The Quotient of a Category by the Action of a Monoidal Category

The Quotien t of a Category b y the Action of a Monoidal Category Brett Milburn ∗ Abstract W e introduce the notion of the quotient of a category C by the action A : M × C − →C of a un ital symmetric monoidal category M . The quotient C / M is a 2-category . W e pro ve its e x istence and uniqueness b y fi rst showing th at every small 2-category has a p resen tation in terms of generators and relations a nd then describing the generators and relations needed for the qu otien t C / M . 1 In tro duction W e sho w that for any generating set X , there is a free 2-category A X on X . F urthermo re, given a generating set X with re la tions C , there is a 2-c a tegory A X,C satisfying a universal pr oper t y . Moreov er , any small 2-catego ry has a pr e sen tatio n in terms of g e ne r ators and r elations. W e sta rt by defining the weak er notio n of a pre-2-categ ory and showing the existence of free pre-2-c a tegories and presentations of pre-2 -categorie s by g enerators a nd relations . W e then a pply the technology of pr e-2-catego ries via gener ators and r elations to attain the s ame r esults for 2-ca tegories. There are v arious versions of free n-categ ories in the literature [2], [4], [5], whic h are s uitable in the appropriate c on texts. Schommer-Prie s, for instance , co nsiders free sym- metric mo noidal bica tegories. Our interest in pre sen ting 2-catego ries in terms of g e ne r ators a nd rela tions is due to its utilit y in taking quotient categ ories. Given a unital, symmetric mo noidal category M and an action A : M × C − →C of M on C , we would like to explain what it means to take the quotient C / M . Our definition of the quotient is motiv ated by a more familiar quo tien t construction. Given the action of a mo noid M o n a space X , the s ma rt notion of quotient X/ M is not a space but a categor y . The ob jects o f X/ M are the po in ts of X , and mor phisms in X/ M are indexed by X × M . Instead o f identifying p oint s x and y = m.x in X which are related by m ∈ M , ther e is a mor phism ζ m x from x to m.x , thu s remembering how x and y are r elated. If M is a symmetric mono idal category acting on a catego ry C , we apply the sa me philosophy . This time, how ever, the quotient Q = C / M is a 2-categ ory . In addition to the 1- morphisms in C , ob jects of M provide 1- morphisms ζ m x : x − → m.x for x ∈ Ob ( C ), m ∈ O b ( M ). W e require that ζ is co nsistent with morphisms in C and M in a sense describ ed by conditions Q4-Q 6 in § 3. Roughly , consistency of ζ with morphisms in M and C means that we requir e certain diagrams to commute–ones that we would exp ect to commute in any re a sonable definitio n of quotient. How ever, instead of asking these diagrams to commute on the nose, w e only r equire them to commute up to some 2-mor phisms. In s e ction 3 we define the quotient C / M and demonstra te its existence and uniqueness up to is o morphism. 2 2-Categories via Generators and Relations W e consider in the sequel only small n-ca tegories and will only b e co ncerned with n -ca tegories for n ≤ 2. In Definition 1 we recall the definition of 2-c a tegory but also define a weaker notion of pre - 2-categor y , which is like a 2-categ ory in that it has 0-ob jects, 1-mor phism, 2- morphism and comp ositions but with none of the ∗ milburn@math.utexas.edu 1 none o f the a sso ciativity or co herence prop erties requir ed of 2-categ ories. It is w or th noting that we div erg e from the standard nomenclature; what we mean b y 2-categ ory is what is often ca lled a bicategory . Additionally , we require morphisms betw een 2-catego ries to resp ect co mpos itio n on the nose rather than up to 2-mo rphism. Definition 1 follows the p oint o f view of Street [5 ]. Instead of viewing an n -catego ry as having 0-morphisms (i.e. ob jects), 1-mo rphisms, etc. as distinct, any k -mor phism x , is identified with the ( k + 1)-morphism id x . In this way , a ll k -mor phisms are on the sa me fo oting as mem b ers of the same set. Definition 1 . 1. Suppo se tha t 0 ≤ n ≤ ∞ . The data fo r a (smal l) st rict n -c ate gory is a set A with maps s i , t i : A− →A for all i < n and maps ∗ i : A × A A− →A , wher e A × A A is the fib ered pro duct ov er maps s i : A− →A and t i : A− →A . Let ρ i , σ i ∈ { s i , t i } denote any source or target map. ( A , s i , t i , ∗ i ) i 2 . No w we define F X = S 1 ≤ n< ∞ S n . Let s 0 ( x • y ) = s 0 y , t 0 ( x • y ) = t 0 x , s 1 ( x • 1 y ) = s 1 y , t 1 ( x • 1 y ) = t 1 x , and σ 1 ( x • 0 y ) = σ 1 x ∗ 0 σ 1 y . T o see that this comp osition ma k es sens e, an eas y inductive pro o f shows that s 1 ( S p ) , t 1 ( S p ) ⊂ C for all p . No te tha t ( F X ) 1 = C . With thes e s ource and target maps, F X is a 2-catego rically graded set. There a re comp osition laws on F X as follows. x ∗ 0 y =  x ∗ 0 y if x, y ∈ C x • 0 y otherwise and x ∗ 1 y = x • 1 y . One ca n chec k that F X ∈ p 2 C at . Suppo se F : X − → D is a map in g r 2 C at such that F restricted to C is a map in p 2 C at . Ha ving defined ˜ F on S k for k ≤ n , define ˜ F o n S n +1 by ˜ F ( x • i y ) = ˜ F x ∗ i ˜ F y for i ∈ { 0 , 1 } . Clearly ˜ F is well defined, and ˜ F ( x ∗ i y ) = ˜ F x ∗ i ˜ F y . F urthermore, as X ⊂ F X in g r 2 C at , it is apparent that ˜ F is the only p ossible extensio n of F to a ma p ˜ F ∈ H om p 2 C at ( F X , D ). Corollary 2 .3. The for getful functor p 2 C at − → g r 2 C at is left ad joint to the functor which sen ds X ∈ g r 2 C at to F ( X \ X 1 ) ∪C X 1 ∈ p 2 C at . Pr o of. This is a sp ecial case of Lemma 2 .2. Supp ose X ∈ g r 2 C at . Let C = C X 1 and X ′ = ( X \ X 1 ) ∪ C . W e take X \ X 1 instead of all of X in order to av oid having r edundan t 1-morphisms. By comp osing s 1 , t 1 : ( X \ X 1 ) ⇒ X 1 with the inclusion X 1 ֒ → C to get maps ( X \ X 1 ) ⇒ C , pa rt 2a of Lemma 2.2 guarantees that X ′ is a 2- categorica lly graded set. Given D ∈ p 2 C at and a map X F − → D in g r 2 C at , we aim to give a map F X ′ − → D in p 2 C at and show that this assignment H om gr 2 C at ( X, D ) − → H om p 2 C at ( F X ′ , D ) is an is o morphism. By Lemma 2.2 part 1, F | X 1 ∈ H om gr 1 C at ( X 1 , D ) ≃ H om p 1 C at ( C X 1 , D ). Here w e consider D as a pre- 1-categor y by forgetting the higher structure ma ps . Note also that C ∈ p 1 C at ֒ → p 2 C at and H om p 1 C at ( C , D ) ≃ H om p 2 C at ( C , D ). Since we hav e extended F from X 1 to C , this a llo ws us to extend F uniquely from X ⊂ X ′ to a map F : X ′ − → D in g r 2 C at such that F |C : C − → D is a map of pre-2-ca tegories. By Lemma 2.2(2b), F : X − → D extends uniquely to a map ˜ F : F X ′ − → D in p 2 C at . By the uniquenes s o f the extens ions, the map H om gr 2 C at ( X, D ) − → H om p 2 C at ( F ( X \ X 1 ) ∪C X 1 , D ) is an inclusio n. Since X ⊂ X ′ ⊂ F X ′ in g r 2 C at , every map G : F X ′ − → D in p 2 C at is an extension of G | X : X − → G in g r 2 C at , whence H om gr 2 C at ( X, D ) ≃ H om p 2 C at ( F ( X \ X 1 ) ∪C X 1 , D ). Definition 3 . 1. As in Lemma 2 .2, given the data X = ( X 1 , X 2 ⇒ C X 1 ) of X 1 ∈ g r 1 C at (which de fines ( C X 1 , s 0 , t 0 , ∗ 0 ) ∈ p 1 C at ) and a set X 2 with ma ps o f sets s 1 , t 1 : X 2 − →C X 1 such that σ 0 s 1 = σ 0 t 1 for all σ 0 ∈ { s 0 , t 0 } , the pr e-2-c ate gory gener ate d by X is the free pre-2-c a tegory F X 2 ∪C X 1 , whic h by abuse of notation we a lso denote by F X . The data X is the gener ating data for the pre-2 -category F X . W e also wr ite X = X 1 ∪ X 2 for brevity . 4 2. A set of c onditions on gener ating data X is a bina ry r elation o n F X . Lemma 2.4. Given gener ating data X and c onditions C , ther e exists an e quivalenc e r elation ∼ on F X such that F X / ∼ ∈ g r 2 C at and has the pr op erty that for any D ∈ p 2 C at and F ∈ H om p 2 C at ( F X , D ) such that xC y implies F ( x ) = F ( y ) for x, y ∈ F X , F factors thr ough F X − →F X / ∼ in g r 2 C at . Pr o of. Let ∼ deno te the fines t relatio n on F X satisfying the following conditions: P0: ∼ is a n equiv alence relation. P1: If xC y , then x ∼ y . P2: If x ∼ y , then σ i x ∼ σ i y for σ i ∈ { s 0 , t 0 , s 1 , t 1 } . P3: If x ∼ x ′ and y ∼ y ′ , then x • y ∼ x ′ • y ′ whenever b oth comp ositions are defined. The notation in P3 is explained at the beg inning of § 2 .1 a nd in the pro of o f 2.2 (2 b). Letting x ∼ y for all x, y ∈ F X is s uc h a re la tion. Because P0-P3 are clos ed under interesctions (i.e. m utual refinements), Zor n’s lemma ensures the existence of a fines t r elation s atisfying P 0-P3. Now supp ose F : F − → D as ab ov e. Then the relatio n xR y if F x = F y satisfies P0 -P3. Thus, F fac to rs through F X /R ∈ g r 2 C at . Since ∼ is the sma llest such relation, F − →F /R factors thro ugh F / ∼ . Hence, F also factors through F / ∼ . W e now show that for a n y generating set X and co nditions C , there is a pr e - 2-categor y F X/C generated by X and sa tisfying C . Theorem 2.5. Given gener ating data X = X 1 ∪ X 2 and c onditions C , ther e exists a unique F X/C ∈ p 2 C at satisfying: 1. Ther e is a map G : F X − →F X/C in p 2 C at su ch that for al l x, y ∈ F X , xC y implies G ( x ) = G ( y ) . 2. F X/C is universal among pr e-2-c ate gories satisfying the ab ove pr op erty in t he sense that for any other map F : F X − → D in p 2 C at for which xC y implies F x = F y for al l x, y ∈ F X , F factors uniquely thr ough G as se en in the diagr am in p 2 C at F X F ✲ D F X/C G ❄ ✲ . Pr o of. First we cons ider only 0-ob jects and 1-morphisms to g et a quotient category C ′ from C = C X 1 . The relation ∼ on F X of Lemma 2.4 restricts to an equiv alence relation on C = ( F X ) 1 . Tha t is to say , for x, y ∈ C , x ∼ y in C if a nd only if x ∼ y in F X . Additionally , C := C / ∼∈ g r 2 C at b ecause ∼ satisfies P 2. Now we define C ′ by taking S 1 = C , S 2 = { x • y | x, y ∈ S 1 and for all x ′ ∼ x , y ′ ∼ y , x ′ ∗ 0 y ′ is not defined } . W e define S n = F 0 2 a nd C ′ = S ∞ n =1 S n . Define s 0 ( x • y ) := s 0 y , t 0 ( x • y ) := t 0 x . Comp osition is defined as x ∗ 0 y =  x ′ ∗ 0 y ′ if x ′ ∗ 0 y ′ ∈ C is defined for some C ∋ x ′ ≡ x , C ∋ y ′ ≡ y x • 0 y = otherwise so that C ′ ∈ g r 1 C at . This comp osition g ives C ′ the structure of a pre-1-ca tegory . W e cla im that any map F : C − → D of pre-1 -categories such that xC y implies F x = F y must factor through C ′ . Such a map F : C − → D m ust factor through C ∈ g r 1 C at , which can be extended to a map ˜ F : C ′ − → D 5 in p 1 C at via ˜ F ( x • 0 y ) = F ( x ) ∗ 0 F ( y ). W e now hav e X 2 ⇒ C − →C ′ , ma king X ′ = X 2 ∪ C ′ a categor ically graded set with a map C ∪ X 2 − →C ′ ∪ X 2 in g r 2 C at . This induces H : F X − →F X ′ in p 2 C at . The next step is to identify all remaining 2-morphis ms related by C . W e ther efore w a nt a rela tio n ∼ on F X ′ which is the finest relation satisfying: P0: ∼ is a n equiv alence relation. P1 ′ : If xC y for x, y ∈ F X , then H x ∼ H y . P2: If x ∼ y , then σ i x ∼ σ i y for σ i ∈ { s 0 , t 0 , s 1 , t 1 } . P3: If x ∼ x ′ and y ∼ y ′ , then x • y ∼ x ′ • y ′ whenever b oth comp ositions are defined. P4: If x, y ∈ C ′ ⊂ F X ′ , then x ∼ y implies x = y . Suppo se there ex ists such a rela tion. Conditions P0 -P4 a re closed under ta king refinements of tw o such relations. Zorn’s lemma implies that there is a minimal s uc h relatio n R . Let F X/C := F X ′ /R . Pr oper ties P2 and P 3 guarantee that F X/C is a pre-2 -category . W e wish to show that F X/C has the sp ecified universal prop erty . T o this end, let F : F X − → D be any map in p 2 C at such that F x = F y whenever xC y . Then F |C : C − → D fa ctors uniquely through C ′ as we hav e already shown, thus inducing a unique map F ′ : F X ′ − → D in p 2 C at . Define a relation Q on F X ′ by xQy if x and y lie in the same fib er of F ′ . Conditions P0 -P3 a bove are satisfied by Q . Clearly , since R is the finest relatio ns satisfying P 0-P4, it is also the finest r elations satisfying P0- P3. Hence, F ′ factors uniquely through F X ′ /Q , which factors uniquely through F X ′ /R in p 2 C at via the map π : F X ′ /R − →F X ′ /Q . Therefor e, F factors uniquely through F X − →F X ′ /R as desire d. This can be expressed in the following commutativ e diag r am in p 2 C at F X ′ F ′ ✲ D F X ′ /R ❄ π ✲ F X ′ /Q. ✻ ✲ It only r emains to s how that there exists a rela tion on F X ′ satisfying P0 -P4. In general, let A ∈ p 2 cat and C = A 1 . Given a subset S ⊂ C × C such that: • C ≃ ∆ C ⊂ S , • σ 0 π 1 = σ 0 π 2 on S for all σ 0 ∈ { s 0 , t 0 } , • if ( f , g ),( h, k ) ∈ S s atisfy t 0 h = s 0 f , then ( f h, g k ) ∈ S , and • ( h, g ) , ( g , f ) ∈ S implies ( h, f ) ∈ S , then S is a pre-2-catego ry with stucture ma ps s 1 = ∆ π 1 , t 1 = ∆ π 2 , s 0 = ∆ s 0 π 1 , t 0 = ∆ t 0 π 2 , ( f , h ) ∗ 0 ( g , k ) = ( f ∗ 0 g , h ∗ 0 k ), and ( f , h ) ∗ 1 ( h, k ) = ( f , k ). The imp ortant p oint is that S has the prop erty that for every f , g ∈ S 1 ≃ C , there exists at most one 2-morphism from f to g . Now, star ting from F X ′ , let S = { ( f , g ) ∈ C ′ | there exists a 2-mo rphism α : f = ⇒ g } . Then ther e is a pro jection π : F X ′ − →S , and the fiber s of π deter mine a r elation satisfying P0 -P4. 2.2 2-Categories In order to apply the previous results to 2-ca tegories, we observe that a 2-category is simply a pr e-2-catego ry with extra data and co nditions. Theorem 2.6. L et X = X 1 ∪ X 2 b e gener ating data and imp ose c onditions C . Ther e exists a un ique (up to isomorphism) 2-c ate gory A X,C e quipp e d with a map G : F X − →A X,C in p 2 C at such that G ( x ) = G ( y ) 6 whenever xC y and such that A X,C is universal with r esp e ct to this pr op erty in the fol lowing sense. Given a 2-c ate gory D and a map F : F X − → D in p 2 C at such that for al l x, y ∈ F X , xC y implies F ( x ) = F ( y ) , F factors uniquely thr ough G in p 2 C at in such a way that t he map H : A X,C − → D such that H G = F is a map of 2-c ate gories. We c al l A X,C the 2-c ate gory gener ate d by X with c onditions C . Pr o of. This is only a slight mo dification of the pro of of Tho rem 2 .5 where the generating data is enlarge d to contain the structure morphisms α f ,g,h , λ f , ρ f and we add to C coherence conditions for 2-categor ie s . W e work under the assumption that the genera ting data X doe s not alre ady c on tain the structure 2-morphisms for a 2 -category . Beginning with F X ′ , the pr e-2-catego ry defined in the third parag raph of the pr o o f of Theorem 2.5, we add to X ′ 2-morphisms λ f : t 0 f ∗ 0 f = ⇒ f and ρ f : f ∗ 0 s 0 f = ⇒ f for ea ch f ∈ C ′ = ( F X ′ ) 1 as well as a 2-mo rphism α h,g , f : h ∗ 0 ( g ∗ 0 f ) = ⇒ ( h ∗ 0 g ) ∗ 0 f for ea c h triple of f , g , h o f 1-mor phisms in C ′ . Also we add 2-morphisms α − 1 h,g , f : ( h ∗ 0 g ) ∗ 0 f = ⇒ h ∗ 0 ( g ∗ 0 f ), ρ − 1 f : f = ⇒ f ∗ 0 s 0 f , a nd λ − 1 f : f = ⇒ t 0 f ∗ 0 f whic h ar e going to be the inv ers es of α h,g , f , λ f , ρ f resp ectively in the 2- category A X,C . Let Y = X ′ F { α f ,h,g , λ f , ρ f , α − 1 f ,g,h , ρ − 1 f , λ − 1 f } f ,g,h ∈C ′ and C ′ = C ∪ { coherence co nditio ns for a 2-ca tegory } ∪ I . Here I deno tes the set of relations { ( α h,g , f • 1 α − 1 h,g , f , h • 0 ( g • 0 f )) , ( α − 1 h,g , f • 1 α h,g , f , ( h • 0 g ) • 0 f ) , ( λ f • 1 λ − 1 f , f ) , ( λ − 1 f • 1 λ f , t 0 f • 0 f ) , ( ρ f • 1 ρ − 1 f , f ) , ( ρ − 1 f • 1 ρ f , f • 0 s 0 f ) } , where w e think of the binary relation C ′ as a s ubset o f O b ( F Y ) × O b ( F Y ). The inclusion (i.e. injective map) X ′ ֒ → Y in g r 2 C at induces a n inclusio n F X ′ ֒ → F Y in p 2 C at . Let R b e the re la tion o n F X ′ (describ ed in the p enultim a te paragr aph of the pro of of Theor em 2 .5) such that F X/C = F X ′ /R . No w w e let R ′ be the finest binary relation on F Y satisfying P0, P2-P4 and having C ′ and R as refinements. The existence of a minimal r elation R ′ is prov en b y the s a me arguments used in the pro o f of Theorem 2.5. The quotient A X,C := F Y /R ′ is a pre-2 -category containing F X/C as a sub category (in the sense that there is an inclusion F X/C ֒ → F Y /R ′ ). The generating data Y contains the extr a data needed to make F X ′ int o a 2- category , a nd the co nditions C ′ are chosen for the purp ose of ens uring that F Y /R ′ satisfies the coherence conditions for 2-catego ries. More pre cisely , in order for A X,C to b e a 2 -category , it m ust contain 2-isomor phism α f ,g,h λ f , and ρ f for all f , g , h ∈ ( A X,C ) 1 , a nd A X,C m ust satisfy the coher ence conditions . One obs tacle to A X,C to b e a 2-catego ry is that we hav e not added enough α ’s ρ ’s and λ ’s. W e have added a n α f ,g,h ρ f and λ f for all f , g , h ∈ C ′ ⊂ ( A X,C ) 1 , but we need one for each f , g , h ∈ ( A X,C ) 1 . This, howev er, is not a problem s ince no tw o 1-morphism are identified in passing from F Y to F Y /R ′ , whence C ′ ≃ ( F X ′ ) 1 = ( F Y ) 1 ≃ ( F Y /R ′ ) 1 . The other po s sible o bstacle for F Y /R ′ to qualify as a 2-categ ory is that there may b e 2-morphisms in F Y /R ′ which ought to b e identified but which a re not, which w ould mean that the cohere nc e co nditions are not satisfied. F or example, ∗ 1 should b e strictly asso ciative. How ever, the choice o f R ′ and the fact that F Y − →F Y /R ′ is surjective preclude this from ha pp ening. Therefor e , F Y /R ′ is a 2- category which comes with a ma p F X − →F Y /R ′ in p 2 C at . It only rema ins to show that A X,C has the desired universal pro per t y . Giv en D ∈ 2 C at a nd F : X − → D in g r 2 C at s uc h that F : F X − → D identifies o b jects r elated by C , then F induces a map F : F X ′ − → D by Theorem 2.5. The map X ′ − → D in g r 2 C at extends uniquely to a map Y − → D b ecause ther e is only one po ssible choice of whe r e to send each α f ,g,h , ρ f , λ f , namely the str ucture maps α F ( f ) ,F ( g ) ,F ( h ) , λ F ( f ) , ρ F ( f ) in D . The map fro m F Y already has the prop erty that F x = F y if xC y (Here we abuse no tation a nd denote all ma ps by F ). The only additiona l relations in C ′ are the coherences conditions for 2- categories. These relations will automatically become equa lities in D beca use D is a 2-categor y . Th us, xR ′ y implies F x = F y , whence F : F Y − → D descends to F Y /R ′ − → D uniquely . This map F Y /R ′ − → D prese r ves the maps α , λ , ρ , so it is a map of 2 -categories . Remark 2.7 . If the original conditions C a re such that no tw o 1-mo rphism in C X are identified in F X by the equiv alence relation ∼ of Lemma 2.4, then we may initially include the 2-catego ry data α f ,g,h , λ f , ρ f 7 and conditions in the o riginal data and co nditions and find that F X/C is alr eady a 2-catego ry . The only obstacle to doing this in general is that there may be morphism in C ′ X which were not in C X . Theorem 2 .6 ha s the unu s ua l prop erty tha t it ma k es reference to pre- 2 -categorie s in the description of A X,C . The fo llo wing corolla ry justifies calling A X the 2-catego ry generated by X . Corollary 2.8. Consider any gener ating data X = ( X 2 ⇒ C X 1 ) . 1. A X has the fol lowing universal pr op erty. Ther e is a c anonic al inclusion ι A : X − →A X of 2-c ate goric al ly gr ade d sets, and for any 2-c ate gory B with an inclusion ι B : X − →B such t hat ( ι B ) |C X 1 is a map in p 2 C at , ther e is a unique map of 2-c ate gorie s F : A X − →B such t hat ι B = F ι A . 2. If X is gener ating data and C is a binary r elation on A X , then ther e exists a 2-c ate gory A X/C satisfying: (a) Ther e is a map of 2-c ate gories G : A X − →A X/C such that xC y implies Gx = Gy , and (b) A n y other m ap F : A X − →B of 2-c ate gories su ch that xC y implies F x = F y factors uniquely thr ough G . 3. Any 2-c ate gory has a pr esentation in terms of gener ators and r elations, i.e. any B ∈ 2 C at is isomorphic to some A X/C for some gener ating data X and binary r elation C on A X . Pr o of. 1. B y Lemma 2.2, to have such a map ι B is the s ame as having a map F X − →B in p 2 C at . The result now follows directly from Theorem 2.6. 2. Let Y = X ⊔ { α f ,g,h , λ f , ρ f } f ,g,h ∈ ( A X ) 1 . W e hav e F X ֒ → F Y π − → A X − →A Y ,π − 1 C . Since F : A X − →B ident ifies o b jects rela ted by C , F π identifies ob jects related by π − 1 C , whence F π factor s uniquely through A X/C := A Y ,π − 1 C via so me map H : A X/C − →B of pr e-2-catego ries. Since π is surjective, the comp osition A X − →A X/C H − → B is F . Note that π − 1 C co n tains the coherence c o nditions for a 2-catego ry , so A X/C is a 2-catego ry . 3. Supp ose B is a 2-ca tegory . Let X = B ∈ g r 2 C at , s o A X p − → B is a s urjection. Let C b e the bina ry relation o n A X which relates every t wo p oints in the same fib er of p . T he n A X/C ≃ B . Theorems 2.5, 2.6 can b e extended to s tr ict 2-ca tegories. Ther e is more than one approach to extending these results. This can be done by mo difying the pr o o fs to get a strict 2-ca tegory given by generator s and relations. At the first s tage, the construc tio n of the fr ee pre-1 -category C X is replaced by the free 1-category , i.e. the pa th ca tegory g enerated by X . The free strict 2 - category F X can be co nstructed in a similar y way . Alternately , we can view a strict 2-ca tegory as a pr e-2-catego ry with extra co nditio ns . W e ca n obse r ve that any 2-categor y is equiv alent to a strict 2 -category and g et a weaker version of 2.6, o r follow the approa c h in [5] to prov e the ex istence of a free ω -ca tegory on a set. 3 The Quotien t of a Category b y the A ction of a M onoidal Cate- gory F or an action of a symmetric mo noidal category M , on a category C , we define the notion of a quotient Q = C / M , which is a 2 -category , and show that such a quotient a lways exists a nd is unique up to isomo rphism. Definition 4. A monoida l category ( M , ⊗ , β , l , r ) consists of a category M , a functor ⊗ : M × M− →M , an o b ject 1 ∈ O b ( M ) and three isomorphisms of functors β a,b,c : ( a ⊗ b ) ⊗ c → a ⊗ ( b ⊗ c ) , 1 ⊗ a l a − → a r a ← − a ⊗ 1; that satisfy 8 • (AA) co nsistency (i.e., s elf-compatibility) of asso ciativity called p entagr am identity (( ab ) c ) d β ab,c,d − − − − → ( ab )( cd ) β a,b,cd − − − − → a ( b ( cd )) =   y 1 a ⊗ β b,c,d x   (( ab ) c ) d β a,b,c ⊗ 1 d − − − − − − → ( a ( bc )) d β a,bc,d − − − − → a (( bc ) d ) • (A U) compa tibility of a s so ciativit y and unital constraints: a ⊗ 1 ⊗ b β a, 1 ,b − − − − → a ⊗ 1 ⊗ b r a ⊗ 1 b   y 1 a ⊗ l b   y a ⊗ b = − − − − → a ⊗ b Definition 5 . Let M b e a small symmetric monoidal catego r y and C a small 1 -categor y . 1. An action o f a monoidal category ( M , ⊗ , β , l , r ) on a ca tegory C co nsists o f a functor A : M × C − →C (also deno ted A : ( m, a ) 7→ m.a ), an ob ject and t wo is omorphisms o f functors β ∗ m,n,a : ( m ⊗ n ) .a → m. ( n.a ) , 1 .a u m − → a ; that sa tisfy • (AA) compatibility) of tw o asso ciativity constraints (again a pe n tagr am identit y), (( lm ) n ) a β ∗ lm,n,a − − − − − → ( lm )( na ) β ∗ l,m,na − − − − − → l ( m ( na )) =   y 1 l ⊗ β ∗ m,n,a x   (( lm ) n ) a β l,m,n ⊗ 1 a − − − − − − − → ( l ( mn )) a β ∗ l,mn,a − − − − − → l (( mn ) a ) • (A U) compatibility of asso ciativity and unita l constraints: ( m ⊗ 1) .a β ∗ m, 1 ,a − − − − → m ⊗ (1 .a ) r m . 1 a   y 1 a .u a   y m.a = − − − − → m.a W e a r e now ready to define the quotient of a catego ry C by a n action of a mo no idal categor y M , but first we recall fro m [3] the defin tition of natura l tr ansformation of functors be t ween 2-ca tegories. Suppo s e that A is 1 - category and B is a 2-categor y . A na tural transfo r mation F = ⇒ G b et ween tw o functor s F, G : C − →D of 2-categ ories consists of a 1-mor phism ζ x : F x − → Gx for each ob ject x ∈ A 0 and a 2- morphism η f : ζ y ∗ 0 F f = ⇒ Gf ∗ 0 ζ x for each 1-mo rphism x f − → y in A sub ject to the following conditions. F or all x ∈ A 0 , η x = ρ − 1 ζ x ∗ 1 λ ζ x , (1) and η is functorial in A . This means that for a ll x f − → y g − → z in A , η gf = α − 1 Gg, Gf ,ζ x ∗ 1 ( Gg ∗ 0 η f ) ∗ 1 α Gg, ζ y ,F f ∗ 1 ( η g ∗ 0 F f ) ∗ 1 α − 1 ζ z ,F g ,F f . (2) 9 Lo osely , this s a ys that the diagra m F x F f ✲ F y F g ✲ F z Gx ζ x ❄ Gf ✲ ⇐ = = = = = = = = = = = η f Gy ζ y ❄ Gg ✲ ⇐ = = = = = = = = = = = = η g Gz ζ z ❄ coincides with F x F ( g f ) ✲ F z Gx ζ x ❄ G ( g f ) ✲ ⇐ = = = = = = = = = = = = η gf Gz . ζ z ❄ These diagrams give the roug h idea, but sinc e comp osition in B is not strictly ass ocia tiv e, the diag rams are ambiguous. The pr ecise s ta temen t is given ab ov e in eq uation (2). Definition 6. A quotien t C / M of an action of M on C co nsists of a tiple ( Q, π , θ ), where Q is a 2-category , π : C − → Q , and θ : π ◦ p 2 = ⇒ π ◦ A is a natural transformation in 2 -Cat, where π p 2 and π A : M × C − → Q . W e ask that for any o ther such ( Q ′ , π ′ , θ ′ ), π ′ factors uniquely thro ugh π via some map F such that F θ = θ ′ . W e now offer an explicit descriptio n of a quotient ( Q, π , θ ). Letting θ = ( η , ζ ), the quotient ( Q, π , η ) is given by Q1 - Q7 listed b elow. Since θ is a morphism with source M × C , a sufficien t condition for functoria lity of θ is that η is functorial in C and M indep endently (Q3 , Q4) and that the η f a ’s are compatible with the η m x ’s (Q6). T o see this, observe that any 1 -morphism ( f , x ) ∈ M × C can b e decomp osed as ( f , 1) ∗ 0 (1 , x ) or (1 , x ) ∗ 0 (1 , f ). Hence, θ is deter mined by its v alues on mo rphisms of the form ( f , 1) a nd (1 , x ). T o be functo- rial, θ must b e functorial in each direction and take the same v alue on b oth p ossible deco mpos itions of ( f , x ). A quotient ( Q, π , θ ) o f C by M is equiv alent to the following data and conditions. • (Q1) a 2 -category Q tog e ther with a functor π : C − → Q . • (Q2) 1-morphisms ζ m a : π ( a ) − → π ( m.a ) in Q for ea c h a ∈ C 0 , m ∈ M 0 . • (Q3) 2- mo rphisms η m x : π ( x ⊗ m ) ∗ 0 ζ m a = ⇒ ζ m b ∗ 0 π x fo r each x ∈ H om C ( a, b ), m ∈ M 0 such that η is functorial in C . In other words, η m x fits int o a diagr am π ( a ) π ( x ) ✲ π ( b ) π ( m.a ) ζ m a ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) . ζ m b ❄ F or η to be functorial in C means simply that g iven a x − → b y − → c in C , π ( a ) π ( x ) ✲ π ( b ) π ( y ) ✲ π ( c ) π ( m.a ) ζ m a ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) ζ m b ❄ π ( y ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = η m y π ( m.c ) ζ m c ❄ 10 coincides with π ( a ) π ( y x ) ✲ π ( c ) π ( m.a ) ζ m a ❄ π ( y x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m y x π ( m.c ) ζ m c ❄ in the sens e of equa tion (2). • (Q4) 2-mo rphisms η f a : π ( a ⊗ f ) ∗ 0 ζ m a = ⇒ ζ m a for each f ∈ H om M ( m, n ), a ∈ Ob ( C ) such that η is functorial in M . In other words, η f a fits int o a dia gram π ( a ) = ✲ π ( a ) π ( m.a ) ζ m a ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( n.a ) ζ n a ❄ such that for all l f − → m g − → n in M , the diagr am π ( a ) = ✲ π ( a ) = ✲ π ( a ) π ( l.a ) ζ l a ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( m.a ) ζ m a ❄ π ( a ⊗ g ) ✲ ⇐ = = = = = = = = = = = = = η g a π ( n.a ) ζ n a ❄ coincides with π ( a ) = ✲ π ( a ) π ( l.a ) ζ l a ❄ π ( a ⊗ g f ) ✲ ⇐ = = = = = = = = = = = = = η gf a π ( n.a ) ζ n a ❄ in the sens e if e quation (2). • (Q5) F or a ∈ C 0 , m ∈ M 0 , η m a of Q 3 and Q 4 are the same, and equation (1) is satisfied. • (Q6) The η ’s are compatible in the se ns e that the following tw o diagra ms of 2- morphisms are “identical” in the sens e of equa tion (2). π ( a ) = ✲ π ( a ) π ( x ) ✲ π ( b ) π ( m.a ) ζ m a ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( n.a ) ζ n a ❄ π ( x ⊗ n ) ✲ ⇐ = = = = = = = = = = = = = η n x π ( n.b ) ζ n b ❄ 11 π ( a ) π ( x ) ✲ π ( b ) = ✲ π ( b ) π ( m.a ) ζ m a ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) ζ m b ❄ π ( b ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f b π ( n.b ) ζ n b ❄ • (Q7) Q is univ er s al with resp ect to these prop erties, i.e. for a ny other 2-categor y ( π ′ : C − → Q ′ , ζ ′ , η ′ ) satisfying (Q1 )-(Q4), π ′ factors uniquely throug h π : C − → Q . As a cor ollary to Theor em 2.6, the existence of a quotient is gua ranteed. Prop osition 3. 1. Given a c ate gory C with an action of a symmet ric monoidal c ate gory M , ther e exists a quotient 2-c ate gory C / M , which is unique up to isomorphism. Pr o of. W e let X b e the union of the following data: 1. C 2. a 1-mor phism ζ m a : a − → m.a for ea c h m ∈ M 0 , a ∈ C 0 3. a 2-mor phism η m x : ζ m b ∗ 0 x = ⇒ x ⊗ a ∗ 0 ζ m a for each a x − → b in C and m ∈ M 0 4. a 2-mor phism η f a : π ( a ⊗ f ) ∗ 0 ζ m a = ⇒ ζ m a for each f ∈ H om M ( m, n ), a ∈ C 0 More co ncretely , let X 1 = C ∪ { ζ m a } ( m,a ) ∈M 0 ×C 0 , let X 2 be the set o f all η ’s, and X = X 1 ∪ X 2 . This generating data pro duces a free pre - 2-categor y F X . W e let C b e the conditions des cribed in Q3-Q6 together with the relations needed to make the pre- 1-categor y gener ated by O b ( C ) ⊂ F X int o a strict 1 -category isomorphic to C . That is to say , we include the following relatio ns . Let ◦ denote comp osition in F X , a nd ∗ 0 denote compos ition in C . F or each f , g ∈ C , the relation f ◦ 0 g = f ∗ 0 g is in C . Also , C contains the relations ( f ◦ 0 g ) ◦ 0 h = f ◦ 0 ( g ◦ 0 h ) for each f , g , h ∈ C for which comp osition is defined. The fina l relations needed are f ◦ 0 s 0 f = f = t 0 f ◦ 0 f as well a s α f ,g,h = ( h ∗ 0 g ) ∗ 0 f , λ f = f , and ρ f = f . With these relations C , we attain the 2-c a tegory Q = A X,C . The conditions in C which rela te mor phisms in Ob ( C ) ⊂ F X are chosen precisely so that O b ( C ) ֒ → F X − →A X,C induces a morphism of 2-ca tegories π : C − →A X,C . Since F X maps to A X,C , A X,C clearly ha s the 1-mor phisms, ζ m a and 2-mor phisms η f a , η m x needed to b e a quo tien t categor y . The co nditio ns C w er e chosen exa ctly so that the rela tio ns describ ed in Q3-Q6 ho ld in A X,C . The universal prop erty of A X,C as the 2-catego ry ge ner ated b y X with rela tions C implies that the universal prop erty Q7 holds for A X,C . The uniqueness of A X,C is a consequence o f the universal prop erty Q7. 3.1 V ariations Definition 6 g iv es the quotient as a sort o f as y mmetrical colimit. Ho wev er , the pro of of Prop ositio n 3.1 can b e mo dified slightly to acco moda te v ariatio ns of Definition 6. F or instance, one can attain a mo r e symmetric version of Q with maps a − → m.a and maps m.a − → a . This can be a ccomplished by asking for another natura l transformation φ : π A = ⇒ π p 2 and mo difications i d π p 2 ⇛ φθ and id π A ⇛ θ φ with inv erses. Alternatively , we could req ue s t that θ and φ are inv er ses of e a c h other a nd get a stricter version. In another v ariation of Definition 6, we may also w a nt to include in Q 2- morphisms ϕ m,n a : ζ mn a = ⇒ β ∗ ∗ 0 ζ m na ∗ 0 ζ n a and ξ l,m,n a : β ∗ ∗ 0 ζ ( lm ) n a = ⇒ ζ l ( mn ) a satisfying a large coher ence diagram. This has the effect of dema nding that the choice of ζ is compatible with the tensor pro duct in M . 12 References [1] Gray , John W., F ormal c ate gory the ory: ad jointness for 2-c ate gories . Springer- V erlag, Berlin, 1 974 [2] Gurski, N., An algebr aic the ory of tric ate gories , www.math.yale.edu/ mg622 /tricats.p df, 2007 [3] Leinster, T., Basic Bicategories , (arXiv:math/98 10017 v1) [4] Schommer-Pries, C., The Cla ssification of Tw o -Dimensional Extended T op ological Field T he o ries, ht tp:// sites.go ogle.com/ site/chrissc hommer priesma th/Home/m y-web-documents/Sc hommer-Pries- Thesis-5-12- 09.pdf , 2 0 09 [5] Street, R., T he Algebra of O rient ed Simplexes, Journ al of Pur e and Applie d A lgebr a v.49 , pp.283-33 6 13