The Quotient of a Category by the Action of a Monoidal Category
aa r X i v : . [ m a t h . C T ] J a n The Quotient of a Category by the Action of a Monoidal Category
Brett Milburn ∗ Abstract
We introduce the notion of the quotient of a category C by the action A : M × C−→C of a unitalsymmetric monoidal category M . The quotient C / M is a 2-category. We prove its existence anduniqueness by first showing that every small 2-category has a presentation in terms of generators andrelations and then describing the generators and relations needed for the quotient C / M . We show that for any generating set X , there is a free 2-category A X on X . Furthermore, given a generatingset X with relations C , there is a 2-category A X,C satisfying a universal property. Moreover, any small2-category has a presentation in terms of generators and relations. We start by defining the weaker notionof a pre-2-category and showing the existence of free pre-2-categories and presentations of pre-2-categoriesby generators and relations. We then apply the technology of pre-2-categories via generators and relationsto attain the same results for 2-categories. There are various versions of free n-categories in the literature[2], [4], [5], which are suitable in the appropriate contexts. Schommer-Pries, for instance, considers free sym-metric monoidal bicategories. Our interest in presenting 2-categories in terms of generators and relations isdue to its utility in taking quotient categories.Given a unital, symmetric monoidal category M and an action A : M × C−→C of M on C , we would liketo explain what it means to take the quotient C / M . Our definition of the quotient is motivated by a morefamiliar quotient construction. Given the action of a monoid M on a space X , the smart notion of quotient X/M is not a space but a category. The objects of
X/M are the points of X , and morphisms in X/M areindexed by X × M . Instead of identifying points x and y = m.x in X which are related by m ∈ M , there isa morphism ζ mx from x to m.x , thus remembering how x and y are related. If M is a symmetric monoidalcategory acting on a category C , we apply the same philosophy. This time, however, the quotient Q = C / M is a 2-category. In addition to the 1-morphisms in C , objects of M provide 1-morphisms ζ mx : x −→ m.x for x ∈ Ob ( C ), m ∈ Ob ( M ). We require that ζ is consistent with morphisms in C and M in a sense describedby conditions Q4-Q6 in §
3. Roughly, consistency of ζ with morphisms in M and C means that we requirecertain diagrams to commute–ones that we would expect to commute in any reasonable definition of quotient.However, instead of asking these diagrams to commute on the nose, we only require them to commute up tosome 2-morphisms. In section 3 we define the quotient C / M and demonstrate its existence and uniquenessup to isomorphism. We consider in the sequel only small n-categories and will only be concerned with n -categories for n ≤
2. InDefinition 1 we recall the definition of 2-category but also define a weaker notion of pre-2-category, which islike a 2-category in that it has 0-objects, 1-morphism, 2-morphism and compositions but with none of the ∗ [email protected] n -category as having 0-morphisms (i.e. objects), 1-morphisms, etc. as distinct, any k -morphism x , is identified with the ( k + 1)-morphism id x . In this way, all k -morphisms are on the same footing asmembers of the same set. Definition 1.
1. Suppose that 0 ≤ n ≤ ∞ . The data for a (small) strict n -category is a set A with maps s i , t i : A−→A for all i < n and maps ∗ i : A × A A−→A , where
A × A A is the fibered product overmaps 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 1, where ( A , s i , t i )is a an object of gr n Cat together with compositions ∗ i for i ≤ n − A , A i := s i A = t i A is the set ofof i-morphisms or alternately i-objects , which has the structure of an strict-i-category, pre-i-category,or i-categorically graded set, respectively.5. A is a pre-2-category A with 2-isomorhphisms α h,g,f : ( h ∗ g ) ∗ f = ⇒ h ∗ ( g ∗ f ) for each f, g, h ∈ A , whenever the compositions are defined, as well as 2-isomorphisms λ f : t f ∗ f = ⇒ f and ρ f : f ∗ s f = ⇒ f for all 1-morphisms f ∈ A . We require A to satisfy the conditions described in[1], [3]. These conditions, which include strict associativity for ∗ , are called coherence conditions for2-categories . The collection of 2-categories are the objects of a category 2 Cat , the morphisms of whichare morphisms of pre-2-categories which preserve the 2-morphisms α , λ , ρ .There are several useful functors relating the above categories, namely • forgetful functors 2 Cat −→ p Cat −→ gr Cat and more generally p n Cat −→ gr n Cat • full embeddings gr n Cat ֒ → gr ( n +1) Cat ֒ → gr ∞ Cat and p n Cat ֒ → p ( n +1) Cat ֒ → p ∞ Cat attained byletting s i = t i = id for i ≥ n . 2 a pair of forgetful functors p ( n +1) Cat −→ p n Cat , the first of which is given by A 7→ A n and the secondforgets the higher structure maps. • Composing the previous forgetful functor n times, we get Ob : p n Cat −→ p Cat ≃ Set , which sends apre-n-category to its underlying set. Similarly, Ob : gr n Cat −→ Set sends an n-categorically graded setto its underlying set. Definition 2. We call the morphisms in 2 Cat , gr n Cat , and p n Cat maps or functors . A map F : C−→D in p n Cat or gr n Cat is called injective or surjective if the underlying map Ob ( f ) of sets is injective or surjectiverespectively. More generally, set-theortic notions such as inclusions, intersections, etc. make sense in gr n Cat and p n Cat by considering the underlying sets. We say, for instance, that C ∈ p n Cat is a sub-pre-n-categoryof D ∈ p n Cat , written C ⊂ D , if Ob ( C ) ⊂ Ob ( D ) and C is closed under all ∗ k , s k , t k in D . Notation : In p Cat , we let the symbol ∗ generically denote “ ∗ or ∗ .” In order to define free pre-2-categories, we will need to have formal strings or words representing composition. With this in mind, wedenote a formal string of two objects in the following way. For Z ∈ gr Cat and subobjects W, Y of Z , let W • Y = { ( w, i, y ) ∈ W × Z / Z × Y | s i x = t i y } and W • j Y = { ( w, i, y ) ∈ W × Z / Z × Y | s i x = t i y and i = j } .For w ∈ W , y ∈ Y , let w • i y = ( w, i, y ) ∈ W • i Y , and let w • y generically denote “ w • y or w • y .” Remark 2.1. Another way to view the notation W • Y is as follows. We can view the correspondences s k , t k : Y ⇒ X over X as a monoidal category with product W × kX Y given by W • k Y defined above.If we were to construct free strict 2-categories from Y over X , we would be interested in taking the freeassociative algebra P n ≥ Y × n , whereas in the construction of free pre-2-categories, we will be describinga refined version the free non-associative algebra P n ≥ Y × n × T r n of such a correspondence (where T r n denotes all trees with n leaves).We now show the existence of free pre-1-categories and pre-2-categories. We will show the existence ofa pre-2-category generated by a 2-categorically graded set X , but we would also like to consider the moregeneral situation of generating a pre-2-category from a 1-categorically graded set X , which generates a freepre-1-category C X described in Lemma 2.2 and two maps of sets s , t : X ⇒ C X . Lemma 2.2. 1. The forgetful functor p Cat −→ gr Cat has a left adjoint X 7→ C X . More explicitly,given X ∈ gr Cat , there exists an object C X ∈ p Cat with the property that there exists an inclusion ι X : X ֒ → C X in gr Cat and for any D ∈ p Cat and F ∈ Hom gr Cat ( X, D ) , F factors uniquelythrough C X , i.e. extends uniquely to a map ˜ F ∈ Hom p Cat ( C X , D ) .2. Given the data of ( C , s , t , ∗ ) ∈ p Cat and a set X together with maps of sets s , t : X ⇒ C suchthat σ s = σ t for all σ ∈ { s , t } ,(a) The disjoint union X = X ∪ C is a 2-categorically graded set.(b) There exists F X ∈ p Cat , called the free 2-pre-category on X , with the following property. Thereis an inclusion ι X : X −→F X in gr Cat , and if D ∈ p cat and F : X −→ D is a morphism in gr Cat such that F |C is a map in p Cat , then F extends uniquely to a map ˜ F : F X −→ D in p Cat .Proof. 1. The pre-category C = C X is going to be built out of chains of length n like the path categoryfor X except that C keeps track of the order of composition, as we no longer require associativity. Wedefine chains of length n recurssively by letting S = X and then defining S n = F ≤ p ≤ n − S p • S n − p ,where S p • S q = { ( x, y ) ∈ S p × S q | s x = t y } . We let C = F ≤ n< ∞ S n .Define s , t on S to agree with the source and target maps already defined on S = X ∈ gr Cat .Now for x • y ∈ S p • S q , define s ( x • y ) = s y and t ( x • y ) = t y . Finally, composition on C isdefined as follows. For x ∈ S p , y ∈ S q such that s x = t y , x ∗ y := x • y ∈ S p + q . One may easily3heck that C ∈ gr Cat and that conditions 1(a)iv and 1(b)iii of Definition 1 so that C ∈ p Cat .Given D ∈ p Cat and F : X −→ D in gr Cat , we would like to extend F to a map ˜ F : C−→ D of pre-2-categories. We must have ˜ F | S = F . Now, having defined ˜ F | S k for k < n , if x • y ∈ S p • S n − p ⊂ S n ,letting ˜ F ( x • y ) = F x ∗ F y defines ˜ F on all of C , and obviously, in order to respect composition, thisis the only possible choice for ˜ F .2. (a) We define s i , t i so that on C , s , t agree with the source and target maps for C ∈ gr Cat ⊂ gr Cat and ( s ) |C , ( t ) |C = id C . On X , we let s , t : X −→C be the maps specified above, and for σ ∈ { s , t } , we let σ | X = σ s : X −→C or equivalently σ t . It is trivial to verify thatproperties 1(a)i, 1(b)i, and 1(b)ii of definition 1 are satisfied.(b) Let S = X F C , let S = ( S • S ) \ C • C , and S n = F S p • S n − p for n > 2. Now we define F X = S ≤ n< ∞ S n . Let s ( x • y ) = s y , t ( x • y ) = t x , s ( x • y ) = s y , t ( x • y ) = t x , and σ ( x • y ) = σ x ∗ σ y . To see that this composition makes sense, an easy inductive proof showsthat s ( S p ) , t ( S p ) ⊂ C for all p . Note that ( F X ) = C . With these source and target maps, F X is a 2-categorically graded set. There are composition laws on F X as follows. x ∗ y = (cid:26) x ∗ y if x, y ∈ C x • y otherwiseand x ∗ y = x • y . One can check that F X ∈ p Cat .Suppose F : X −→ D is a map in gr Cat such that F restricted to C is a map in p Cat . Havingdefined ˜ F on S k for k ≤ n , define ˜ F on S n +1 by ˜ F ( x • i y ) = ˜ F x ∗ i ˜ F y for i ∈ { , } . Clearly ˜ F iswell defined, and ˜ F ( x ∗ i y ) = ˜ F x ∗ i ˜ F y . Furthermore, as X ⊂ F X in gr Cat , it is apparent that˜ F is the only possible extension of F to a map ˜ F ∈ Hom p Cat ( F X , D ). Corollary 2.3. The forgetful functor p Cat −→ gr Cat is left adjoint to the functor which sends X ∈ gr Cat to F ( X \ X ) ∪C X ∈ p Cat .Proof. This is a special case of Lemma 2.2. Suppose X ∈ gr Cat . Let C = C X and X ′ = ( X \ X ) ∪ C .We take X \ X instead of all of X in order to avoid having redundant 1-morphisms. By composing s , t : ( X \ X ) ⇒ X with the inclusion X ֒ → C to get maps ( X \ X ) ⇒ C , part 2a of Lemma 2.2guarantees that X ′ is a 2-categorically graded set.Given D ∈ p Cat and a map X F −→ D in gr Cat , we aim to give a map F X ′ −→ D in p Cat and showthat this assignment Hom gr Cat ( X, D ) −→ Hom p Cat ( F X ′ , D ) is an isomorphism. By Lemma 2.2 part 1, F | X ∈ Hom gr Cat ( X , D ) ≃ Hom p Cat ( C X , D ). Here we consider D as a pre-1-category by forgettingthe higher structure maps. Note also that C ∈ p Cat ֒ → p Cat and Hom p Cat ( C , D ) ≃ Hom p Cat ( C , D ).Since we have extended F from X to C , this allows us to extend F uniquely from X ⊂ X ′ to a map F : X ′ −→ D in gr Cat such that F |C : C−→ D is a map of pre-2-categories. By Lemma 2.2(2b), F : X −→ D extends uniquely to a map ˜ F : F X ′ −→ D in p Cat . By the uniqueness of the extensions, the map Hom gr Cat ( X, D ) −→ Hom p Cat ( F ( X \ X ) ∪C X , D ) is an inclusion. Since X ⊂ X ′ ⊂ F X ′ in gr Cat , everymap G : F X ′ −→ D in p Cat is an extension of G | X : X −→ G in gr Cat , whence Hom gr Cat ( X, D ) ≃ Hom p Cat ( F ( X \ X ) ∪C X , D ). Definition 3. 1. As in Lemma 2.2, given the data X = ( X , X ⇒ C X ) of X ∈ gr Cat (which defines( C X , s , t , ∗ ) ∈ p Cat ) and a set X with maps of sets s , t : X −→C X such that σ s = σ t forall σ ∈ { s , t } , the pre-2-category generated by X is the free pre-2-category F X ∪C X , which by abuseof notation we also denote by F X . The data X is the generating data for the pre-2-category F X . Wealso write X = X ∪ X for brevity. 4. A set of conditions on generating data X is a binary relation on F X . Lemma 2.4. Given generating data X and conditions C , there exists an equivalence relation ∼ on F X suchthat F X / ∼ ∈ gr Cat and has the property that for any D ∈ p Cat and F ∈ Hom p Cat ( F X , D ) such that xCy implies F ( x ) = F ( y ) for x, y ∈ F X , F factors through F X −→F X / ∼ in gr Cat .Proof. Let ∼ denote the finest relation on F X satisfying the following conditions:P0: ∼ is an equivalence relation.P1: If xCy , then x ∼ y .P2: If x ∼ y , then σ i x ∼ σ i y for σ i ∈ { s , t , s , t } .P3: If x ∼ x ′ and y ∼ y ′ , then x • y ∼ x ′ • y ′ whenever both compositions are defined.The notation in P3 is explained at the beginning of § x ∼ y for all x, y ∈ F X is such a relation. Because P0-P3 are closed under interesctions (i.e. mutual refinements), Zorn’slemma ensures the existence of a finest relation satisfying P0-P3.Now suppose F : F−→ D as above. Then the relation xRy if F x = F y satisfies P0-P3. Thus, F factorsthrough F X /R ∈ gr Cat . Since ∼ is the smallest such relation, F−→F /R factors through F / ∼ . Hence, F also factors through F / ∼ .We now show that for any generating set X and conditions C , there is a pre-2-category F X/C generatedby X and satisfying C . Theorem 2.5. Given generating data X = X ∪ X and conditions C , there exists a unique F X/C ∈ p Cat satisfying:1. There is a map G : F X −→F X/C in p Cat such that for all x, y ∈ F X , xCy implies G ( x ) = G ( y ) .2. F X/C is universal among pre-2-categories satisfying the above property in the sense that for any othermap F : F X −→ D in p Cat for which xCy implies F x = F y for all x, y ∈ F X , F factors uniquelythrough G as seen in the diagram in p Cat F X F ✲ D F X/C G ❄ ✲ . Proof. First we consider only 0-objects and 1-morphisms to get a quotient category C ′ from C = C X . Therelation ∼ on F X of Lemma 2.4 restricts to an equivalence relation on C = ( F X ) . That is to say, for x, y ∈ C , x ∼ y in C if and only if x ∼ y in F X . Additionally, C := C / ∼∈ gr Cat because ∼ satisfies P2. Now wedefine C ′ by taking S = C , S = { x • y | x, y ∈ S and for all x ′ ∼ x , y ′ ∼ y, x ′ ∗ y ′ is not defined } . Wedefine S n = F C ′ = S ∞ n =1 S n . Define s ( x • y ) := s y , t ( x • y ) := t x .Composition is defined as x ∗ y = (cid:26) x ′ ∗ y ′ if x ′ ∗ y ′ ∈ C is defined for some C ∋ x ′ ≡ x , C ∋ y ′ ≡ yx • y = otherwiseso that C ′ ∈ gr Cat . This composition gives C ′ the structure of a pre-1-category.We claim that any map F : C−→ D of pre-1-categories such that xCy implies F x = F y must factor through C ′ . Such a map F : C−→ D must factor through C ∈ gr Cat , which can be extended to a map ˜ F : C ′ −→ D p Cat via ˜ F ( x • y ) = F ( x ) ∗ F ( y ).We now have X ⇒ C−→C ′ , making X ′ = X ∪ C ′ a categorically graded set with a map C ∪ X −→C ′ ∪ X in gr Cat . This induces H : F X −→F X ′ in p Cat . The next step is to identify all remaining 2-morphismsrelated by C . We therefore want a relation ∼ on F X ′ which is the finest relation satisfying:P0: ∼ is an equivalence relation.P1 ′ : If xCy for x, y ∈ F X , then Hx ∼ Hy .P2: If x ∼ y , then σ i x ∼ σ i y for σ i ∈ { s , t , s , t } .P3: If x ∼ x ′ and y ∼ y ′ , then x • y ∼ x ′ • y ′ whenever both compositions are defined.P4: If x, y ∈ C ′ ⊂ F X ′ , then x ∼ y implies x = y .Suppose there exists such a relation. Conditions P0-P4 are closed under taking refinements of two suchrelations. Zorn’s lemma implies that there is a minimal such relation R . Let F X/C := F X ′ /R . PropertiesP2 and P3 guarantee that F X/C is a pre-2-category. We wish to show that F X/C has the specified universalproperty. To this end, let F : F X −→ D be any map in p Cat such that F x = F y whenever xCy . Then F |C : C−→ D factors uniquely through C ′ as we have already shown, thus inducing a unique map F ′ : F X ′ −→ D in p Cat . Define a relation Q on F X ′ by xQy if x and y lie in the same fiber of F ′ . Conditions P0-P3 aboveare satisfied by Q . Clearly, since R is the finest relations satisfying P0-P4, it is also the finest relationssatisfying P0-P3. Hence, F ′ factors uniquely through F X ′ /Q , which factors uniquely through F X ′ /R in p Cat via the map π : F X ′ /R −→F X ′ /Q . Therefore, F factors uniquely through F X −→F X ′ /R as desired.This can be expressed in the following commutative diagram in p Cat F X ′ F ′ ✲ D F X ′ /R ❄ π ✲ F X ′ /Q. ✻ ✲ It only remains to show that there exists a relation on F X ′ satisfying P0-P4. In general, let A ∈ p cat and C = A . Given a subset S ⊂ C × C such that: • C ≃ ∆ C ⊂ S , • σ π = σ π on S for all σ ∈ { s , t } , • if ( f, g ),( h, k ) ∈ S satisfy t h = s f , then ( f h, gk ) ∈ S , and • ( h, g ) , ( g, f ) ∈ S implies ( h, f ) ∈ S ,then S is a pre-2-category with stucture maps s = ∆ π , t = ∆ π , s = ∆ s π , t = ∆ t π , ( f, h ) ∗ ( g, k ) =( f ∗ g, h ∗ k ), and ( f, h ) ∗ ( h, k ) = ( f, k ). The important point is that S has the property that forevery f, g ∈ S ≃ C , there exists at most one 2-morphism from f to g . Now, starting from F X ′ , let S = { ( f, g ) ∈ C ′ | there exists a 2-morphism α : f = ⇒ g } . Then there is a projection π : F X ′ −→S , and thefibers of π determine a relation satisfying P0-P4. In order to apply the previous results to 2-categories, we observe that a 2-category is simply a pre-2-categorywith extra data and conditions. Theorem 2.6. Let X = X ∪ X be generating data and impose conditions C . There exists a unique (upto isomorphism) 2-category A X,C equipped with a map G : F X −→A X,C in p Cat such that G ( x ) = G ( y )6 henever xCy and such that A X,C is universal with respect to this property in the following sense. Givena 2-category D and a map F : F X −→ D in p Cat such that for all x, y ∈ F X , xCy implies F ( x ) = F ( y ) , F factors uniquely through G in p Cat in such a way that the map H : A X,C −→ D such that HG = F is amap of 2-categories. We call A X,C the 2-category generated by X with conditions C .Proof. This is only a slight modification of the proof of Thorem 2.5 where the generating data is enlargedto contain the structure morphisms α f,g,h , λ f , ρ f and we add to C coherence conditions for 2-categories. Wework under the assumption that the generating data X does not already contain the structure 2-morphismsfor a 2-category.Beginning with F X ′ , the pre-2-category defined in the third paragraph of the proof of Theorem 2.5,we add to X ′ λ f : t f ∗ f = ⇒ f and ρ f : f ∗ s f = ⇒ f for each f ∈ C ′ = ( F X ′ ) aswell as a 2-morphism α h,g,f : h ∗ ( g ∗ f ) = ⇒ ( h ∗ g ) ∗ f for each triple of f, g, h of 1-morphisms in C ′ . Also we add 2-morphisms α − h,g,f : ( h ∗ g ) ∗ f = ⇒ h ∗ ( g ∗ f ), ρ − f : f = ⇒ f ∗ s f , and λ − f : f = ⇒ t f ∗ f which are going to be the inverses of α h,g,f , λ f , ρ f respectively in the 2-category A X,C . Let Y = X ′ F { α f,h,g , λ f , ρ f , α − f,g,h , ρ − f , λ − f } f,g,h ∈C ′ and C ′ = C ∪ { coherence conditions for a 2-category } ∪ I .Here I denotes the set of relations { ( α h,g,f • α − h,g,f , h • ( g • f )) , ( α − h,g,f • α h,g,f , ( h • g ) • f ) , ( λ f • λ − f , f ) , ( λ − f • λ f , t f • f ) , ( ρ f • ρ − f , f ) , ( ρ − f • ρ f , f • s f ) } , where we think of the binary relation C ′ as a subset of Ob ( F Y ) × Ob ( F Y ).The inclusion (i.e. injective map) X ′ ֒ → Y in gr Cat induces an inclusion F X ′ ֒ → F Y in p Cat . Let R be the relation on F X ′ (described in the penultimate paragraph of the proof of Theorem 2.5) such that F X/C = F X ′ /R . Now we 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 relation R ′ is proven by the same arguments used in the proofof Theorem 2.5. The quotient A X,C := F Y /R ′ is a pre-2-category containing F X/C as a subcategory (in thesense that there is an inclusion F X/C ֒ → F Y /R ′ ). The generating data Y contains the extra data neededto make F X ′ into a 2-category, and the conditions C ′ are chosen for the purpose of ensuring that F Y /R ′ satisfies the coherence conditions for 2-categories.More precisely, in order for A X,C to be a 2-category, it must contain 2-isomorphism α f,g,h λ f , and ρ f for all f, g, h ∈ ( A X,C ) , and A X,C must satisfy the coherence conditions. One obstacle to A X,C to be a2-category is that we have not added enough α ’s ρ ’s and λ ’s. We have added an α f,g,h ρ f and λ f for all f, g, h ∈ C ′ ⊂ ( A X,C ) , but we need one for each f, g, h ∈ ( A X,C ) . This, however, is not a problem sinceno two 1-morphism are identified in passing from F Y to F Y /R ′ , whence C ′ ≃ ( F X ′ ) = ( F Y ) ≃ ( F Y /R ′ ) .The other possible obstacle for F Y /R ′ to qualify as a 2-category is that there may be 2-morphisms in F Y /R ′ which ought to be identified but which are not, which would mean that the coherence conditions arenot satisfied. For example, ∗ should be strictly associative. However, the choice of R ′ and the fact that F Y −→F Y /R ′ is surjective preclude this from happening. Therefore, F Y /R ′ is a 2-category which comeswith a map F X −→F Y /R ′ in p Cat .It only remains to show that A X,C has the desired universal property. Given D ∈ Cat and F : X −→ D in gr Cat such that F : F X −→ D identifies objects related by C , then F induces a map F : F X ′ −→ D byTheorem 2.5. The map X ′ −→ D in gr Cat extends uniquely to a map Y −→ D because there is only onepossible choice of where to send each α f,g,h , ρ f , λ f , namely the structure maps α F ( f ) ,F ( g ) ,F ( h ) , λ F ( f ) , ρ F ( f ) in D . The map from F Y already has the property that F x = F y if xCy (Here we abuse notation and denoteall maps by F ). The only additional relations in C ′ are the coherences conditions for 2-categories. Theserelations will automatically become equalities in D because D is a 2-category. Thus, 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 preserves the maps α , λ , ρ ,so it is a map of 2-categories. Remark 2.7. If the original conditions C are such that no two 1-morphism in C X are identified in F X bythe equivalence relation ∼ of Lemma 2.4, then we may initially include the 2-category data α f,g,h , λ f , ρ f F X/C is already a 2-category. The onlyobstacle to doing this in general is that there may be morphism in C ′ X which were not in C X .Theorem 2.6 has the unusual property that it makes reference to pre-2-categories in the description of A X,C . The following corollary justifies calling A X the 2-category generated by X . Corollary 2.8. Consider any generating data X = ( X ⇒ C X ) .1. A X has the following universal property. There is a canonical inclusion ι A : X −→A X of 2-categoricallygraded sets, and for any 2-category B with an inclusion ι B : X −→B such that ( ι B ) |C X is a map in p Cat , there is a unique map of 2-categories F : A X −→B such that ι B = F ι A .2. If X is generating data and C is a binary relation on A X , then there exists a 2-category A X/C satisfying:(a) There is a map of 2-categories G : A X −→A X/C such that xCy implies Gx = Gy , and(b) Any other map F : A X −→B of 2-categories such that xCy implies F x = F y factors uniquelythrough G .3. Any 2-category has a presentation in terms of generators and relations, i.e. any B ∈ Cat is isomorphicto some A X/C for some generating data X and binary relation C on A X .Proof. 1. By Lemma 2.2, to have such a map ι B is the same as having a map F X −→B in p Cat . Theresult now follows directly from Theorem 2.6.2. Let Y = X ⊔ { α f,g,h , λ f , ρ f } f,g,h ∈ ( A X ) . We have F X ֒ → F Y π −→ A X −→A Y,π − C . Since F : A X −→B identifies objects related by C , F π identifies objects related by π − C , whence F π factors uniquelythrough A X/C := A Y,π − C via some map H : A X/C −→B of pre-2-categories. Since π is surjective,the composition A X −→A X/C H −→ B is F . Note that π − C contains the coherence conditions for a2-category, so A X/C is a 2-category.3. Suppose B is a 2-category. Let X = B ∈ gr Cat , so A X p −→ B is a surjection. Let C be the binaryrelation on A X which relates every two points in the same fiber of p . Then A X/C ≃ B .Theorems 2.5, 2.6 can be extended to strict 2-categories. There is more than one approach to extendingthese results. This can be done by modifying the proofs to get a strict 2-category given by generators andrelations. At the first stage, the construction of the free pre-1-category C X is replaced by the free 1-category,i.e. the path category generated by X . The free strict 2-category F X can be constructed in a similary way.Alternately, we can view a strict 2-category as a pre-2-category with extra conditions. We can observe thatany 2-category is equivalent to a strict 2-category and get a weaker version of 2.6, or follow the approach in[5] to prove the existence of a free ω -category on a set. For an action of a symmetric monoidal 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 always exists and is unique up to isomorphism. Definition 4. A monoidal category ( M , ⊗ , β, l, r ) consists of a category M , a functor ⊗ : M × M−→M ,an object 1 ∈ Ob ( M ) and three isomorphisms of functors β a,b,c : ( a ⊗ b ) ⊗ c → a ⊗ ( b ⊗ c ) , ⊗ a l a −→ a r a ←− a ⊗ (AA) consistency (i.e., self-compatibility) of associativity called pentagram identity (( ab ) c ) d β ab,c,d −−−−→ ( ab )( cd ) β a,b,cd −−−−→ a ( b ( cd )) = y a ⊗ β b,c,d x (( ab ) c ) d β a,b,c ⊗ d −−−−−−→ ( a ( bc )) d β a,bc,d −−−−→ a (( bc ) d ) • (AU) compatibility of associativity and unital constraints: a ⊗ ⊗ b β a, ,b −−−−→ a ⊗ ⊗ b r a ⊗ b y a ⊗ l b y a ⊗ b = −−−−→ a ⊗ b Definition 5. Let M be a small symmetric monoidal category and C a small 1-category.1. An action of a monoidal category ( M , ⊗ , β, l, r ) on a category C consists of a functor A : M × C−→C (also denoted A : ( m, a ) m.a ), an object and two isomorphisms of functors β ∗ m,n,a : ( m ⊗ n ) .a → m. ( n.a ) , .a u m −→ a ;that satisfy • (AA) compatibility) of two associativity constraints (again a pentagram identity),(( lm ) n ) a β ∗ lm,n,a −−−−−→ ( lm )( na ) β ∗ l,m,na −−−−−→ l ( m ( na )) = y l ⊗ β ∗ m,n,a x (( lm ) n ) a β l,m,n ⊗ a −−−−−−−→ ( l ( mn )) a β ∗ l,mn,a −−−−−→ l (( mn ) a ) • (AU) compatibility of associativity and unital constraints:( m ⊗ .a β ∗ m, ,a −−−−→ m ⊗ (1 .a ) r m . a y a .u a y m.a = −−−−→ m.a We are now ready to define the quotient of a category C by an action of a monoidal category M , butfirst we recall from [3] the defintition of natural transformation of functors between 2-categories. Supposethat A is 1-category and B is a 2-category. A natural transformation F = ⇒ G between two functors F, G : C−→D of 2-categories consists of a 1-morphism ζ x : F x −→ Gx for each object x ∈ A and a 2-morphism η f : ζ y ∗ F f = ⇒ Gf ∗ ζ x for each 1-morphism x f −→ y in A subject to the following conditions.For all x ∈ A , η x = ρ − ζ x ∗ λ ζ x , (1)and η is functorial in A . This means that for all x f −→ y g −→ z in A , η gf = α − Gg,Gf,ζ x ∗ ( Gg ∗ η f ) ∗ α Gg,ζ y ,F f ∗ ( η g ∗ F f ) ∗ α − ζ z ,F g,F f . (2)9oosely, this says that the diagram F x F f ✲ F y F g ✲ F zGxζ x ❄ Gf ✲ ⇐ = = = = = = = = = = = η f Gyζ y ❄ Gg ✲ ⇐ = = = = = = = = = = = = η g Gzζ z ❄ coincides with F x F ( gf ) ✲ F zGxζ x ❄ G ( gf ) ✲ ⇐ = = = = = = = = = = = = η gf Gz.ζ z ❄ These diagrams give the rough idea, but since composition in B is not strictly associative, the diagrams areambiguous. The precise statement is given above in equation (2). Definition 6. A quotient C / M of an action of M on C consists of a tiple ( Q, π, θ ), where Q is a 2-category, π : C−→ Q , and θ : π ◦ p = ⇒ π ◦ A is a natural transformation in 2-Cat, where πp and πA : M × C−→ Q .We ask that for any other such ( Q ′ , π ′ , θ ′ ), π ′ factors uniquely through π via some map F such that F θ = θ ′ .We now offer an explicit description of a quotient ( Q, π, θ ). Letting θ = ( η, ζ ), the quotient ( Q, π, η ) isgiven by Q1-Q7 listed below. Since θ is a morphism with source M×C , a sufficient condition for functorialityof θ is that η is functorial in C and M independently (Q3, Q4) and that the η fa ’s are compatible with the η mx ’s (Q6). To see this, observe that any 1-morphism ( f, x ) ∈ M × C can be decomposed as ( f, ∗ (1 , x ) or(1 , x ) ∗ (1 , f ). Hence, θ is determined by its values on morphisms of the form ( f, 1) and (1 , x ). To be functo-rial, θ must be functorial in each direction and take the same value on both possible decompositions of ( f, x ).A quotient ( Q, π, θ ) of C by M is equivalent to the following data and conditions. • (Q1) a 2-category Q together with a functor π : C−→ Q . • (Q2) 1-morphisms ζ ma : π ( a ) −→ π ( m.a ) in Q for each a ∈ C , m ∈ M . • (Q3) 2-morphisms η mx : π ( x ⊗ m ) ∗ ζ ma = ⇒ ζ mb ∗ πx for each x ∈ Hom C ( a, b ), m ∈ M such that η isfunctorial in C . In other words, η mx fits into a diagram π ( a ) π ( x ) ✲ π ( b ) π ( m.a ) ζ ma ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) .ζ mb ❄ For η to be functorial in C means simply that given a x −→ b y −→ c in C , π ( a ) π ( x ) ✲ π ( b ) π ( y ) ✲ π ( c ) π ( m.a ) ζ ma ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) ζ mb ❄ π ( y ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = η m y π ( m.c ) ζ mc ❄ π ( a ) π ( yx ) ✲ π ( c ) π ( m.a ) ζ ma ❄ π ( yx ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m y x π ( m.c ) ζ mc ❄ in the sense of equation (2). • (Q4) 2-morphisms η fa : π ( a ⊗ f ) ∗ ζ ma = ⇒ ζ ma for each f ∈ Hom M ( m, n ), a ∈ Ob ( C ) such that η isfunctorial in M . In other words, η fa fits into a diagram π ( a ) = ✲ π ( a ) π ( m.a ) ζ ma ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( n.a ) ζ na ❄ such that for all l f −→ m g −→ n in M , the diagram π ( a ) = ✲ π ( a ) = ✲ π ( a ) π ( l.a ) ζ la ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( m.a ) ζ ma ❄ π ( a ⊗ g ) ✲ ⇐ = = = = = = = = = = = = = η g a π ( n.a ) ζ na ❄ coincides with π ( a ) = ✲ π ( a ) π ( l.a ) ζ la ❄ π ( a ⊗ gf ) ✲ ⇐ = = = = = = = = = = = = = η g f a π ( n.a ) ζ na ❄ in the sense if equation (2). • (Q5) For a ∈ C , m ∈ M , η ma of Q3 and Q4 are the same, and equation (1) is satisfied. • (Q6) The η ’s are compatible in the sense that the following two diagrams of 2-morphisms are “identical”in the sense of equation (2). π ( a ) = ✲ π ( a ) π ( x ) ✲ π ( b ) π ( m.a ) ζ ma ❄ π ( a ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f a π ( n.a ) ζ na ❄ π ( x ⊗ n ) ✲ ⇐ = = = = = = = = = = = = = η n x π ( n.b ) ζ nb ❄ ( a ) π ( x ) ✲ π ( b ) = ✲ π ( b ) π ( m.a ) ζ ma ❄ π ( x ⊗ m ) ✲ ⇐ = = = = = = = = = = = = = = η m x π ( m.b ) ζ mb ❄ π ( b ⊗ f ) ✲ ⇐ = = = = = = = = = = = = = η f b π ( n.b ) ζ nb ❄ • (Q7) Q is universal with respect to these properties, i.e. for any other 2-category ( π ′ : C−→ Q ′ , ζ ′ , η ′ )satisfying (Q1)-(Q4), π ′ factors uniquely through π : C−→ Q .As a corollary to Theorem 2.6, the existence of a quotient is guaranteed. Proposition 3.1. Given a category C with an action of a symmetric monoidal category M , there exists aquotient 2-category C / M , which is unique up to isomorphism.Proof. We let X be the union of the following data:1. C 2. a 1-morphism ζ ma : a −→ m.a for each m ∈ M , a ∈ C 3. a 2-morphism η mx : ζ mb ∗ x = ⇒ x ⊗ a ∗ ζ ma for each a x −→ b in C and m ∈ M 4. a 2-morphism η fa : π ( a ⊗ f ) ∗ ζ ma = ⇒ ζ ma for each f ∈ Hom M ( m, n ), a ∈ C More concretely, let X = C ∪ { ζ ma } ( m,a ) ∈M ×C , let X be the set of all η ’s, and X = X ∪ X . Thisgenerating data produces a free pre-2-category F X . We let C be the conditions described in Q3-Q6 togetherwith the relations needed to make the pre-1-category generated by Ob ( C ) ⊂ F X into a strict 1-categoryisomorphic to C . That is to say, we include the following relations. Let ◦ denote composition in F X , and ∗ denote composition in C . For each f , g ∈ C , the relation f ◦ g = f ∗ g is in C . Also, C contains the relations( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) for each f, g, h ∈ C for which composition is defined. The final relations neededare f ◦ s f = f = t f ◦ f as well as α f,g,h = ( h ∗ g ) ∗ f , λ f = f , and ρ f = f .With these relations C , we attain the 2-category Q = A X,C . The conditions in C which relate morphismsin Ob ( C ) ⊂ F X are chosen precisely so that Ob ( C ) ֒ → F X −→A X,C induces a morphism of 2-categories π : C−→A X,C . Since F X maps to A X,C , A X,C clearly has the 1-morphisms, ζ ma and 2-morphisms η fa , η mx needed to be a quotient category. The conditions C were chosen exactly so that the relations described inQ3-Q6 hold in A X,C . The universal property of A X,C as the 2-category generated by X with relations C implies that the universal property Q7 holds for A X,C . The uniqueness of A X,C is a consequence of theuniversal property Q7. Definition 6 gives the quotient as a sort of asymmetrical colimit. However, the proof of Proposition 3.1can be modified slightly to accomodate variations of Definition 6. For instance, one can attain a moresymmetric version of Q with maps a −→ m.a and maps m.a −→ a . This can be accomplished by asking foranother natural transformation φ : πA = ⇒ πp and modifications id πp ⇛ φθ and id πA ⇛ θφ with inverses.Alternatively, we could request that θ and φ are inverses of each other and get a stricter version. In anothervariation of Definition 6, we may also want to include in Q ϕ m,na : ζ mna = ⇒ β ∗ ∗ ζ mna ∗ ζ na and ξ l,m,na : β ∗ ∗ ζ ( lm ) na = ⇒ ζ l ( mn ) a satisfying a large coherence diagram. This has the effect of demanding thatthe choice of ζ is compatible with the tensor product in M .12 eferences [1] Gray, John W., Formal category theory: adjointness for 2-categories . Springer-Verlag, Berlin, 1974[2] Gurski, N., An algebraic theory of tricategories