The club of simplicial sets

The club of simplicial sets Dennis Boriso v dennis.bo riso v@gmail.com Max-Planc k Institute for Mathematics, Bonn, G erman y No v em ber 9, 2018 Abstract A club structure is defined on the categ ory of simplicial sets. This club genera lizes the op erad of as so ciative rings by a dding “amalga - mated” pro ducts. 1 In tro duction There is a straigh tforw ard wa y to define op erads in the monoidal category ( C at, × ): just apply the standard definition. How ev e r, since C at is not ju st a category , bu t a 2-catego ry , suc h a d efinition is of very limitied v alue. F or one thing, the action of sy m metric groups thr ough fun ctors would not b e the correct action in most applications. Instead, one w ould need symmetric groups to act by morph isms. In this p ap er we make u se of another wa y , the 2-catego rical structure of C at makes itself f elt. In the d efinition of op erads one parameterizes the pro cedure of ta king sev eral p oints in a set and comp osing them into one p oint. Of course sets can b e substituted with ob jects in an y other symmetric monoidal category , but th e p rinciple r emains the same: w e comp ose strin gs of elemen ts. Let M ∈ C at b e a catego ry . A string of ob jects { A 1 , . . . , A n } ⊆ M is the same as a diagram n → M , wher e n is a d iscrete category on n -ob jects. Here of course w e can tak e an y diagram D → M , where D is not n ecessarily discrete, and try to “comp ose” it. This kin d of comp ositions cannot be describ ed in terms of op erads. W e need the notion of a club instead. In their full generalit y clubs w ere dev elop ed by G.M.Kelly in [KG 74], and they are the wa y to enco de asso ciativit y of comp ositions wh en w e com- 1 p ose arbitrary diagrams, and not just the ones parameterized b y discrete catego ries. Recall that the main axiom of op erads is associativit y of comp ositions of op er ations. Club s pro vide a formalizatio n for the same axiom, but in the more general case, where op erations can h a v e arbitrary diagrams as inp uts, and not just strin gs. Of cours e, op erads are clubs of a particular kind . In this pap er we define a club structure on the category of simplicial sets S S et . This club generalizes the op erad of asso ciativ e rings by adding comp ositions of element s r elative to other elemen ts. Discrete sim p licial sets give just an asso ciativ e p ro du ct, non-discrete simplicial sets giv e “amalgamat ed” asso ciativ e p ro du cts. An example of suc h amalgamate d p ro ducts are monoidal globular categories in [Ba98]. Here is the structure of the pap er: in section 2 w e recall the definition of clubs. The general d efinition, giv en in [KG74], tak es place in an arbitrary 2-cate gory . W e do not n eed this generalit y , and we consider only clubs in C at . As an example we sh o w that set-theoretic op er ad s are clubs in C at of a particular kind . W e also u se a different notation from [KG74]. A club in C at is a monoid in the catego ry of diagrams in C at . This categ ory has a ve ry imp ortan t monoidal pro d uct, which is a straigh tforw ard generalization to catego ries of the semidirect pro duct of groups. Therefore w e use th e symb ol ⋉ to designate this monoidal stru cture. F or any diagram D in C at , the fun ctor D ⋉ − : C at → C at is an instance of what is called a familial 2 -functor ([WM07], [SR00]). In sectio n 3, s tarting f rom the cate gory S S et of simp licial sets, w e define the stru cture of a club on S S et , where S S et is a diagram in C at , p arameter- ized b y S S et , with every sim p licial set mapp ed to its category of simplices. In fact, we defi ne t w o clubs: one on the en tire category of simp licial sets, and another on the su b category , consisting of injectiv e morphisms. The latter is imp ortant in applications, when w e hav e a category M , with an S S et -algebra str u cture on it, and w e wan t to ha v e an S S et -algebra structure on the sub catego ry of mono-morph isms in M . A note on notation: When working with sets we use the approac h of unive rses ([SGA4]), in p articular we s p eak of a small category S et of sets, meaning sets in a giv en universe. C onsequen tly we ha v e a sm all category S S et of sim p licial sets. 2 2 Semi-direct pro duc t and clubs in C at Definition 1 L et C at b e the fol lowing c ate gory: • An obje ct D ∈ C at is a p air { D , R } , wher e D is a smal l c ate gory, and R : D → C at is a f u nctor. • A morphism F : D → D ′ in C at is a p air { F , ρ } wher e F : D → D ′ is a fu nctor, and ρ is a natur al tr ansformation , making the f ol lowing diagr am c ommutative: D F / / R B B B B B B B B B B B B B B B B D ′ R ′ } } | | | | | | | | | | | | | | | | | o o ρ C at No w we are going to d efine a monoidal structure { ⋉ , 1 } on C at . W e start with pro d ucts of ob jects. Let D , D ′ ∈ C at b e ob jects, let d ∈ D b e an ob ject, an d let R ( d ) ψ → D ′ b e a fun ctor. W e define a catego ry R ( d ) ⋉ ψ D ′ as follo ws: • Ob jects of R ( d ) ⋉ ψ D ′ are p airs { a, b } , where a ∈ R ( d ) and b ∈ R ′ ψ ( a ). • Morph isms of R ( d ) ⋉ ψ D ′ are pairs { α, β } , where α : a 1 → a 2 is a morphism in R ( d ), and β : R ′ ψ ( α )( b 1 ) → b 2 is a morphism in R ′ ψ ( a 2 ). • Comp osition of { a 1 , b 1 } { α 1 ,β 1 } / / { a 2 , b 2 } { α 2 ,β 2 } / / { a 3 , b 3 } is { α 2 α 1 , β 2 R ′ ψ ( α 2 )( β 1 ) } . It is easy to c hec k that R ( d ) ⋉ ψ D ′ is indeed a category , and similarit y b et we en this construction and semi-direct pr o duct of group s is obvio us. Differen t from the case of group s, we can put together all R ( d ) ⋉ ψ D ′ ’s for all d ∈ D and all ψ : R ( d ) → D ′ to get a diagram D ⋉ D ′ ∈ C at . First we describ e the parameterizing category . Let D ⋉ D ′ b e the small catego ry , d efined as follo ws: • Ob jects of D ⋉ D ′ are pairs { d, ψ d } , where d ∈ D is an ob ject, and ψ d : R ( d ) → D ′ is a functor. 3 • Morph isms in D ⋉ D ′ are pairs { f , φ } , where f : d 1 → d 2 is a morphism in D , and φ is a natural transformation, making the follo wing diagram comm u tativ e: R ( d 1 ) R ( f ) / / ψ d 1 ! ! C C C C C C C C C C C C C C C C C R ( d 2 ) ψ d 2 } } { { { { { { { { { { { { { { { { { φ / / D ′ No w w e define R ⋉ R ′ : D ⋉ D ′ → C at . As we said ab ov e, w e wo uld lik e to collect all R ( d ) ⋉ ψ d D ′ ’s in to one diagram, so on ob jects R ⋉ R ′ is clear: R ⋉ R ′ : { d, ψ d }  / / R ( d ) ⋉ ψ d D ′ . It is straigh tforward then to define the action of R ⋉ R ′ on morph ism s of D ⋉ D ′ b y comp osing fun ctors and n atural transformations in an ob vious w a y . Here is the explicit description: for a m orp hism { d 1 , ψ d 1 } { f ,φ } → { d 2 , ψ d 2 } w e define the fun ctor R ⋉ R ′ ( { f , φ } ) : R ( d 1 ) ⋉ ψ d 1 D ′ / / R ( d 2 ) ⋉ ψ d 2 D ′ as follo ws: let { α, β } : { a 1 , b 1 } → { a 2 , b 2 } b e a morphism in R ( d 1 ) ⋉ ψ d 1 D ′ , then w e define R ⋉ R ′ ( { f , φ } )( { α, β } ) to b e { R ( f )( a 1 ) , R ′ ( φ a 1 )( b 1 ) } { R ( f )( α ) ,R ′ ( φ a 2 )( β ) } / / { R ( f )( a 2 ) , R ′ ( φ a 2 )( b 2 ) } F unctorialit y of this construction is ob vious, since all we do here is comp osing functors and natural transformations. Definition 2 L et D , D ′ ∈ C at b e obje cts. We define their semi-direct pro duct D ⋉ D ′ to b e { D ⋉ D ′ , R ⋉ R ′ } ∈ C at as ab ove. No w we describ e the un it ob jects for ⋉ . Let 1 b e a d iscrete categ ory on one ob ject, an d let 1 ∈ C at consist of 1 , mapp ed to itself in C at . It is easy to see that for an y D ∈ C at we hav e canonically D ⋉ 1 ∼ = D ∼ = 1 ⋉ D . W e do n ot pr ov e the follo wing p rop osition, since it is a consequence of the general result, pr o ved in [K G74]. 4 Prop osition 1 L et C at b e the c ate gory of smal l diagr ams in C at (Defini- tion 1). The semi- dir e ct pr o duct ⋉ (Definition 2) to gether with 1 define a monoidal structur e on C at . W e w ould like to n ote that ⋉ is not symmetric. This is easy to see f rom the follo wing simp le example: let D = 1 , D ′ = 2 (discrete categories on one and tw o ob jects resp ectiv ely); let R : D → C at b e defined b y by mapping the only ob ject to 2 ∈ C at , and let R ′ : D ′ → C at b e d efined by mapping ev ery ob ject to 1 ∈ C at . Then D ⋉ D ′ ≇ D ′ ⋉ D . Definition 3 A club in C at is a monoid i n ( C at, ⋉ , 1 ) . As with most monoids , we will b e intereste d in mo dules o v er a club in C at . Giv en a club C , it is straigh tforw ard to define a C -mo d ule in C at , but in practice we would lik e clubs to act on categories, i.e. ob jects of C at , rather than C at . F or that w e need a bit of notation. Let D b e a s mall category . T here are is a natural w a y to asso ciate an ob ject in C at to D . Let D := D 7→ 1 b e the d iagram in C at , h aving D as the parameterizing category , s.t. ev ery ob j ect in D is mapp ed to 1 ∈ C at . Notice that the assignment D 7→ D is a f unctor f rom C at to C at , and it is left adjoin t to the forgetful functor C at → C at , that maps eve ry diagram to its parameterizing category . Definition 4 L et C b e a club i n C at , and let M ∈ C at b e a c ate gory. We define C ( M ) to b e the p ar ameterizing c ate gory of C ⋉ M . A C -algebra is a c ate gory M , to gether with a functor C ( M ) → M , satisfying the usual asso ciativity c onditions. No w we are ready to consider examples. 1. Let P b e a set- theoretic non-symmetric op erad, i.e. w e h a ve a se- quence of sets { P n } n ≥ 0 , a chosen elemen t e ∈ P 1 , and a sequence of comp ositions γ m 1 ,...,m n : P n × P m 1 × . . . × P m n → P m 1 + ... + m n , satisfying the usual conditions of asso ciativit y and u nitalit y . No w we construct a diagram in C at , starting with P , and for every n > 0 a c hoice of a discrete catego ry n having an ordered set of n 5 ob jects. The parameterizing catego ry P is the discrete category ha ving ` n ≥ 0 P n as the set of ob jects. Ev ery p ∈ P n ⊆ P is mapp ed to n . W e will denote the resulting diagram by P . This construction works for any N -collect ion in S et , in particular for P ◦ P , where ( P ◦ P ) k = a m 1 + ... + m n = k P n × P m 1 × . . . × P m n . Pro of of the follo w ing p rop osition is straigh tforw ard. Prop osition 2 1. F or any c ol le ction P in S et we have P ◦ P ∼ = P ⋉ P . (1) 2. The c orr esp ondenc e (1 ) defines a bije ction b etwe en the set of op- er adic c omp ositions { γ m 1 ,...,m n } on P , and the set of ⋉ -monoidal structur es { F , ρ } : P ⋉ P → P , { I , ι } : 1 → P , s.t. ρ, ι ar e natur al e quivalenc es, pr eserving the or der on n ’s. Let S b e a set, and supp ose P acts on it. Then it is easy to see h o w to translate such an ac tion in to th e structure of a P -algebra on S , considered as a discrete category . 2. No w let P b e an op erad in S et . Here, in add ition to c ho osing a discrete catego ry n on n ob jects, for eac h n ≥ 1 we fi x an isomorphism S n ∼ = Aut ( n ) . Then w e can d efine a diagram P ∈ C at as follo w s : the parameterizing catego ry P h as ` n ≥ 0 P n as the set of ob jects, and ∀ p, q ∈ P n ⊆ P , w e put H om ( p, q ) to b e the set of all σ n ∈ S n , s.t. σ n ( p ) = q ; the f unctor R : P → C at maps eve ry p ∈ P n ⊆ P to n , and ev ery morphism p σ → q to the corresp onding endofu nctor on n . It is clear that P is indeed an ob ject in C at , and we can app ly th e same tec hnique to ev ery Σ-collect ion in S et . Different from the non- symmetric case, w e hav e that in general P ⋉ P ≇ P ◦ P . How ev er, we ha v e a n atural inclusion P ⋉ P → P ◦ P , and hence w e can conclude the follo wing. 6 Prop osition 3 F or any Σ -c ol le ction P in S et , ther e is a bije ction b etwe en op er adic structur es on P , and ⋉ - monoidal structur es { F , ρ } : P ⋉ P → P , { I , ι } : 1 → P , s.t. ρ, ι ar e natur al e quivalenc es, that pr eserve or der on n ’s. Pro of: The only d ifference h ere from th e non-sym metric case is the action of sym metric groups. Since P n × P m 1 × . . . × P m n carries the action of only S n × S m 1 × . . . × S m n , and hence in general it is not an S m 1 + ... + m n -set, w e ha v e that P ⋉ P ≇ P ◦ P , and w e cannot pro ceed as in Prop osition 2. In d efining op erads one extends P n × P m 1 × . . . × P m n b y tensorin g it with S m 1 + ... + m n o ver S n × S m 1 × . . . × S m n . Ho wev er, while this mak es definition of an op erad cleaner, it is not really needed, and it is enough to p ostulate equiv ariance only with resp ect to S n × S m 1 × . . . × S m n .  Also here it is easy to see ho w to translate the n otion of a P -algebra in S et in to a P -algebra in C at . 3 The club of simplicial sets In th e p revious section we ha v e considered t wo examples of ⋉ -monoids of a sp ecial kind. I n general a ⋉ -monoid is giv en by an ob ject D in C at , together with morphisms { F , ρ } : D ⋉ D → D , { I , ι } : 1 → D in C at , satisfying the usu al asso ciativit y and un it axioms. In the case of set-theoretic op erads we hav e required that ρ and ι are not just n atural transformations, b ut n atural e quivalenc es . This requirement w as a consequen ce of the wa y we represen ted op erads: all op eradic comp o- sitions w ere enco d ed in the parameterizing category P , i.e. o p erations are represent ed as ob jects in P . The f u nctor P → C at wa s th ere only to kee p trac k of the arit y of th ese op erations. No w w e consider a case where ρ is not requ ired to b e inv ertible. Th is case is the main example for a “d iagrammatic” op eradic actio n on categories: here we d o not comp ose strings of ob jects, but diagrams of ob jects, and hence categories in the image of P → C at stop b eing ju st a b o okke eping device, but carry inform ation of their o wn . 7 The diagrams in question here are give n by simplicial sets. W e start with defining a pr o cedure that pro d u ces a category out of a sim p licial set. Let S S et b e a sm all category of simplicial sets. F or an y S ∈ S S et , S = {S n } n ≥ 0 , w e defin e a category S as follo ws: • Th e set of ob jects in S is ` n ≥ 0 S n . • Given t wo ob jects s m ∈ S m ⊆ S , s n ∈ S n ⊆ S , H om ( s m , s n ) is the set of all s implicial op erators S m → S n , that map s m to s n . It is clear that for any S ∈ S S et , S is a small category . It is also clear that for an y morphism f : S → S ′ in S S et there is a fun ctor R ( f ) : S → S ′ , and that th is assignment f 7→ R ( f ) is functorial. Therefore we h a ve an ob ject S S et := { S S et, R } ∈ C at . Prop osition 4 Ther e is a structur e of ⋉ -monoid on S S et . Pro of: W e need to define { ∆ , δ } : S S et ⋉ S S et → S S et, (2) where ∆ : S S et ⋉ S S et → S S et is a fun ctor, and δ is a natural tr an s formation S S et ⋉ S S et ∆ / / R ⋉ R $ $ I I I I I I I I I I I I I I I I I I I I S S et R | | z z z z z z z z z z z z z z z z z o o δ C at W e start with defin ing ∆ on ob jects. Let {S , ψ } b e an ob ject in S S et ⋉ S S et , i.e. S is a simplicial set, and ψ : S → S S et is a functor. The set of ob jects in the catego ry S ⋉ ψ S S et is graded b y Z ≥ 0 × Z ≥ 0 , indeed, ob jects in S are Z ≥ 0 -graded b y dimension of s implices, and similarly for categories in the image of R ◦ ψ . It is easy to see that the category S ⋉ ψ S S et can b e obtained from a bisimplicial set T by considering simp lices as ob j ects and bisimp licial op er- ators as morphism s. Let T ∈ S S et b e the d iagonal in T . W e set ∆( {S , ψ } ) := T , δ : T → S ⋉ ψ S S et , with δ b eing give n by the diagonal. Thus we h a ve d efined (2) on ob jects. 8 Let { f , φ } : {S , ψ } → {S ′ , ψ ′ } b e a morphism in S S et ⋉ S S et , w here f : S → S ′ is a map of simplicial sets, and φ is a n atural transformation S f / / ψ ! ! B B B B B B B B B B B B B B B B B S ′ ψ ′ } } { { { { { { { { { { { { { { { { { φ / / S S et The pair { f , φ } induces a fun ctor S ⋉ ψ S S et → S ′ ⋉ ψ ′ S S et that pr eserv es the Z ≥ 0 × Z ≥ 0 -grading, and defines a map of bisimplicial sets T → T ′ . Consequent ly w e get a functor T → T ′ and a corresp onding map of simplicial sets T → T ′ . This completes defin ition of (2). No w we turn to asso ciativit y of { ∆ , δ } . This is rather s tr aigh tforward, essen tially it amoun ts to asso ciativit y of taking diagonals in bisimp licial sets. It remains to define the un it. There is an ob vious 1 → S S et , with 1 going to the 1-p oint s implicial set.  Ha ving the ⋉ -monoid structur e on S S et , we can talk ab ou t S S et -algebras in C at . F or example, if a categ ory M is closed with resp ect to taking colim- its, w e ha v e a canonical stru cture of a S S et -algebra on M : giv en a sim p licial diagram in M , take its colimit. Sometimes, ha ving a S S et -algebra M w e migh t lik e to wo rk only with the su b category of M , consisting of mono-morp h isms. It might happ en th at the action of the enti re S S et do es not preserve the c hosen s ub categ ory . In these cases the follo wing d efinition is useful. Let M b e a category , and let I b e a set of generators of M . F or example, if M = S S et , I is the set of standard simp lices { ∆[ n ] } n ≥ 0 . Let D S : S → M b e an ob ject in S S et ( M ) (Definition 4). Definition 5 L et I ∈ I . An I -p oin t in D S , is a morphism I [ n ] → D S in S S et ⋉ M , wher e I [ n ] i s I , c onsider e d as a c onstant diagr am over ∆[ n ] . It is clear that for eac h s uc h I ∈ I w e obtain a simplicial set I ( D S ) of I - p oints. If ( F , φ ) : D S → D T is a morphism in S ( M ), it induces a morp hism of simplicial sets ( F , φ ) I : I ( D S ) → I ( D T ). Definition 6 We wil l say that ( F , φ ) i s a fibration , if F is inje ctive, and for e ach I ∈ I the morphism of simplicial sets ( F , φ ) I is a fibr ation. 9 It is straigh tforw ard to c heck th at { I [ n ] } I ∈I ,n ≥ 0 is a set of generators for S S et ( M ), and hence w e can iterate this d efinition to get the notion of a fibration in S S et k ( M ) for any k ≥ 1. Let sset ⊂ S S et b e the s ub categ ory consisting of injectiv e morphisms, and let sset b e the corresp onding ob ject in C at . Let sset ◦ ss et ⊂ sset ⋉ ss et to b e th e sub category of fibrations. S im ilarly , w e d efine ss et ◦ k for an y k ≥ 1. Prop osition 5 The se quenc e { sset ◦ k } k ≥ 1 is stable with r esp e ct to the ⋉ - monoid structur e on S S et . Using this p rop osition we can regard ss et itself as a m onoid, and h ence consider sset -algebras. Usually it h app en s that if M is an S S et -algebra, then the category of mono-morph isms in M is an sset -algebra. References [Ba98] M.A.Batanin. M onoidal g lobular c ate gories as a natur al envir onment for the the ory of we ak n-c ate gories. Adv ances in Mathematics 136, pp. 39-103 (1998). [K G74] G.M.Kelly . On clubs and do ctrines. In Cate gory seminar. Sydney 1972/ 1973. Sp ringer L NM 420, p p. 181-257 (1974). [SGA4] M.Artin, A.Grothend iec k, J.L.V erdier. Th´ eorie des top os e t c oho- molo gie ´ etale des sch´ emas - T ome 1. Lecture Notes in Mathematics 269, Sprin ger V erlag, Berlin, (1972). [SR00] R.Street. The p etit top os of globular sets. , J ournal of pu re and ap- plied algebra, 154, pp. 299-315 (2000). [WM07] M.W eb er. F amilial 2-functors and p ar ametric right adjoints. , The- ory and applications of categories, V ol. 18, No. 22, pp . 665-732 (2007) 10