aa r X i v : . [ m a t h . C T ] J a n ON 2-FINAL 2-FUNCTORS
JUN MAILLARD
A final functor between categories F : A → B is a functor that allows the restric-tion of diagrams on B to A without changing their colimits. More precisely, thefunctor F is final if, for any diagram D : B → B , there is a canonical isomorphism colim B D ∼ = colim A D ◦ F where either colimit exists whenever the other one does. There is a classical criterionfor final functors [Mac71, §IX.3]: a functor F : A → B is final if and only if, for anyobject b ∈ B , the slice category b/ F is nonempty and connected. Such a criterionalso exists for ( ∞ , -categories [Lur09, §4.1]: an ( ∞ , -functor F : A → B is final(with respect to any ∞ -diagram) if and only if for any object b ∈ B , the slice ∞ -category b/ F is weakly contractible. One would expect a similar result for anydimension: an ( n, -functor F : A → B is final (with respect to any n -diagram)if and only if, for any object b ∈ B , the slice ( n, -category b/ F is nonemptyand has trivial homotopy groups π k for ≤ k ≤ n − . Note that these arenot consequences of the known criterion for -functors and ( ∞ , -functors (seeremark 2.8 and remark 3.3). This paper presents a combinatorial proof in the case n = 2 (theorem 3.4). An application of this criterion will appear in my Ph.D.thesis [Mai]. 1. Bicategorical notions
We will follow the naming conventions of [JY21] for bicategories. In particular,the terms -category and -functor denote the strict ones. We will use the term (2 , -category for a -category with only invertible -morphisms, and the term (2 , -functor for a -functor between (2 , -categories.We recall some usual constructions and properties of -categories we will use,and introduce some notations.1.1. Notation.
The symbol ≃ denotes an isomorphism between two objects (in the -categorical sense).The symbol ∼ = denotes an equivalence between two objects (in the -categoricalsense).1.2. Definition.
Let C be a 2-category. The opposite 2-category of C , written C op , is the 2-category with • Objects: the objects of C • Hom-categories: C op ( A, B ) = C ( B, A ) Date : January 22, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Final functors, 2-functors, higher connectivity.Author supported by Project ANR ChroK (ANR-16-CE40-0003) and Labex CEMPI (ANR-11-LABX-0007-01).
Notation.
We write [ A , B ] for the -category of pseudofunctors, pseudonaturaltransformations and modifications, between two -categories A and B .1.4. Notation.
Let I , C be -categories and T be an object of C . We denote by ∆ T the constant functor I → C with value T .1.5. Definition.
Let I , C be -categories and D : I → C be a -functor. A (pseudo)bicolimit of D is an object L of C and a family of equivalences Ψ T : C ( L, T ) ∼ = [ I , C ]( D , ∆ T ) pseudonatural in T . When it exists, the bicolimit of D is unique up to equivalenceand the object L is noted bicolim I D .1.6. Notation.
We will use the term -diagram to denote a -functor we introducewith the intent to take its bicolimit.We will use the term cone under D with vertex T to denote objects of the categoryof pseudonatural transformations and modifications [ I , C ]( D , ∆ T ) .1.7. Definition.
Let C be a (2 , -category. Fix a (2 , -functor F : I → C and anobject c of C . The slice c/ F is the (2 , -category with: • Objects: the pairs ( i, f ) consisting of an object i of I and a morphism f : c → F i • Morphisms ( i, f ) → ( i ′ , f ′ ) : the pairs ( u, µ ) consisting of a morphism u : i → i ′ of I and a 2-isomorphism µ : f ′ → F ( u ) f of C : F i F i ′ c F uf f ′ µ • ( u, µ ) ⇒ ( v, ν ) : the 2-morphisms α : u ⇒ v satisfying: F i F i ′ c F v F uf f ′ µ F α = F i F i ′ c F vf f ′ ν • Compositions are induced by the compositions of I and C .A slice -category c/ F is endowed with a canonical forgetful -functor: c/ F → I ( i, f ) i ( u, µ ) uα α Combinatorial paths and homotopies
A 2-category C has an associated CW-complex | C | , defined using the Duskinnerve [JY21, §5.4], which maps objects of C to vertices, -morphisms to 1-simplicesand -morphisms to 2-simplices. There are thus notions of paths and homotopies N 2-FINAL 2-FUNCTORS 3 of paths in C . We give in this section a combinatorial approach to these, for (2 , -categories.We fix a (2 , -category C .2.1. Definition. A path (of 1-morphism) in C is a finite sequence of objects ( a i ) ≤ i ≤ n and a family of pairs ( ε i , f i ) ≤ i ≤ n of a sign ε ∈ {− , } and a morphism f i : (cid:26) a i − → a i if ε i = 1 a i → a i − if ε i = − Such a path is said to have source a and target a n .2.2. Notation.
We write p : a a n to denote a path with source a and target a n .A path can be pictured as a zig-zag of morphisms (potentially with consecutivemorphisms in the same direction): a f −→ a f ←− a f ←− . . . f n −→ a n Following the usual conventions, left-to-right arrows represents pairs with ε = 1 and right-to-left arrows pairs with ε = − . The empty path (at an object a ) shouldbe represented by a .There is an obvious notion of concatenation of paths with compatible target andsource, given by the concatenation of the sequence of morphisms.2.3. Definition.
We say two path p, p ′ of C are elementary homotopic , written p ∼ elem p ′ , in any of the following cases:(1) a Id −→ a ∼ elem a , for any object a (2) a Id ←− a ∼ elem a , for any object a (3) a f −→ a f −→ a ∼ elem a f f −−−→ a , for any composable pair f , f ofmorphisms(4) a f ←− a f ←− a ∼ elem a f f ←−−− a , for any composable pair f , f ofmorphisms(5) a u ←− a v −→ a ∼ elem a u ′ −→ a ′ v ′ ←− a , for any 2-isomorphism a ′ a a a u ′ v ′ u v We then define a homotopy relation ∼ on paths as the smallest congruent (for theconcatenation of paths), reflexive, symmetric and transitive relation encompassingthe relation ∼ elem .2.4. Remark.
We should pause to consider two consequences of (5):
JUN MAILLARD • a 2-morphism a a f f can be arranged into the following square: a a a a f IdId f This shows, together with (1) and (2), that a f −→ a ∼ a f −→ a , as onewould expect. • a 1-morphism a f −→ a can be used to form the square: a a a a f f Once again using (1) and (2), this proves that ( a f ←− a f −→ a ) ∼ a . Asimilar argument (putting f on the upper side of the square) shows that ( a f −→ a f ←− a ) ∼ a . Hence, up to homotopy, the paths a f −→ a and a f ←− a are mutual inverses.Note that for two paths to be homotopic, they must have the same source andthe same target.It is natural to look for a category of paths up-to homotopy:2.5. Definition.
The (algebraic) fundamental groupoid Π ( C ) of C is the 1-categorywith: • Objects: the objects of C . • Morphisms: the classes of paths between objects modulo the homotopyrelation. • Composition is induced by the concatenation of paths.2.6.
Definition.
The (2,1)-category C is said to be connected if for any pair ofobjects a, a ′ there is a path with source a and target a ′ .The (2,1)-category C is said to be simply connected if p ∼ p ′ for any pair ofpaths p, p ′ with same source and same target.2.7. Remark.
A (2,1)-category C is nonempty, connected and simply connected ifand only if its fundamental groupoid Π ( C ) is equivalent to , the category withexactly one object and one morphism.2.8. Remark.
Given a (2,1)-category C which is nonempty, connected and simplyconnected, its nerve | C | is not necessarily weakly contractible. Indeed higher ho-motopy groups may be nontrivial. For instance, one can realize the sphere S as N 2-FINAL 2-FUNCTORS 5 the nerve of the (2 , -category with two objects, two parallel 1-morphisms betweenthese objects, and two parallel 2-isomorphisms between these 1-morphisms.2.9. Remark.
For any algebraic path p in C , there is an associated topologicalpath | p | : I → | C | . The following assertions, which should result from simplicialapproximation, motivate the definitions of this section:The -category C is connected (resp. simply connected) if and only if the CW-complex | C | is connected (resp. simply connected).Two algebraic paths p, p ′ in C are homotopic if and only if the topological paths | p | , | p ′ | are homotopic.The categories Π ( C ) and Π ( | C | ) are equivalent.3. A criterion for 2-final 2-functors
Definition. A -functor F : A → B between (2 , -categories is -final if forany 2-diagram D : B → E , the pseudo bicolimits bicolim B D and bicolim A D ◦ F each exists if and only if the other one exists, and the canonical comparison mor-phism bicolim A D ◦ F → bicolim B D is an equivalence.3.2. Remark.
In the above definition, E is only assumed to be a -category. However,since B is a (2 , -category, the pseudo bicolimits can be equivalently computed in E g , the (2 , -category with the objects of E , the 1-morphisms of E , and the invert-ible E . Hence we could assume E to be a (2 , -category, withoutchanging the meaning of the definition.3.3. Remark. A -final -functor F : A → B between -categories is a functor suchthat, for any diagram D : B → E , the colimits colim B D and colim A D ◦ F eachexists if and only if the other one exists, and the canonical comparison morphism colim A D ◦ F → colim B D is an isomorphism.A -final -functor F : A → B between -categories (seen as -categories withonly the identities as -morphisms) is -final, since any diagram is also a -diagram.The converse is not true, though: there are -final functors which are not -final.3.4. Theorem.
Let A , B be two (2 , -categories. A 2-functor F : A → B is 2-final(definition 3.1) if and only if, for any object b ∈ B , the slice (2,1)-category b/ F isnonempty, connected and simply connected (definition 2.6). We will first prove the backward implication.Fix a 2-functor D : B → E . We will construct a pseudoinverse to the canonicalcomparison morphism bicolim A D ◦ F → bicolim B D This morphism correspond to a family of functors, pseudonatural in e : K : [ B , E ]( D , ∆ e ) → [ A , E ]( D ◦ F , ∆ e ) . We will construct a pseudoinverse L to K . Given a cone φ : D ◦ F ⇒ ∆ e , we obtaina cone L ( φ ) : D ⇒ ∆ e as follows: JUN MAILLARD • objects in the slice categories b/ F define the 1-morphism components L ( φ ) b (see definition 3.5), • paths in b/ F define natural transformations between the components (seedefinition 3.7), • homotopies between paths ensure the cohesion of these constructions (seeproposition 3.8).Consider an arbitrary cone under D ◦ F with vertex e ∈ E , that is, a pseudonatural transformation φ : D ◦ F → ∆( e ) . We first want to define a cone ψ under D with vertex e , using the cone φ .As a first step, we fix an object b and we want to define the component at bψ b : D ( b ) → e . Since the slice (2,1)-category b/ F is nonempty, we consider thefollowing candidate.3.5. Definition.
We fix an object in b/ F , that is an object a ( b ) ∈ A and a mor-phism α ( b ) : b → F ( a ( b )) . Define ψ ( a ( b ) ,α ( b )) : D ( b ) D ( F ( a ( b ))) e D ( α ( b )) φ a ( b ) We then consider the dependence of ψ ( a ( b ) ,α ( b )) on ( a ( b ) , α ( b )) . Fix anotherobject ( a ′ ( b ) , α ′ ( b )) ∈ b/ F . Since b/ F is connected there is a path p : ( a , α ) = ( a ( b ) , α ( b )) ( a n , α n ) = ( a ′ ( b ) , α ′ ( b )) which can be pictured as: F ( a ) F ( a ) F ( a ) F ( a n ) b F u F u · · · α α α α n µ µ Applying the 2-functor D and using the cone φ , we obtain the pasting diagram:(3.6) DF ( a ) D ( b ) DF ( a ) e DF ( a ) DF ( a n ) DF ( u ) DF ( u ) ... D ( µ ) φ u D ( µ ) − φ − u We can thus define:
N 2-FINAL 2-FUNCTORS 7
Definition.
Any path p : ( a, α ) ( a ′ , α ′ ) in b/ F defines a 2-isomorphism in E j ( p ) : ψ ( a,α ) → ψ ( a ′ ,α ′ ) as given by the above pasting diagram 3.6.3.8. Proposition.
For any paths p, p ′ : ( a, α ) ( a ′ , α ′ ) in b/ F with same sourceand target, j ( p ) = j ( p ′ ) . Proof.
We first prove that two elementary homotopic paths p ∼ elem p ′ induce thesame 2-isomorphism j ( p ) = j ( p ′ ) . The four first cases are immediate consequencesof the pseudonaturality of φ . We can thus assume that p = F ( a ) F ( a ) F ( a ) b F u F vµ ν p ′ = F ( a ) F ( a ′ ) F ( a ) b F u ′ F v ′ µ ′ ν ′ and that there is a 2-isomorphism ζ : u ′ u ⇒ v ′ v such that F ( a ′ ) b F ( a ) F ( a ) F ( a ) µ ′ F u ′ µ F ζ F v ′ F u F v = F ( a ′ ) b F ( a ) F ( a ) ν ′ ν F v ′ F v We can apply the functor D and express this relation using string diagrams (see[JY21, §3.7]):(3.9) D α ′ D α DF v DF v ′ D µ ′ D µ DF ζ = D α ′ D α DF v DF v ′ D ν ′ D ν JUN MAILLARD
Similarly, the pseudonaturality of φ gives the relation:(3.10) DF u DF u ′ φ a ′ φ a φ u ′ φ u = DF u DF u ′ φ a ′ φ a DF ζ φ v ′ φ v We can now compute j ( p ) : j ( p ) = D α φ a D α φ a D µ φ u φ − v D ν − (3.10) = D µ φ − u ′ DF ζ φ v ′ D ν − = φ − u ′ D µ DF ζ D ν − φ v ′ (3.9) = D α φ a D α φ a φ − u ′ D µ ′− D ν ′ φ v ′ = j ( p ′ ) We now show that, for two homotopic paths p ∼ p ′ , we have j ( p ) = j ( p ′ ) . Itsuffices to show that the relation R on paths defined by p R p ′ ⇐⇒ j ( p ) = j ( p ′ ) N 2-FINAL 2-FUNCTORS 9 is reflexive, symmetric, transitive and congruent, since we have already proved thatit contains ∼ elem . The three first properties are obviously satisfied. The last oneis a direct consequence of the compatibility of j with the concatenation of paths: j ( p · p ′ ) = j ( p ′ ) j ( p ) .Since b/ F is simply connected by hypothesis and j is homotopy invariant, the -isomorphism j ( p ) only depends on the source and the target of p . Hence for any twoobjects ( a, α ) and ( a ′ , α ′ ) in b/ F , there is a unique -isomorphism ψ ( a,α ) ⇒ ψ ( a ′ ,α ′ ) in E induced by a path in b/ F . (cid:3) Definition.
Given a morphism u : b → b ′ in B , there is a base change functor: u ∗ : b ′ / F → b/ F ( x, χ ) ( x, χ ◦ u )( v, ν ) ( v, ν · u ) Note that this functor also extends to a function between the respective sets ofpaths.3.12.
Proposition.
The application j maps base change to whiskering: j ( u ∗ p ) = j ( p ) · D u We can now use the above properties to construct a cone ψ under D with ver-tex e . For any b ∈ B , fix an arbitrary object ( a ( b ) , α ( b )) in b/ F . This defines thecomponents ψ b = ψ ( a ( b ) ,α ( b )) , as stated in definition 3.5. For a morphism u : b → b ′ ,note that ψ ( a ( b ′ ) ,α ( b ′ ) ◦ D u = ψ u ∗ ( a ( b ′ ) ,α ( b ′ )) ; hence we can define ψ u as the unique 2-isomorphism j ( p ) induced by any path p : u ∗ ( a ( b ′ ) , α ( b ′ )) ( a ( b ) , α ( b )) . We mustcheck that ψ is indeed a pseudonatural transformation. The compatibility of j withthe whiskering and the concatenation of paths implies the required compatibility of ψ with the composition of morphisms. It remains to check the compatibility with2-morphisms. Let u, u ′ : b → b ′ be two parallel 1-morphisms and δ : u ⇒ u ′ be a2-morphism in B . By unicity of 2-morphisms induced by a path (proposition 3.8),it suffices to check that the pasting(3.13) D b ′ e D b D u D u ′ D δ ψ u ′ is induced by a path. Indeed, fix a path p : ( u ′ ) ∗ ( a ( b ′ ) , α ( b ′ )) ( a ( b ) , α ( b )) andrecall that, by definition, ψ u ′ = j ( p ) . We consider the path p ′ of length one: p ′ = ( a ( b ′ ) , u ′ α ( b ′ )) (Id ,α ( b ′ ) δ ) ←−−−−−−− ( a ( b ′ ) , u ′ α ( b ′ )) = a ( b ′ ) a ( b ′ ) b ′ b ′ b u u ′ δ The above pasting (3.13) is then induced by the concatenation p ′ · p of p ′ and p .Through similar arguments, we can see that any other choice of the objects ( a ( b ) , α ( b )) b ∈ B leads to an isomorphic cone. From now on, we assume that the objects ( a ( b ) , α ( b )) b ∈ B are fixed and we write L ( φ ) for the cone ψ under D induced from the cone φ under D ◦ F . Since we willnot work with a single fixed cone φ anymore, we should write j φ instead of j .We would like to extend this mapping φ
7→ L φ to a functor L : [ A , E ]( D ◦ F , ∆ e ) → [ B , E ]( D , ∆ e ) . We use the proposition:3.14.
Proposition.
Let m : φ → φ ′ be a modification between two cones φ, φ ′ : D ◦ F ⇒ ∆ e For any path p : ( a, α ) ( a ′ , α ′ ) in b/ F , we have the following equality: (3.15) e e DF ( a ) DF ( a ′ ) D ( b ) φ a φ a ′ φ ′ a ′ m a ′ j φ ( p ) = e e DF ( a ) DF ( a ′ ) D ( b ) φ a φ ′ a m a φ a ′ j φ ′ ( p ) Proof.
For an empty path p , the proposition reduce to the tautology m a = m a . Fora path p = ( a, α ) ( u,µ ) −−−→ ( a ′ , α ′ ) of length 1, we can decompose the equation as: e e DF ( a ) DF ( a ′ ) D ( b ) φ a φ a ′ φ ′ a ′ φ − u m a ′ DF u D µ − = e e DF ( a ) DF ( a ′ ) D ( b ) φ a φ ′ a m a φ a ′ φ ′− u DF u D µ − The lower parts of these diagrams are the same and the upper parts are equal,by the property of the modification m . Hence eq. (3.15) holds for a path p =(( a, α ) ( u,µ ) −−−→ ( a ′ , α ′ )) . A similar decomposition of the diagrams shows that it alsoholds for a path p = (( a, α ) ( u,µ ) ←−−− ( a ′ , α ′ )) of length one in the reverse direction.Since j φ and j φ ′ are compatible with paths concatenation, if eq. (3.15) holds fortwo composable paths p and p ′ , it also holds for their concatenation pp ′ . We canthus conclude that it holds for any path p , as the path p is generated by paths oflength 1. (cid:3) This property directly implies that the components L ( m ) b = D ( b ) D ( a ( b )) e m a ( b ) define a 2-morphism L ( φ ) → L ( φ ′ ) . The functoriality of L is straightforward. N 2-FINAL 2-FUNCTORS 11
Now consider the canonical functor K : [ B , E ]( D , ∆ e ) → [ A , E ]( D ◦ F , ∆ e ) ψ • ψ F ( • ) m • m F ( • ) sending cones under D with vertex e to cones under D ◦ F with vertex e . We arenow ready to show that L and K are mutual pseudo-inverses.3.16. Proposition.
There is a natural isomorphism η : Id ⇒ KL .Proof. Let φ ∈ [ A , E ]( D ◦ F , ∆ e ) be a cone under D ◦ F with vertex e . We willwrite j = j φ .We want to define the component η φ : φ → KL φ of η at φ . Since η φ must be amodification, we have to define its components at each object a ∈ A : η φ,a : φ a ⇒ φ a ( F a ) ◦ D α F a . Both ( a , Id F a ) and ( a ( F a ) , α ( F a )) are objects of F a / F , which is connected.Hence, there is a path in F a / F : p : ( a , Id F a ) ( a ( F a ) , α ( F a )) We have to check that η φ is a modification. That is, for any morphism f : a → a in A , we have to check the commutativity of: φ a φ a ( F a ) ◦ D α ( F a ) φ a ◦ DF f φ a ( F a ) ◦ D α ( F a ) ◦ DF f η φ,a φ f L ( φ ) F f η φ,a ·DF f We first remark that there is a path p = (( a , Id) ( f, Id) −−−→ ( a , F f )) in F a / F andthe induced 2-isomorphism is φ f = j ( p ) . Moreover, expanding the definitions, wehave η φ,a = j ( p ) for some p : ( a , Id) ( a ( F a ) , α ( F a )) η φ,a = j ( p ) for some p : ( a , Id) ( a ( F a ) , α ( F a )) L ( φ ) F f = j ( p ) for some p : ( a ( F a ) , α ( F a )) F ( f ) ∗ ( a ( F a ) , α ( F a )) where p and p are paths in F a / F , and p is a path in F a / F . We can checkthat p · p and p · F ( f ) ∗ p are paths ( a , Id) F ( f ) ∗ ( a ( F a ) , α ( F a )) . Hence ( η φ,a · DF f ) ◦ φ f = j ( F ( f ) ∗ p ) ◦ j ( p )= j ( p · F ( f ) ∗ p )= j ( p · p )= j ( p ) ◦ j ( p )= L ( φ ) F f ◦ η φ,a We also have to check the naturality of η . For any modification m : φ → φ ′ , wewant the commutativity of the square: φ KL φφ ′ KL φ ′ η φ m KL mη φ ′ That is, for any object a ∈ A : φ a φ a ( F a ) ◦ D α ( F a ) φ ′ a KL φ ′ a j φ ( p ) m a m F a j φ ′ ( p ) where p : ( a , Id) ( a ( F a ) , α ( F a )) is a path in F a / F . This last square com-mutes by proposition 3.14. (cid:3) In the reverse direction we show:3.17.
Proposition.
There is a natural isomorphism ǫ : LK ⇒ Id .Proof. Fix a cone ψ : D ⇒ ∆ e under D . Write ψ ′ = LK ( ψ ) . For any b ∈ B , wehave: ψ ′ b = K ( ψ ) a ( b ) ◦ D ( α ( b )) = ψ F ( a ( b )) ◦ D ( α ( b )) Hence we can define a 2-morphism ǫ ψ,b : ψ ′ b ⇒ ψ b in E by: ǫ ψ,b = ψ α ( b ) When b ranges over all objects of B , these morphisms then form a modification ǫ ψ : ψ ′ → ψ . Indeed for any morphism ( u, µ ) : ( a, α ) → ( a ′ , α ′ ) in b/ F , we have: e DF ( a ′ ) DF ( a ) D ( b ) ψ F ( u ) ψ α D µ = e DF ( a ′ ) D ( b ) ψ α ′ N 2-FINAL 2-FUNCTORS 13 which implies a similar formula for any path p : ( a ′ , α ′ ) ( a, α ) in b/ F : e DF ( a ′ ) DF ( a ) D ( b ) j K ( ψ ) ( p ) ψ α = e DF ( a ′ ) D ( b ) ψ α ′ This in turn implies that ǫ ψ is a modification. Fix a morphism u : b → b ′ andconsider a path p : u ∗ ( a ( b ′ ) , α ( b ′ )) ( a ( b ) , α ( b )) (hence we have ψ ′ u = j K ( ψ ) ( p ) ).We check the modification axiom at u : e e DF ( a ( b ′ )) DF ( a ( b )) D ( b ′ ) D ( b ) ψ ′ u ǫ ψ,b = e DF ( a ( b ′ )) DF ( a ( b )) D ( b ) j K ( ψ ) ( p ) ψ α ( b ) = e DF ( a ( b ′ )) D ( b ) ψ α ( b ′ ) u = e e DF ( a ( b ′ )) D ( b ′ ) D ( b ) ψ u ǫ ψ,b ′ Finally we have to check the naturality of ǫ : LK → Id , that is, for any modificationof cones m : ψ → ψ ′ , the commutativity of the square: LK ψ ψ LK ψ ′ ψ ′ ǫ ψ LK m mǫ ψ ′ Indeed, for any object b ∈ B : ǫ ψ ′ ,b ◦ ( LK m ) b = ψ ′ α ( b ) ◦ m a ( b ) α ( b ) = m b ◦ ψ α ( b ) . (cid:3) Putting together proposition 3.16 and proposition 3.17 we deduce:3.18.
Proposition.
The canonical functor K : [ B , E ]( D , ∆ e ) → [ A , E ]( D ◦ F , ∆ e ) is an equivalence. Since this is true for any object e of E , clearly bicolim D exists if and only if bicolim D ◦ F exists and, if it is the case, they are canonically equivalent.We have thus proved one implication of theorem 3.4:3.19.
Proposition.
Let F : A → B be a (2 , -functor. If for any object b ∈ B ,the slice (2 , -category b/ F is nonempty, connected and simply connected, then the (2 , -functor F is -final. The reverse implication is proved by observing the following fact:3.20.
Proposition.
Let F : A → B be a (2 , -functor. Then Π ( b/ F ) ∼ = bicolim a ∈ A B ( b, F a ) . Proof.
The wanted equivalence can be proved by constructing a family of equiva-lences, pseudonatural in the category T : C T : [ A , Cat ]( B ( b, F− ) , ∆ T ) ∼ = [Π ( b/ F ) , T ] . Fix ψ : B ( b, F− ) ⇒ ∆ T a pseudonatural transformation. We want to define afunctor C T ( ψ ) : Π ( b/ F ) → T .For any object ( a, α : b → F ( a )) of Π ( b/ F ) , set: C T ( ψ )( a, α ) = ψ a ( α ) For any morphism ( u, µ : uα ⇒ α ′ ) : ( a, α ) → ( a ′ , α ′ ) of b/ F , define the compositeisomorphism: C T ( ψ )( u, µ ) : ψ a ( α ) ψ a ′ ( u ◦ α ) ψ a ′ ( α ′ ) ( ψ u ) α ψ a ′ ( µ ) This can be extended to paths using the relations: C T ( ψ )(( a ′ , α ′ ) ( u,µ ) ←−−− ( a, α )) = C T ( ψ )(( a, α ) ( u,µ ) −−−→ ( a ′ , α ′ )) − C T ( ψ )(( a, α )) = Id ψ a ( α ) C T ( ψ )( p · p ′ ) = C T ( ψ )( p ′ ) ◦ C T ( ψ )( p ) On can check that such a definition is homotopy invariant, and gives a well-definedfunctor C T ( ψ ) : Π ( b/ F ) → T .For a modification m : ψ → ψ ′ , we define a natural transformation C T ( m ) : C T ( ψ ) ⇒ C T ( ψ ′ ) with components:(3.21) C T ( m ) ( a,α ) = ( m a ) α To show that C T is an equivalence, we show that it is a fully faithful and essen-tially surjective functor.Indeed it is clear that (3.21) defines a bijection between modifications ψ → ψ ′ and natural transformations C T ( ψ ) ⇒ C T ( ψ ′ ) . Hence C T is fully faithful.Moreover, given any functor F : Π ( b/ F ) → T , one can define a pseudonaturaltransformation ψ : B ( b, F− ) → ∆ T by: ψ a : B ( b, F a ) → Tα F ( a, α ) ν F (Id a , ν ) N 2-FINAL 2-FUNCTORS 15 ( ψ u ) α : F ( a, α ) F ( u, Id) −−−−−→ F ( a ′ , u ◦ α ) for any object a and morphism u : a → a ′ of A . It is straightforward to check: F = C T ( ψ ) (cid:3) We can now prove:3.22.
Proposition.
Let F : A → B be a -final (2 , -functor. Then, for any object b in B : Π ( b/ F ) ∼ = 1 Proof.
We have a chain of equivalences: Π ( b/ F ) 3 . ∼ = bicolim a ∈ A B ( b, F a ) (1) ∼ = bicolim b ′ ∈ B B ( b, b ′ ) (2) ∼ = 1 The equivalence (1) is an application of the -finality of F . The equivalence (2) isa consequence of the Yoneda lemma for -categories. Indeed we have the chain ofequivalences, for any -category T , and pseudonatural in T : [ B , Cat ]( B ( b, − ) , ∆ T ) ∼ = ∆ T ( b ) ∼ = T ∼ = Cat (1 , T ) (cid:3) By combining proposition 3.22 and remark 2.7, we have:3.23.
Proposition.
Let F : A → B be a -final (2 , -functor. Then, for any object b in B , the (2 , -category b/ F is nonempty, connected and simply connected. There is a dual notion of , with a dual criterion, proven by aduality argument.3.24.
Definition.
Let F : A → B be a -functor between (2 , -categories. The 2-functor is said to be if, for any -diagram D : B → E , each of the bilimits bilim B D and bilim A D ◦ F exists whenever the other one exists, and the canonicalcomparison -morphism bilim B D → bilim A D ◦ F is an equivalence.3.25.
Proposition.
Let F : A → B be a 2-functor between (2 , -categories. The2-functor F is initial if and only if the 2-functor F op : A op → B op is final. Theorem.
Let F : A → B be a 2-functor between (2 , -categories. The 2-functor F is initial if and only if, for any object b ∈ B , the slice (2 , -category F /b is nonempty, connected and simply connected. Further directions
There are various direction in which one may try to improve the finality criterionpresented in the previous section.The most straightforward one is to work in the context of bicategories, wherecomposition of -morphisms is only associative up to isomorphism. One shouldnote that the correct notions of -finality for pseudofunctors between bicategorieswith invertible -morphisms should be weakened to include any pseudofunctor asdiagram, and not only the strict ones as we do in definition 3.1. Since not all pseudofunctors can be strictified ([Lac07]), the analogous result for pseudofunctorsis not a direct corollary of theorem 3.4.Another natural route is to prove it for higher dimensions n . An analogous com-binatorial proof would require a combinatorial presentation of higher homotopies inan n -category, and probably to set up a machinery for working inductively on thedimension. An alternative, potentially more reasonable approach may be to adaptLurie’s topological proof to finite dimensions. References [JY21] Niles Johnson and Donald Yau. . New York:Oxford University Press, 2021. isbn : 978-0-19-887137-8.[Lac07] Stephen Lack. “Bicat Is Not Triequivalent to Gray”. In:
Theory and Ap-plications of Categories
18 (2007), No. 1, 1–3.[Lur09] Jacob Lurie.
Higher Topos Theory . Annals of Mathematics Studies no.170. Princeton, N.J: Princeton University Press, 2009. 925 pp. isbn : 978-0-691-14049-0.[Mac71] Mac Lane Saunders.
Categories for the Working Mathematician . Gradu-ate Texts in Mathematics. New York: Springer-Verlag, 1971. isbn : 978-0-387-90036-0.[Mai] Jun Maillard. Ph.D. Thesis (In preparation).
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
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