seve ral cases suc h as K ( k o ), K ( k u ) and A ( B B G ), our inv olution is non-trivial. The adv an tage of our construction of an in v olution is th at it is relativ ely easy to describ e: it is of a pu r ely com binatorial nature th at mimics the construction o f t he inv olution on the algebraic K -theory of rings w ith an ti-in volution. 2. K -theor y of bimon oidal c a tegories Roughly s p eaking, a (strict) bimonoidal cat egory R is a category with t wo binary op erations, ⊗ and ⊕ , that let R b eha ve like a rig – a ring without additiv e inv erses. More pr ecisely , for eac h pair of ob j ects A, B in R there are ob j ects A ⊕ B and A ⊗ B in R and w e assume s tr ict asso ciativit y for b oth o p erations. T here a re ob jects 0 R ∈ R and 1 R ∈ R that are strict ly neutral with resp ect to ⊕ resp. ⊗ and there are isomorphisms c A,B ⊕ : A ⊕ B → B ⊕ A with c B ,A ⊕ ◦ c A,B ⊕ = id . E verything in sigh t is n atur al and satisfies co herence conditions. The ad d ition ⊕ and the multiplicati on ⊗ are related via distributivit y la ws. The complete list of axioms can b e found in [EM, defin ition 3.3], with the sligh t difference that we d emand the left d istributivit y map d ℓ : A ⊗ B ⊕ A ′ ⊗ B → ( A ⊕ A ′ ) ⊗ B to b e the iden tit y and d r to b e a natural isomorphism. Similarly to [M, VI, P rop osition 3.5] one can see th at ev ery bimonoidal category is equiv alen t to a strict one, so there it is no loss of generalit y to assum e strictness. The ring-lik e features of b imonoidal categories allo w it to consid er matrices and a lgebraic K -theory of suc h categ ories. In th e follo wing we recall some definitions and resu lts from [BDR]. Definition 2.1. [BDR, definition 3.2] Th e c ate gory of n × n -matric es over R , M n ( R ), is defined as follo ws. The ob jects of M n ( R ) are matrices A = ( A i,j ) n i,j =1 of ob jects of R and morph isms from A = ( A i,j ) n i,j =1 to C = ( C i,j ) n i,j =1 are matrices φ = ( φ i,j ) n i,j =1 where eac h φ i,j is a morphism in R from A i,j to C i,j . Lemma 2.2. [BDR, prop osition 3.3 ] F or a bimonoidal c ate gory ( R, ⊕ , 0 R , c ⊕ , ⊗ , 1 R ) the c ate- gory M n ( R ) is a mono idal c ate gory with r esp e ct to the matr ix multiplic ation bifunctor M n ( R ) × M n ( R ) · − → M n ( R ) ( A i,j ) n i,j =1 · ( B i,j ) n i,j =1 = ( C i,j ) n i,j =1 with C i,j = n M k =1 A i,k ⊗ B k ,j The unit of this structur e is given by the unit matrix obje ct E n which has 1 R as diagonal entries and 0 R in the other plac es. In the follo win g w e will assu me that the category R is small. As R is b imonoidal, its set of path comp onen ts π 0 ( R ) h as a structure of a rig, and its group completion, Gr ( π 0 ( R )) = ( − π 0 R ) π 0 R , is a ring. Definition 2.3. [BDR, d efinition 3.4] W e defin e the monoid of invertible n × n -matric es over π 0 ( R ), GL n ( π 0 ( R )), to b e the n × n -matrices o v er π 0 ( R ) that are inv ertible as matrices o v er Gr ( π 0 ( R )). Note that GL n ( π 0 ( R )) is the pullbac k in the d iagram GL n ( π 0 R ) / /     GL n ( Gr ( π 0 R ))     M n ( π 0 R ) / / M n ( Gr ( π 0 R )) 2 F or in stance, if π 0 ( R ) is the r ig of n atural num b ers in cluding zero, N 0 , th en the element s in GL n ( N 0 ) are n × n -matrices o v er N 0 that are inv ertible if they are considered as matrices with integ ral entries, i . e. , GL n ( N 0 ) = M n ( N 0 ) ∩ GL n ( Z ) consists of matrices in M n ( N 0 ) with determinan t ± 1. Definition 2.4. [BDR, defi n ition 3.6] The c ate gory of we akly invertible n × n -matric es over R , GL n ( R ), is the full sub categ ory of M n ( R ) with ob jects all m atrices A = ( A i,j ) n i,j =1 ∈ M n ( R ) whose matrix of π 0 -classes [ A ] = ([ A i,j ]) n i,j =1 is cont ained in GL n ( π 0 ( R )). Matrix m ultiplication is compatible with the p rop erty of b eing wea kly inv ertible and hence the category GL n ( R ) inherits a monoidal structure from M n ( R ). W e recall the defin ition of the b ar construction of monoidal categories from [BDR, definition 3.8]. Definition 2.5. Let ( C , · , 1) b e a monoidal category . Th e b ar c onstruction of C , B ( C ), is a simplicial catego ry . Let [ q ] b e the ordered set [ q ] = { 0 < 1 < . . . < q } . An ob ject A in B q ( C ) consists of the follo wing data. (a) F or eac h 0 6 i < j 6 q there is an ob ject A ij in C . (b) F or eac h 0 6 i < j < k 6 q there is an isomorphism φ ij k : A ij · A j k → A ik in C suc h that for all 0 6 i < j < k < l 6 q the f ollo wing diagram commutes ( A ij · A j k ) · A k l φ ij k · id   ∼ = / / A ij · ( A j k · A k l ) id · φ j kl   A ik · A k l φ ikl / / A il A ij · A j l . φ ij l o o A morphism f : A → B in B q C consists of m orp hisms f ij : A ij → B ij in C su c h that for all 0 6 i < j < k 6 q f ik φ ij k = ψ ij k ( f ij · f j k ) : A ij · A j k → B ik . Here, the ψ ij k : B ij · B j k → B ik denote the s tructure maps of B . The simplicial structure is as follo w s : if ϕ : [ q ] → [ p ] ∈ ∆ the fu nctor ϕ ∗ : B p ( C ) → B q ( C ) is obtained by precomp osing with ϕ . In order to allo w for degeneracy maps s i w e us e the con ven tion that all ob j ects of the f orm A ii are the un it of th e monoidal str u cture. The K -theory of the bimonoidal catego ry R can no w b e defi n ed as usual. W e tak e the bar constructions of the monoidal categories GL n R for all n > 0, r ealise them, tak e the disj oin t union of all of these and group complete. Definition 2.6. [BDR, definition 3.12] F or any bimonoidal category R its K - the ory is K ( R ) = Ω B ( G n > 0 | B GL n R| ) . Note that K ( R ) is w eakly equiv alent to K f 0 (( − π 0 R ) π 0 R ) × | B GL R| + , where K f 0 (( − π 0 R ) π 0 R ) denotes th e free K -theory of the ring ( − π 0 R ) π 0 R = Gr ( π 0 R ). The main r esu lt of [BDRR, theorem 1.1] is the id entificatio n of K ( R ) with the algebraic K -theory of the ring sp ectrum asso ciated to R , H R , if R is a small top ologic al bimonoidal catego ry satisfying the follo wing conditions: • All morphisms in R are isomorphism s. • F or ev ery ob j ect X ∈ R the translation functor X ⊕ ( − ) is faithful. 3 3. Bimonoidal ca tegories with anti-inv olution In ord er to define an inv olution of K ( R ) we need to assu me s ome extra structur e on our bimonoidal category R , n amely the existence of an anti-i n v olution on R . Da vid Barnes considers in v olutions on monoidal categ ories in [B, section 7]. W e ha v e to incorp orate the full bimonoidal structure, but s ome of our axioms b elo w relate to h is. Definition 3.1. An anti-involution in a strictly bimonoidal catego ry R consists of a functor ζ : R → R w ith ζ ◦ ζ = id and suc h that there are n atural isomorph isms (1) µ A,B : ζ ( A ⊗ B ) → ζ ( B ) ⊗ ζ ( A ) for all A, B ∈ R . In addition, the functor ζ and the isomorphisms µ hav e to satisfy the follo wing prop erties. (a) T he functor ζ is strictly s y m metric monoidal w ith resp ect to ( R , ⊕ , 0 R , c ⊕ ) [ML, XI.2]. (b) T he m ultiplicativ e u nit 1 R is fixed und er ζ , i. e. , ζ (1 R ) = 1 R and µ 1 R ,A = id ζ ( A ) = µ ( A, 1 R ). (c) T he isomorphisms µ are associativ e in the sense that the d iagram ζ ( A ⊗ B ⊗ C ) µ A ⊗ B,C / / µ A,B ⊗ C   ζ ( C ) ⊗ ζ ( A ⊗ B ) id ⊗ µ A,B   ζ ( B ⊗ C ) ⊗ ζ ( A ) µ B,C ⊗ id / / ζ ( C ) ⊗ ζ ( B ) ⊗ ζ ( A ) comm u tes for all A, B , C ∈ R . (d) T he distrib utivit y isomorph isms d ℓ and d r and the isomorphisms µ render th e follo wing diagrams comm utativ e ζ ( A ⊗ B ⊕ A ⊗ C ) ζ ( d r ) / / µ A ⊗ B ⊕ µ A ⊗ C   ζ ( A ⊗ ( B ⊕ C )) µ A,B ⊕ C   ζ ( B ) ⊗ ζ ( A ) ⊕ ζ ( C ) ⊗ ζ ( A ) d ℓ / / ( ζ ( B ) ⊕ ζ ( C )) ⊗ ζ ( A ) , ζ ( A ⊗ C ⊕ B ⊗ C ) ζ ( d ℓ ) / / µ A ⊗ C ⊕ µ B ⊗ C   ζ (( A ⊕ B ) ⊗ C ) µ A ⊕ B,C   ζ ( C ) ⊗ ζ ( A ) ⊕ ζ ( C ) ⊗ ζ ( B ) d r / / ζ ( C ) ⊗ ( ζ ( A ) ⊕ ζ ( B )) . F or a bimonoidal category w ith anti-in vo lution ( R , ζ , µ ) the ob jects that are fixed un der the an ti-in volutio n ζ do not form a b imonoidal category in general. They carry a p erm utativ e structure with r esp ect to ⊕ . Remark 3.2. In th e case of rings an ant i-in v olution is a map fr om a ring R to the r ing R o where R o has the same additiv e structur e as R bu t has rev ersed multiplica tion. In a similar sp irit one can d efine a bimonoidal category R o for an y bim on oidal category R where the m ultiplicativ e structure is r eversed. Ho wev er, the left d istributivit y in R o is th en no iden tity an y longer b ecause it corresp onds to the right d istributivit y la w in R . F or a bimonoidal category with an ti-in volutio n, R , ζ can b e view ed as a lax morphism s of bimonoidal categories from R to R o in adaptation of [BDRR , defin ition 2.6] to a setting with d ℓ 6 = id. Definition 3.3. A morphism of bimonoidal c ate gories with anti-involution , F : ( R , ζ , µ ) → ( R ′ , ζ ′ , µ ′ ), is a lax bimonoidal functor F : R → R ′ with the additional prop erties that F ◦ ζ = ζ ′ ◦ F and that µ and µ ′ are compatible with th e transformations (2) λ A,B : F ( A ) ⊗ F ( B ) → F ( A ⊗ B ) 4 in the sen s e that the diagram F ( ζ ( A )) ⊗ F ( ζ ( B )) λ A,B / / F ( ζ ( A ) ⊗ ζ ( B )) F ( ζ ( B ⊗ A )) F ( µ ) o o ζ ′ ( F ( A )) ⊗ ζ ′ ( F ( B )) ζ ′ ( F ( B ) ⊗ F ( A )) ζ ′ ( λ ) / / µ ′ o o ζ ′ ( F ( B ⊗ A )) comm u tes for all A, B in R . W e prolong the an ti-inv olution ζ to the catego ry of matrices M n ( R ) co ordin atewise, so for an y A = ( A i,j ) i,j ∈ M n ( R ) ζ (( A i,j ) i,j ) = ( ζ ( A i,j )) i,j . If the matrix A is an elemen t in GL n ( R ) then so is ζ ( A ) and ζ ( E n ) = E n . 4. The anti-inv olution on K ( R ) Regardless of the sp ecial form of the bimonoidal category with anti-in vo lution ( R , ζ , µ ), the com b inatorial nature of the bar constru ction B GL ( R ) allo ws for a canonica l in volutio n map. In the follo wing R is alwa ys a fixed b im on oidal category with an ti-in volution. Definition 4.1. F or a matrix of ob jects A ∈ M n ( R ) the tr ansp ose of A , A t , has A t i,j = A j,i as en tries. F or a m orp hism φ : A → C in M n ( R ) we d efi ne φ t as φ t i,j := φ j,i : A j,i = A t i,j → C t i,j = C j,i . F or a general bimonoidal cat egory , the form ula that w e are used to, namely ( A · B ) t = B t · A t do es not h old on the nose, bu t only up to a t w ist. W e h a v e ( A · B ) t i,j = ( A · B ) j,i = n M k =1 A j,k ⊗ B k ,i whereas ( B t · A t ) i,j = n M k =1 B t i,k ⊗ A t k ,j = n M k =1 B k ,i ⊗ A j,k . Using the structure maps µ of the anti -in v olution on R , w e can then defin e µ = L n k =1 µ A j,k ,B k,i and obtain a n atural m ap from ( ζ ( A · B )) t to ζ ( B ) t · ζ ( A ) t . The map µ b eha v es well on morphisms. Lemma 4.2. F or morphisms φ : A → C and ψ : B → D in M n ( R ) the fol lowing diagr am c ommutes ( ζ ( A · B )) t µ / / ( ζ ( φ · ψ )) t   ζ ( B ) t · ζ ( A ) t ζ ( ψ ) t · ζ ( φ ) t   ( ζ ( C · D )) t µ / / ζ ( D ) t · ζ ( C ) t Pr o of. The ( i, j ) matrix comp onent of the diagram ab o ve is L n k =1 ζ ( A j,k ⊗ B k ,i ) L k µ A j,k ,B k,i / / L k ζ ( φ j,k ⊗ ψ k,i )   L n k =1 ζ ( B k ,i ) ⊗ ζ ( A j,k ) L k ζ ( ψ k,i ) ⊗ ζ ( φ j,k )   L n k =1 ζ ( C j,k ⊗ D k ,i ) L k µ C j,k ,D k,i / / L n k =1 ζ ( D k ,i ) ⊗ ζ ( C j,k ) and this commutes b ecause µ is n atural.  5 Definition 4.3. Let A 0 , 1 . . . A 0 ,q . . . . . . A q − 1 ,q together with coherent isomorphisms φ i,j,k : A i,j · A j,k → A i,k , 0 6 i < j < k 6 q , b e an element in B q GL n ( R ). W e define τ : B q GL n ( R ) → B q GL n ( R ) via τ : A 0 , 1 . . . A 0 ,q . . . . . . A q − 1 ,q 7→ ( ζ ( A q − 1 ,q )) t . . . ( ζ ( A 0 ,q )) t . . . . . . ( ζ ( A 0 , 1 )) t . Let B i,j denote ( ζ ( A q − j,q − i )) t . T he corresp ondin g isomorphisms τ ( φ ) i,j,k : B i,j · B j,k → B i,k for 0 6 i < j < k 6 q are giv en by (3) τ ( φ ) i,j,k : ζ ( A q − j,q − i )) t · ( ζ ( A q − k ,q − j )) t µ − 1   ( ζ ( A q − k ,q − j · A q − j,q − i )) t ( ζ ( φ q − k ,q − j,q − i )) t / / ( ζ ( A q − k ,q − i )) t . Let α = α A,B ,C : A · ( B · C ) − → ( A · B ) · C b e the natural associativit y isomorph ism in th e monoidal str u cture of ( GL n R , · , E n ). W e can express α in terms of d istributivit y m aps and additiv e t wist m aps as follo w s: let σ b e the add itiv e t wist σ : n M k =1 n M ℓ =1 A i,k ⊗ B k ,ℓ ⊗ C ℓ,j − → n M ℓ =1 n M k =1 A i,k ⊗ B k ,ℓ ⊗ C ℓ,j that exc hanges th e p riorit y of s ummation of the tw o sums. T h en (4) α = d ℓ ◦ σ ◦ d − 1 r = σ ◦ d − 1 r . Here, the distribu tivit y la w is applied to sums of n en tries. This do es not cause problems as addition is assum ed to b e strictly asso ciativ e. Th e fact that α satisfies MacLane’s p enta gon axiom [ML, VI I.1(5)] can b e seen b y brute-force comparison of terms using the axioms [EM, definition 3.3]. Lemma 4.4. The asso ci ativity isomorp hism for matrix multip lic ation, α , and the isomorphisms µ ar e c omp atible, i.e. , they satisfy (5) (id · µ ) ◦ µ ◦ ζ ( α ) t = α − 1 ◦ ( µ · id) ◦ µ . Pr o of. T o ease notation, w e will abbreviate A ⊗ B to AB . The ( i, j ) matrix comp onen t of the equation (id · µ ) ◦ µ ◦ ζ ( α ) t = α − 1 ◦ ( µ · id) ◦ µ : ζ ( A · ( B · C )) t − → ζ ( C ) t · ( ζ ( B ) t · ζ ( A ) t ) 6 that we wan t to ha v e is part of th e diagram L n k =1 L n ℓ =1 ζ ( A j,k B k ,ℓ C ℓ,i ) σ / / ζ ( d r )   L L µ   L n ℓ =1 L n k =1 ζ ( A j,k B k ,ℓ C ℓ,i ) ζ ( d ℓ )   L L µ   ζ ( L n k =1 A j,k ( L n ℓ =1 B k ,ℓ C ℓ,i )) ζ ( α ) / / µ   ζ ( L n ℓ =1 ( L n k =1 A j,k B k ,ℓ ) C ℓ,i ) µ   ⊳ d ℓ / / L L µ ⊗ id ) ) L n k =1 ( L n ℓ =1 ζ ( B k ,ℓ C ℓ,i )) ζ ( A j,k ) µ · id   L n ℓ =1 ζ ( C ℓ,i ) ( L n k =1 ζ ( A j,k B k ,ℓ )) id · µ   ⊲ d r o o L L id ⊗ µ u u L n k =1 ( L n ℓ =1 ζ ( C ℓ,i ) ζ ( B k ,ℓ )) ζ ( A j,k ) α − 1 / / L n ℓ =1 ζ ( C ℓ,i ) ( L n k =1 ζ ( B k ,ℓ ) ζ ( A j,k )) L n k =1 L n ℓ =1 ζ ( C ℓ,i ) ζ ( B k ,ℓ ) ζ ( A j,k ) d ℓ O O σ − 1 = σ / / L n ℓ =1 L n k =1 ζ ( C ℓ,i ) ζ ( B k ,ℓ ) ζ ( A j,k ) d r O O Here, the symb ol ⊳ on the left hand side stand s for L n k =1 L n ℓ =1 ζ ( B k ,ℓ ⊗ C ℓ,i ) ⊗ ζ ( A j,k ) and the ⊲ on the right hand side is sh ort for L n ℓ =1 L n k =1 ζ ( C ℓ,i ) ⊗ ζ ( A j,k ⊗ B k ,ℓ ) . F rom the d efinition of an anti- in v olution w e kno w that the top triangles and the outer diagram comm ute. Naturalit y of the d istributivit y transformations makes the b ottom triangles commute and th erefore the square in the middle comm utes as well.  Lemma 4.5. The isomorphisms τ ( φ ) i,j,k as in (3) ar e c oher ent. Pr o of. Recall that the φ i,j,k are the co herence isomorphisms for the tr iangle of matrices ( A i,j ) i,j and that B ij = ( ζ ( A q − j,q − i )) t . W e ha v e to pro v e that the follo wing diagram comm utes. (6) B ij · ( B j k · B k ℓ ) α / / id · τ ( φ ) j,k,ℓ   ( B ij · B j k ) · B k ℓ τ ( φ ) i,j,k · id   B ij · B j ℓ τ ( φ ) i,j,ℓ # # G G G G G G G G B ik · B k ℓ τ ( φ ) i,k,ℓ { { x x x x x x x x B iℓ As τ ( φ ) j,k ,ℓ is the comp osition ( ζ ( φ q − ℓ,q − k ,q − j )) t ◦ µ − 1 , as we kn o w from naturalit y of µ that µ − 1 ◦ (( ζ ( φ q − k ,q − j, q − i )) t · id ) = µ − 1 ◦ (( ζ ( φ q − k ,q − j, q − i )) t · id t ) = (id · ζ ( φ q − k ,q − j, q − i )) t ◦ µ − 1 and as we h a ve Lemma 4.4, it suffices to show that the diagram ζ ( A q − ℓ,q − k · ( A q − k ,q − j · A q − j,q − i )) t ζ ( α ) t / / ζ (id · ( φ q − k ,q − j,q − i )) t   ζ (( A q − ℓ,q − k · A q − k ,q − j ) · A q − j,q − i ) t ( ζ ( φ q − ℓ,q − k ,q − j ) · id) t   ζ ( A q − ℓ,q − k · A q − k ,q − i ) t ζ ( φ q − ℓ,q − k ,q − i ) t ) ) R R R R R R R R R R R R R R ζ ( A q − ℓ,q − j · A q − j,q − i ) t ζ ( φ q − ℓ,q − j,q − i ) t u u l l l l l l l l l l l l l l ζ ( A q − ℓ,q − i ) t comm u tes. As b oth transp osition and ζ a re fu nctors, the comm utativit y of th is diagram is equiv alen t to th e equalit y φ q − ℓ,q − j, q − i ◦ ( φ q − ℓ,q − k ,q − j · id ) ◦ α = φ q − ℓ,q − k ,q − i ◦ (id · φ q − k ,q − j, q − i ) and this holds b ecause the isomorphism s ( φ q − ℓ,q − j, q − i ) are coheren t.  7 Remark 4.6. If G is a group, then the in v erse map induces a map on the lev el of classifying spaces B ι : B G → B G op . Here, G op is the group G with opp osite multiplicati on. This map is homotopic to the map κ : B G → B G op whic h sends (( g 1 , . . . , g q ) , ( t 0 , . . . , t q )) ∈ B q G to (( g q , . . . , g 1 ) , ( t q , . . . , t 0 )) ∈ B G op (see [BF, p. 206] for an explicit homotop y). Note that κ can b e d efi ned for monoids as we ll, in particular it app lies to the monoid of w eakly in v ertible matrices o v er a b imonoidal category R . Let r : ∆ op → ∆ op [St, (3.14)] b e the f ollo wing functor: on ob jects r is the identi t y . If f = g op : [ p ] → [ q ] is a morp hism in ∆ op then r ( f ) is the opp osite of the monotone map th at is giv en b y i 7→ p − g ( q − i ) , for all 0 6 i 6 q . If δ i : [ q ] → [ q + 1] d enotes the map th at is the inclusion that misses i and is strictly monotone ev er y w here else and if σ i : [ q ] → [ q − 1] is the surjection th at sends i an d i + 1 to i and is strictly monotone elsewhere, then note that r ( s i ) = r (( σ i ) op ) = ( σ q − i − 1 ) op = s q − i − 1 , r ( d i ) = r (( δ i ) op ) = ( δ q − i − 1 ) op = d q − i − 1 . Let CA T den ote the catego ry of small categories. Lemma 4.7. If ˜ B GL n ( R ) denotes the b ar c onstruction of GL n R with r esp e ct to the simplicial structur e ∆ op r / / ∆ op B GL n R / / CA T , then τ induc es a wel l-define d map of simplicial c ate gories τ : B GL n R → ˜ B GL n R for al l n . Pr o of. The argument is straightforw ard for s i = ( σ i ) op , using the fact that ζ resp ects u nit matrices. W e hav e to pro v e that the diagram B q GL n R τ / / d i   ˜ B q GL n R d q − i − 1   B q − 1 GL n R τ / / ˜ B q − 1 GL n R comm u tes. Mo ving an ti-clockwise sends a triangle of ob jects ( A k ℓ ) first to the triangle ( C k ℓ ) with C k ℓ = A δ i ( k ) ,δ i ( ℓ ) and then app lies τ to yield as ( k, ℓ )-entry ( ζ ( A δ i ( q − 1 − ℓ ) ,δ i ( q − 1 − k ) )) t . W alking clo c kwise means to app ly the in v olution τ first to obtain the triangle ( B k ℓ ) with B k ℓ = ( ζ ( A q − ℓ,q − k )) t . Afterwards the application of d q − i − 1 sends this tr iangle to ( D k ℓ ) with D k ℓ = ζ ( A q − δ q − i − 1 ℓ,q − δ q − i − 1 ( k ) ) t . As we h a v e δ i ( q − 1 − s ) = q − δ q − i − 1 ( s ) for all 0 6 s 6 q − 1, the claim f ollo ws.  Theorem 4.8. The involution τ gives rise to an involution on K ( R ) for every bimonoidal c ate gory with anti-involution ( R , ζ , µ ) . Pr o of. W e saw that the inv olution τ is a morp h ism of simplicial categories τ : B GL n R → ˜ B GL n R , th us it r emains to sh o w that the r ealizat ion of ˜ B GL n R , | ˜ B GL n R| is homeomorph ic to | B GL n R| and that the inv olution passes to the group completion. The first claim is easy to see, b ecause the self-map r on ∆ op amoun ts to a map on the realizatio n that reverses the co ord inates in the standard simplices. 8 As K ( R ) = Ω B ( F n > 0 | B GL n R| ), we ha v e to sh o w that τ is compatible w ith the monoid structure on F n > 0 | B GL n R| . Note, that the follo win g diagram comm utes B q GL n R × B q GL m R ⊕ / / ( τ ,τ )   B q GL n + m R τ   ˜ B q GL n R × ˜ B q GL m R ⊕ / / ˜ B q GL n + m R and therefore we obtain on the lev el of classifying spaces that | B GL n R| × | B GL m R| ⊕ / / ( | τ | , | τ | )   | B GL n + m R| τ   | ˜ B GL n R| × | ˜ B GL m R| ⊕ / / | ˜ B GL n + m R| comm u tes.  Prop osition 4.9. If F : ( R , ζ , µ ) → ( R ′ , ζ ′ , µ ′ ) is a morphism of b imonoidal c ate gories with anti-involution, then F c ommutes with the involutions on KR and K R ′ , i.e. , F ◦ τ = τ ◦ F , KR τ / / F   KR F   KR ′ τ / / KR ′ . Let R b e a asso ciativ e ring with unit. An anti-involution on R (called inv olution in [BF, definition 1.1]) is a f unction ι : R → R with ι ( ι ( a )) = a , ι ( a + b ) = ι ( a ) + ι ( b ) and ι ( ab ) = ι ( b ) ι ( a ) for all a, b ∈ R . Definition 4.10. If R is a ring or a rig, then the catego ry w h ic h has the elemen ts of R as ob jects and only iden tit y morph isms is a bim on oidal category . W e denote this category by R R and call it the discr ete c ate gory asso c iate d to the ring or rig R . I f R is comm utativ e, then R R is bip ermutativ e. If R is a ring then the sp ectrum asso ciated to R R is the Eilenberg-Mac Lane s p ectrum of the ring R . F or a rig R , we obtain the Eilenb er g-MacLane s p ectrum of the group completion Gr ( R ). Note that f or a small b imonoidal category with ant i-in v olution ( R , ζ , µ ), the set of path comp onent s π 0 ( R ), is a rig with an ti-in volutio n. Corollary 4.11. F or a smal l bimonoidal c ate gory with anti-involution ( R ′ , ζ , µ ) , the map K ( R ′ ) → K ( R π 0 ( R ′ ) ) c ommutes with the involutions on K ( R ′ ) and K ( R π 0 ( R ′ ) ) ≃ K f ( Gr ( π 0 ( R ′ ))) . Prop osition 4.12. F or a ring with anti-involution the involution c onstructe d on K ( R R ) agr e e s with the standar d involution on K i ( R ) , i > 1 . Pr o of. As GL m R R is a strict monoidal category , the b ar constru ction fr om Section 2 is equiv- alen t to the ordinary b ar construction [BDRR, corollary 8.5] and the isomorphism from the ordinary bar constru ction to the one in the monoidal setting is giv en b y send ing a q -simplex of the ordinary bar construction ( B 0 , . . . , B q ) to the triangle in B q GL n R R A 0 , 1 . . . A 0 ,q . . . . . . A q − 1 ,q with en tries A i,i +1 = B i on the diagonal. The other en tries a re giv en b y iterated matrix m ultiplication of the B i s and the isomorphisms φ ij k are c h osen to b e iden tit y maps . On the diagonal th e inv olution τ sends ( B 0 , . . . , B q ) to ( ζ ( B q ) t , . . . , ζ ( B 0 ) t ) and this is pr ecisely what the standard in volutio n in alg ebraic K -theory does (compare for instance [BF, definition 1.1 2]).  9 Note, that if one is willing to w ork a w a y from the prim e 2, then in v olutions giv e rise to splittings K ( R ) ∼ K ( R ) a × K ( R ) s of K ( R ) in to an an tisymm etric part, K ( R ) a , and a symmetric part, K ( R ) s . Corollary 4.11 tells us that su c h splittings are compatible with the path comp onen t map. Remark 4.13. There is no straigh tforward w a y to mimic Burghelea’s and Fiedoro wicz’s con- struction of hermitian K -theory in th e setting of b imonoidal categories w ith anti -in v olution. There are t w o main obstacles: m atrix m ultiplication is n ot a sso ciativ e an y lo nger and we do not demand that the s tructure isomorsphism µ is the ident it y . This has th e effect that the analog ue of their category ε O n [BF, 1.2] in th e bimonoidal w orld do es not giv e a strict category . Similarly , their b ar construction description of ε O n do es not hav e a d irect analog ue. In order to form the one-sided bar construction B 1 (Sym 1 n ( R ) , GL n ( R ) , ∗ ) in the spirit of [BF , 1.3] one has to ha v e an action of the monoidal category GL n ( R ) on the category of sym m etric matrices Sym 1 n ( R ). Here, t he ob jects of Sym 1 n ( R ) are matrices A ∈ GL n ( R ) with ζ ( A ) t = A and morphisms are morph isms in GL n ( R ) that are unto uc hed by ζ . But for M ∈ S ym 1 n ( R ) and A ∈ GL n ( R ) the ob ject ( ζ ( A ) t · M ) · A will only b e symmetric u p to isomorphism in general. The inv olution on her m itian K -theory [BF, 4.1] is induced b y the map that sends a sy m metric matrix A to its n egativ e. W e kno w fr om [BDRR] that K ( R ) is equiv alen t to K ( ¯ R ) f or some m ultiplicativ e group completion ¯ R of R and matrices o ver ¯ R ha v e additiv e in v er s es on the lev el of path comp onen ts. It is straigh tforw ard to co ok up other examples of bimonoidal categories with anti- in v olution along the follo wing lines. F or a discrete group G let ∨ G E b e the category with ob jects n g with n ∈ N 0 and g ∈ G . W e iden tify all ob jects 0 g to 0 whic h stand s for the empty set and the n g should b e thought of as the set { 1 , . . . , n } labelled b y g ∈ G . Morph isms are giv en by ∨ G E ( n g , m h ) =  ∅ n 6 = m Σ n n = m, g = h or n = m = 0 . The classifying sp ace of ∨ G E is B ( ∨ G E ) = _ G B ( E ) = _ G   G n > 0 B Σ n   . W e define a bimonoidal structure on ∨ G E as follo w s. Ob jects can only b e added if their indices agree: n g ⊕ m h =  ( n + m ) g g = h 0 g 6 = h and we define the m u ltiplication to b e n g ⊗ m h = ( n ⊗ m ) g h . The add itive t wist, c ⊕ on E G , is inherited from E , 0 is th e zero ob ject and 1 e is the m ultiplicativ e unit, if e d enotes the neutral elemen t of the group G . With this str ucture ∨ G E is a bimonoidal category; if G is ab elian, then ∨ G E is actually bip ermutativ e. W e can defin e an an ti-in vol ution on ∨ G E for an y d iscr ete G via ζ ( n g ) = n g − 1 . Note, that the isomorp h isms µ are not trivial in this case, but ζ ( n g ⊗ m h ) = ζ (( n ⊗ m ) g h ) = ( n ⊗ m ) ( gh ) − 1 = ( n ⊗ m ) h − 1 g − 1 6 = ζ ( m h ) ⊗ ζ ( n g ) = ( m ⊗ n ) h − 1 g − 1 10 so we d efine µ to b e c ⊗ where c ⊗ is the multi plicativ e t wist in the b ip ermutativ e structur e of E . W e h a ve that ζ (1 e ) = 1 e and condition (7 ) follo ws from the equation d r ◦ ( c ⊗ ⊕ c ⊗ ) = c ⊗ ◦ d ℓ in bip erm utativ e cate gories and the asso ciativit y of µ is a consequence of Lemma 5.3. The path comp onen ts of ∨ G E constitute th e monoid ring N 0 [ G ] and therefore w e obtain with Corollary 4.11 that the in duced map on K -theory K ( ∨ G E ) → K ( N 0 [ G ]) is compatible with the in v olutions on b oth sides. Note that K ( N 0 [ G ]) ∼ K ( Z [ G ]). 5. Braided bimonoidal and bipermut a tive ca tegor ies W e will sh ow that braided bimon oidal, and therefore in particular bip ermutativ e cate gories, pro vide examples of bimonoidal categorie s with anti -in v olution. Definition 5.1. A br aide d bimonoidal c ate gory ( R , ⊕ , 0 R , c ⊕ , ⊗ , 1 R , β ) consists of a p er mutativ e catego ry ( R , ⊕ , 0 R , c ⊕ ) and a strict b raided monoidal category ( R , ⊗ , 1 R , β ) (see [ML , XI.1]) where β is the br aiding β = β A,B : A ⊗ B − → B ⊗ A. These tw o structures interact via distrib utivit y la ws. W e assume that the left distrib utivit y isomorphism d ℓ : A ⊗ B ⊕ A ′ ⊗ B − → ( A ⊕ A ′ ) ⊗ B is th e identit y and that the right distribu tivit y isomorph ism is giv en in terms of d ℓ and β , su c h that the f ollo wing diagram comm utes (7) A ⊗ B ⊕ A ⊗ C β ⊕ β   d r / / A ⊗ ( B ⊕ C ) β   B ⊗ A ⊕ C ⊗ A β ⊕ β   d ℓ / / ( B ⊕ C ) ⊗ A β   A ⊗ B ⊕ A ⊗ C d r / / A ⊗ ( B ⊕ C ) . In addition w e wa n t that R satisfies the remaining axio ms of a bip erm utativ e categ ory in the sense of [EM, definition 3.6]. Note, that condition (7) implies that β ◦ β ◦ d ℓ = d ℓ ◦ ( β ⊕ β ) ◦ ( β ⊕ β ) is satisfied. Gerald Dunn studied braided b imonoidal categ ories and the reader might w ant to compare the ab o v e definition with [Du1, defin ition 3.1]. As a class of examples of b raided bimonoidal catego ries Dunn considered the category of what he called free crossed G -sets for a discrete group G [Du2, example 2.3]. F or every p erm utativ e category ( C , ⊕ , 0 C , c ⊕ ) one can construct the free b r aided b imonoidal catego ry B r ( C ) along the lines of the construction in [EM, theorem 10.1]. Consider the trans- lation category E B r n of the n -th b raid group B r n . Then B r ( C ) := G i > 0 E B r n × B r n C n is a braided bimonoidal category (see [Du1, prop osition 3.5]). W e present a different class of examples in S ection 7, 7.2. In order to c hec k that braided bimonoidal categ ories actually are bim on oidal categ ories with an ti-in volutio n and that they fit in th e setting of our d efi nition of K ( R ) in Section 2 w e will need t w o tec hnical r esults. 11 Lemma 5.2. Pr op erty (7) implies that the fol lowing diagr am c ommutes (8) A ⊗ B ⊗ C ⊕ A ⊗ B ′ ⊗ C d r   d ℓ / / ( A ⊗ B ⊕ A ⊗ B ′ ) ⊗ C d r ⊗ id   A ⊗ ( B ⊗ C ⊕ B ′ ⊗ C ) id ⊗ d ℓ / / A ⊗ ( B ⊕ B ′ ) ⊗ C Pr o of. W e embed d iagram (8) into the follo wing d iagram. In order to sav e space we use AB for A ⊗ B and A + B for A ⊕ B . AB C + AB ′ C d ℓ / / β ⊕ β ~ ~ } } } } } } } } } } } } } } } } ( I ) d r   ( AB + AB ′ ) C d r ⊗ id   ( β ⊕ β ) ⊗ id A A A A A A A A A A A A A A A A ( I V ) A ( B C + B ′ C ) β ~ ~ } } } } } } } } } } } } } } } } ( I I ) id ⊗ d ℓ / / A ( B + B ′ ) C β                  β ⊗ id A A A A A A A A A A A A A A A A ( I I I ) B C A + B ′ C A d ℓ   ( B A + B ′ A ) C d ℓ ⊗ id   ( B C + B ′ C ) A d ℓ ⊗ id / / ( B + B ′ ) C A ( B + B ′ ) AC id ⊗ β o o The leftmost sub d iagram ( I ) corresp ond s precisely to pr op erty (7). Diagram ( I I ) commutes b ecause β is natural and diagram ( I I I ) disp lays one of th e axioms for a braided monoidal catego ry and s ub d iagram ( I V ) again corresp onds to prop ert y (7 ). As th e left d istributivit y maps are identit ies, the ou ter diagram again corresp onds to the pr op ert y used in ( I I I ). T h erefore the embedd ed sub d iagram (8) comm utes as well.  This result ensures th at the set of axioms used in [BDRR] is fulfilled in the sett ing of braided bimonoidal categories. The next r esult is the k ey ingredien t that allo ws us to interpret braided bimonoidal categories as bim on oidal cate gories with an ti-in volutio n. Lemma 5.3. L et R b e a br aide d bimonoidal c ate gory. Then the br aiding β satisfies (id ⊗ β A,B ) ◦ β A ⊗ B ,C = ( β B ,C ⊗ id) ◦ β A,B ⊗ C . Pr o of. Consid er the follo wing diagram. A ⊗ C ⊗ B β A,C ⊗ id ( ( P P P P P P P P P P P P A ⊗ B ⊗ C id ⊗ β B,C 7 7 n n n n n n n n n n n n β A,B ⊗ C A A A A A A A A A A A A A A A A A A A β A ⊗ B,C / / β A,B ⊗ id   C ⊗ A ⊗ B id ⊗ β A,B   B ⊗ A ⊗ C id ⊗ β A,C ( ( P P P P P P P P P P P P C ⊗ B ⊗ A B ⊗ C ⊗ A β B,C ⊗ id 7 7 n n n n n n n n n n n n The t w o triangles d ispla y a coherence relation for b raided monoidal categories and thus they comm u tes. T he outer diagram is the Y ang-B axter equation for the braiding and th u s the whole diagram is commuta tiv e.  Prop osition 5.4. Every br aide d bimonoida l c ate gory is a bimonoidal c ate gory with anti-involution if one defines ζ to b e the identity and µ = β . In p articular, every b i p ermutative c ate gory i s a bimonoidal c ate gory with anti-involution with ζ = id and µ = c ⊗ . Pr o of. The claim f ollo ws directly from Lemma 5.3, b ecause all other p arts of the structure of a bimonoidal category with an ti-inv olution are trivial.  12 Note, that a morph ism of bimonoidal catego ries with anti- in v olution as in Defin ition 3.3 sp ecializes to the requirement of b eing a lax symmetric b imonoidal functor in the case of b ip er- m utativ e catego ries. 6. Group actions Let G b e a discrete group. Definition 6.1. (a) L et R b e a b imonoidal category and let G b e a discrete group . A G -action on R consists of a f unctor φ g : R → R for every g ∈ G , su ch that ev ery φ g is a strict bimon oidal f unctor and φ 1 = id , φ g ◦ φ h = φ g h , for all g , h ∈ G. (b) F or a bimonoidal catego ry w ith an ti-inv olution we require eac h φ g in addition to b e a morphism of bimonoidal cat egories with an ti-in vo lution according to Definition 3.3. Example 6.2. Th e b ip ermutativ e category of complex v ector spaces, V C , with ob jects the natural num b ers with zero and morp hisms V C ( n, m ) =  ∅ n 6 = m U ( n ) n = m carries a Z / 2 Z -actio n. On ob jects the action is trivial, and on morphism s it is g iv en b y co mplex conjugation of unitary matrices. Note that the action is n on-trivial on the endomorphisms U (1) of the multiplica tiv e unit. Example 6.3. Let A → B b e a G -Ga lois extension of comm u tativ e r ings in the sen se of [CHR]. W e can consid er the discrete bip ermutativ e categories R A and R B as in 4.10. Th en R B is a bip ermutativ e category with G -action. Definition 6.4. F or a bimonoidal category R with G -action, the G -fixe d c ate gory is the sub- catego ry of R contai ning all ob jects and morp hisms that are fixed under every φ g , g ∈ G . W e denote this category b y R G . The follo wing result is straigh tforward to see. Lemma 6.5. The G -fixe d c ate gory of a strict G -action on a b i monoidal c ate gory (with anti- involution) is again a bimonoidal c ate gory (with anti-i nvolution). Example 6.6. If R is a ring with a G -action, then the G -fixed ca tegory of R R is the bim on oidal catego ry associated to the G -fixed su bring of R . Example 6.7. F or the category V C the Z / 2 Z -fixed category is the bip erm utativ e category of real v ecto r s p aces, V R , wh ose ob jects are again th e natur al num b ers, but wh ose morp hisms are giv en b y V R ( n, m ) =  ∅ n 6 = m O ( n ) n = m. Note, that th e homotop y fixed p oint sp ectrum H V h Z / 2 Z C is k u h Z / 2 Z and this is not equiv alen t to the asso ciated sp ectrum k o = H V R . In the case of Eilen b erg-Mac Lane sp ectra, ho w ever, w e obtain that H R hG = H R hG R ≃ H ( R G ) = H R R G . Moreo ver, if A → B is a G -Galoi s extension of comm u tative rin gs, then H A = H R A → H R B = H B is a G -Galois extension of comm utativ e S -algebras in the sens e of Rognes [Ro , p rop osition 4.2.1]. Prop osition 6.8. L et R b e a (symmetric) bimonoidal c ate gory with G -action. Then the we ak e quivalenc e [BDRR, theorem 1.1] K ( R ) ≃ K ( H R ) is G -e quivariant. 13 Pr o of. All constructions in v olv ed in the pro of of [BDRR, th eorem 1.1] are natural with resp ect to lax (symmetric) b imonoidal fun ctors.  Remark 6.9. As G -actio ns on bimonoidal categories w ith an ti-inv olution are give n in terms of morphisms of su c h categories, they can b e com bined with the external inv olution on the bar construction for K ( R ). 7. Examples 7.1. Endomorphisms of a p e rmutativ e category . Let ( C , ⊕ , 0 C , c ⊕ ) b e any p ermuta tiv e catego ry . C onsider the catego ry of all lax symmetric monoidal fun ctors from C to itself. El- mendorf and Mandell [EM, p. 176] describ e ho w to imp ose a bimonoidal stru cture on this catego ry . W e denote this categ ory b y E n d( C ). The addition is giv en “p oint wise”, i.e. , for t w o lax symmetric m onoidal fun ctors F , G : C → C one defin es ( F ⊕ G )( C ) = F ( C ) ⊕ G ( C ) . The multiplic ativ e stru cture is giv en by comp osition. If w e consider the full sub categ ory of End( C ) of in vertible lax sym metric monoidal functors and w e tak e the bimonoidal sub category of End( C ) generated b y these under direct sum and comp osition whic h w e call In v( C ). One might th in k of In v ( C ) as the gr oup-rig of the categ ory C . W e can define an inv olution on I n v( C ) b y sending a generator F ∈ In v ( C ) to its in v ers e ζ ( F ) = F − 1 and prolonging this inv olution to finite w ords (under ⊕ and ◦ ) in suc h functors. F or instance, w e ha v e ζ ( G 1 ⊕ G 2 ) = G − 1 1 ⊕ G − 1 2 . As w e hav e ( G ◦ F ) − 1 = F − 1 ◦ G − 1 w e can choose µ to b e the identit y . Group actions on (symmetric) bimonoidal catego ries pro vide non-trivial examples. If a dis- crete group G acts on a (sym metric) bimonoidal category R , then the element s of th e grou p are ob jects of th e category Inv( R ). F or instance the category of complex vec tor sp aces V C with its Z / 2 Z -action give s rise to a non-trivial category In v( V C ). If R is a ring with G -action, then the category F ( R ) with ob jects n ∈ N 0 and morphisms the R -automorphisms of R n is a b imonoidal catego ry w ith G -action. The actio n is tr ivial on ob jects and it sends an automorph ism ϕ to g ϕ for g ∈ G wh ere g ϕ is the morph ism that sends v ∈ R n to g ϕ ( v ). 7.2. Hopf-bimo dules. C ategories of Hopf-bimo dules p ro vide a class of examples of (non- strict) braided bimonoidal categories. Consider a Hopf algebra H in a symmetric monoidal catego ry . An ob ject M is an H Hopf-bimo d ule if it is a bimo d ule o v er H and simultaneously a H righ t- and left-comod ule su c h that the como dule str ucture maps are morph isms of H -bimo d ules. Here, the diagonal on H giv es th e H -bim o dule structure on H ⊗ M and M ⊗ H . S c hauen burg sho w ed [Sch, theorem 6.3] th at the cat egory of H -Hopf-bimo dules, H H M H H , is a braided monoidal catego ry with the tensor pro du ct o v er H , if the an tip o de of the Hopf algebra H is inv ertible, and th at the category H H M H H is equiv alent to the category of right Y etter-Drinfel’d H -mo dules [Sc h, theorem 5.7 (3)] if the u n derlying catego ry has equalizers. Let us consider the sym metric monoidal catego ry of k -mo dules for some comm utativ e ring with unit, k , and the d irect su m as the additive s tr ucture. Unadorned tens or p ro du cts are tensor pr o ducts o v er k . The category of H -bimo dules, H M H , o v er a Hopf algebra H is th en a (non strict) bim on oidal category with th e direct su m of k -mo du les as additiv e and th e tensor pro du ct o ver H as m u ltiplicativ e stru cture. The dir ect su m of tw o k -mo du les A, B ∈ H M H is an H -bimo dule if w e declare the structure m ap s to b e H ⊗ ( A ⊕ B ) d − 1 r / / H ⊗ A ⊕ H ⊗ B / / A ⊕ B 14 and ( A ⊕ B ) ⊗ H d − 1 ℓ / / A ⊗ H ⊕ B ⊗ H / / A ⊕ B . Here, d r and d ℓ denote the distributivity isomorp hisms in the u nderlying category of k -mo dules, i.e. , d r : A ⊗ B ⊕ A ⊗ B ′ → A ⊗ ( B ⊕ B ′ ) , d ℓ : A ⊗ B ⊕ A ′ ⊗ B → ( A ⊕ A ′ ) ⊗ B . Similarly , the left and r igh t como d ule structures on A and B , ψ A , ψ A resp. ψ B , ψ B , giv e rise to a left an d a r igh t como dule structure on the su m via A ⊕ B ψ A ⊕ ψ B / / H ⊗ A ⊕ H ⊗ B d r / / H ⊗ ( A ⊕ B ) and A ⊕ B ψ A ⊕ ψ B / / A ⊗ H ⊕ B ⊗ H d ℓ / / ( A ⊕ B ) ⊗ H . It is tedious but straigh tforward to c h ec k th at the coherence isomorphism s of the bimonoidal catego ry of H -bimo dules are actually morphisms of como dules. The explicit form of the braiding from [Sch, theorem 6.3] allo ws it to chec k that condition (7) of Definition 5.1 is ind eed satisfied and that Laplaza’s distributivit y axioms [L, section 1] are sat isfied with the braiding β replacing the multiplica tiv e t wist. 7.3. In v olutions on A ( ∗ ) and A ( B B G ) . L et E denote the b ip ermutativ e catego ry of finite sets w hose ob j ects are th e fi nite sets n = { 1 , . . . , n } for n ∈ N 0 . By conv ent ion 0 is the empt y set. The morph isms in E are E ( n , m ) =  ∅ n 6 = m Σ n n = m. F or the full structur e see [M, VI, Examp le 5.1] or [BDRR, Example 2.4]. Its asso ciated sp ectrum is th e sp here sp ectrum and th us the equiv alence from [BDRR, theorem 1.1] identifies K ( E ) with the algebraic K -theory of th e sphere sp ectrum, K ( S ), whic h in turn is equiv alen t to W aldh ausen’s A -theory of a p oin t, A ( ∗ ). Steiner constructed an inv olution on A ( X ) for all spaces X in [St, theorem 3.10] where he u s ed the m o del for A ( X ) that consists of the algebraic K -theory of the spherical group ring of Ω X , K ( S [Ω X ]). He defined his in v olution a s th e comp osition of loop inv ersion, matrix transp osition and reversal of m ultiplication whic h in our con text is tak en care of b y the reflection map on the bar constru ction. Th us our definition of the in volutio n on K ( E ) yields a definition that resem bles his. Another description of inv olutions on W aldhausen’s K -theory of sp aces is due to V ogell [V]. F or a construction of sp ectrum level in v olutions on S [Ω M ] f or manifolds M see [K]. John Rognes drew m y atten tion to the example of fi nite f ree G -sets and G -equiv ariant bijec- tions. F or a group G w e consider the follo wing s mall v ersion of this category . W e define the catego ry E G whose ob jects are again the finite sets n = { 1 , . . . , n } for n ∈ N 0 with 0 = ∅ and whose morphism s are giv en b y E G ( n , m ) =  ∅ n 6 = m G ≀ Σ n n = m The classifying sp ace B ( E G ) is (9) G n > 0 B ( G ≀ Σ n ) = G n > 0 B G n × Σ n E Σ n . F or an ab elian group G w e defin e a b ip ermutativ e structure on E G as f ollo ws. O n ob jects, w e tak e the b ip ermuta tiv e structure [BDRR, example 2.4]), and on m orphisms we d efine ( g 1 , . . . , g n , σ ) ⊕ ( g ′ 1 , . . . , g ′ m , σ ′ ) = ( g 1 , . . . , g n , g ′ 1 , . . . , g ′ m , σ ⊕ σ ′ ) . for ( g 1 , . . . , g n , σ ) ∈ G ≀ Σ n and ( g ′ 1 , . . . , g ′ m , σ ′ ) ∈ G ≀ Σ m . There are natural isomorphisms c G ⊕ : ( g 1 , . . . , g n , σ ) ⊕ ( g ′ 1 , . . . , g ′ m , σ ′ ) → ( g ′ 1 , . . . , g ′ m , σ ′ ) ⊕ ( g 1 , . . . , g n , σ ) 15 for all ( g 1 , . . . , g n , σ ) and ( g ′ 1 , . . . , g ′ m , σ ′ ) that u se the additiv e t wist c ⊗ from the structur e of E and that shuffle the g i and g ′ j . It is straight forwa rd to c hec k, that ( E G, ⊕ , 0 , c G ⊕ ) is a p erm utativ e catego ry . As a multiplica tiv e stru cture we defi n e ( g 1 , . . . , g n , σ ) ⊗ ( g ′ 1 , . . . , g ′ m , σ ′ ) = ( g 1 g ′ 1 , . . . , g 1 g ′ m , . . . , g n g ′ 1 , . . . , g n g ′ m , σ ⊗ σ ′ ) . Note that for this constru ction to b e natural, the group G has to b e ab elian. W e can compare ( g 1 , . . . , g n , σ ) ⊗ ( g ′ 1 , . . . , g ′ m , σ ′ ) with ( g ′ 1 , . . . , g ′ m , σ ′ ) ⊗ ( g 1 , . . . , g n , σ ) by us in g n atural isomor- phisms c G ⊗ whic h are built out of the multiplicativ e t wist c ⊗ from E and which reorder arra ys lik e ( g 1 g ′ 1 , . . . , g 1 g ′ m , . . . , g n g ′ 1 , . . . , g n g ′ m ) to ( g ′ 1 g 1 , . . . , g ′ 1 g n , . . . , g ′ m g 1 , . . . , g ′ m g n ). With this m ultiplicativ e stru cture ( E G, ⊗ , 1 , c G ⊗ ) is a p erm utativ e category and the m ulti- plicativ e and additiv e structure com b ine to tu rn E G into a bip erm utativ e cat egory . W e can define an anti -in v olution on E G b y declaring ζ to b e the ident it y on ob jects and on morph isms w e defi ne ζ ( g 1 , . . . , g n , σ ) = ( g − 1 1 , . . . , g − 1 n , σ ) for all g i ∈ G and p ermuta tions σ . Then ζ is strictly additiv e and w e can u se the multiplica tiv e twist c G ⊗ in E as µ in order to obtain natural isomorphisms µ from ζ (( g ′ 1 , . . . , g ′ m , σ ′ ) ⊗ ( g 1 , . . . , g n , σ )) to ζ ( g 1 , . . . , g n , σ ) ⊗ ζ ( g ′ 1 , . . . , g ′ m , σ ′ ). Barratt [Ba] defined his f unctor Γ + for based simplicial sets X and identified its geometric realizatio n with Ω ∞ Σ ∞ | X | . F or | X | = B G + w e obtain that Ω ∞ Σ ∞ B G + is the infi nite lo op space asso ciated to the sp ectrum H ( E G ) and therefore this rin g sp ectrum is the spherical group ring S [ B G ] = Σ ∞ + ( B G ). Its algebraic K -theory is W aldhausen’s K -theory A ( B B G ) = K ( S [Ω B B G ]) ∼ K ( S [ B G ]). F or G ab elian, the in v erse map on G induces the inv erse map on B G and via the map of H -spaces B G ∼ → Ω B B G this is r elated to lo op inv ersion. Hence in th is s ense th e induced in v olution on K ( E G ) ≃ A ( B B G ) corresp onds to Steiner’s inv olution on A ( B B G ). 8. Non-triviality F arr ell and Hsiang [FH, L emm a 2.4] calculated th e effect of the inv olution of K i ( Z ) ⊗ Q : elemen ts in p ositiv e degrees are sen t to their add itiv e inv erse. W e use this fact to p ro v e the follo wing. Prop osition 8.1. The involutions on K ( V ) ∼ K ( k u ) , K ( V R ) ∼ K ( k o ) , K ( E G ) ∼ A ( B B G ) ( G ab elian) ar e non-trivial. Pr o of. W e can mo del th e map π : k u → H Z of commutativ e ring sp ectra on the lev el of bip er- m utativ e categories π : V → R Z b y sending an ob ject n to the natural n um b er n and pro jecting the set U ( n ) of endomorphisms of n to the set { id } . This is a morp h ism of bip ermuta tiv e catego ries with an ti-in v olution. On the leve l of K -theory we obtain an ind uced map K ( ku ) ≃ K ( V ) K ( π ) − → K ( R Z ) ≃ K f ( Z ) . Ausoni and Rognes show [AR, Theorem 2.5 (a)] that rationally the map K ( π ) : K ( k u ) → K ( Z ) is split. As the inv olution is non-trivial on K ∗ > 0 ( Z ) ⊗ Q , it is non-trivial on K ( k u ). Similarly , they sho w that rationally K ( Z ) splits off K ( k o ). Consider the follo wing diagram of bip ermutativ e categories with an ti-inv olution: E G   E π ! ! C C C C C C C C U U / / ( ( V π   V R π } } | | | | | | | | R Z 16 The maps from E mod el the u nit map from th e sp here sp ectrum S ∼ H E to A ( B B G ), k o and k u and are giv en by th e identi t y on ob jects an d the inclusion of Σ n in to the resp ectiv e endomorphisms of n . Rationally , A ( ∗ ) ∼ K ( E ) agrees with K ( Z ) and it splits off A ( B B G ), so th e inv olution is not trivial on A ( B B G ).  Ausoni and Rognes also pr o ved in [AR] that rationally A ( K ( Z , 3)) is equiv alen t to K ( k u ). A map of r in g sp ectra A ( K ( Z , 3)) → K ( k u ) is giv en b y u sing the s tr ing of maps B U (1) → B U ⊗ → GL 1 ( k u ) → Ω ∞ ( k u ) and taking the adjoint wh ich is a map from the susp ension ring sp ectrum Σ ∞ + B U (1) ∼ S [ B U (1)] to k u . T h is yields an ind uced map on algebraic K -theory K ( S [ B U (1)]) → K ( k u ). W e can mo del this via a f unctor of categories F : E S 1 → V . Here, F sen d s n to n and maps a morp hism ( z 1 , . . . , z n , σ ) ∈ S 1 ≀ Σ n to the matrix diag( z 1 , . . . , z n ) · E σ ∈ U ( n ) wh ere diag denotes the corresp onding d iagonal matrix and E σ is the p ermuta tion matrix asso ciated to σ . Th e fact that E σ · d iag( w 1 , . . . , w n ) = diag( w σ − 1 (1) , . . . , w σ − 1 n ) · E σ for w i ∈ S 1 ensures the n atur alit y of F . As the diagram ( S 1 ≀ Σ n ) × ( S 1 ≀ Σ m ) / / ⊕   U ( n ) × U ( m ) ⊕   S 1 ≀ Σ n + m / / U ( n + m ) comm u tes, w e see that F resp ects addition. I f ( e 1 , . . . , e n ) and ( f 1 , . . . , f m ) are ordered bases for C n resp ectiv ely C m , then we choose ( e 1 ⊗ f 1 , . . . , e 1 ⊗ f m , . . . , e n ⊗ f 1 , . . . , e n ⊗ f m ) as an ordered basis for C nm . With this con ve n tion, F resp ects ⊗ as w ell. Ho wev er, F is not a functor of b imonoidal catego ries with ant i-in v olution if w e c h o ose the an ti-in volutio n (id , c ⊗ ) on V coming from its bip ermutat iv e structure. Consider the Z / 2 Z = h ξ i -action on V from Example 6.2. Lemma 8.2. The c omp osition ¯ ζ of the anti-involution (id , c ⊗ ) on V with the gr oup action of Z / 2 Z is an anti-involution on V . Pr o of. If we set ¯ ζ := ξ ◦ ζ , then ¯ ζ ◦ ¯ ζ = ξ ◦ ζ ◦ ξ ◦ ζ = ξ 2 ◦ ζ 2 = id b ecause ξ an d ζ comm ute. F or t w o matrices A ∈ U ( n ) and B ∈ U ( m ) we ha ve that c ⊗ sends ¯ ζ ( A ⊗ B ) = ¯ A ⊗ ¯ B to ¯ B ⊗ ¯ A . The distributivity constraint fr om Defin ition 3.1 ju st express the fact that d r is giv en in terms of d ℓ in V . Th e r emaining axioms are easy to c h ec k.  Corollary 8.3. The f unctor F : E S 1 → V is a morphism of bimonoidal c ate gories with anti- involution F : ( E S 1 , ζ , c G ⊗ ) − → ( V , ¯ ζ , c ⊗ ) . Referen ces [AR] C. A usoni, J. Rognes, R ational algebr aic K -the ory of top olo gic al K -the ory , preprint arXiv :0708.2 160 [BDR] Nils A. Baas, Bjørn Ian Dundas, John Rognes, Two-ve ctor bund les and forms of el liptic c ohomolo gy , T op ology , geometry and qu antum field theory . Proceedings of the 2002 Oxford symp osium in honour of the 60th birthday of Graeme S egal, Oxford, UK, June 24-29, 2002. Cambridge Univ ersit y Press, London Ma thematical So ciety Lecture Note Series 308, (2004) 18 –45. [BDRR] Nils A. Baas, Bjørn I an Dundas, Birg it Rich t er, John Rognes, Two-ve ctor-bund les define a form of el l iptic c ohomolo gy the ory , preprint 17 [B] Da vid Barnes, R ational Equivariant Sp e ctr a , Thesis Universit y of Sheffield (2008), arXiv:0802.095 4 [Ba] Mic hael G. 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