Interchange of filtered 2-colimits and finite 2-limits

In terc hange of filtered 2-coli mits and finite 2-limits Delphine Dup ont Abstract In this paper we go in to the study of 2-limit and 2-colimit in the 2- category C AT the category of small categories. In particular w e show the comm u tation of filtered 2-col imits and finite 2-limits. It is a generalization of a classical result in category theory (see for example [1]). 1 In tro du ction Recently the 2-catego ry theor y has developed rapidly . It gives a very useful language in many field of mathematics. One of the main notions is the 2-limits and 2-colimits. F or example the stalk of a stack is a 2 -colimit. In this pap er we genralize a classica l r esult in categor y theory . In a first part we re c all the definition of 2-limit and 2 -colimit in C AT , the 2-catego ry of small ca teg ories. W e g ive a n explicit description of a filtered 2- colimit a nd of a 2 -limit. But we do no t expres s the mo rphisms of these categories in a classica l wa y . W e give them as elements o f a colimit or a limit. In a second part we prov e the in terchange of filtered 2 -colimits and finite 2-limits. T o prove this theorem we use the explicit expression of the ca tegories 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ) and 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ). W e do not recall the definitions of a 2-c ategory , a 2 -functor (some times called pseudo functor in the literature), a 2-natur a l transfor mation of functors and a 2 -mo dification. The reader can find them in the c hapter 7 of [1 ], the app endix of the pap er [3] and the pa p er [2 ]. Let us recall the definition o f a filtered categ ory : Definition 1. A c ate gory I is filter e d if it satisfies the c onditions (i)-(iii) b elow. (i) I is non empty, (ii) for any i and j in I , ther e exist k ∈ I and morphisms i → k , j → k , (iii) for any p ar al lel morp hisms s, s ′ : i / / / / j ther e ex ists a morphism h : j → k s uch that h ◦ s = h ◦ s ′ . 2 2 -colimits and 2 -limits Let us first reca ll the definitions of a 2-limit and a 2-co limit. This part is inspired by the app endix of [3]. W e cite [2] as a cla ssical refere nc e . 1 Definition 2. L et I b e a smal l c ate gory, and b : I → C AT a 2 -functor The system b admits a 2 -c olimit if and only if ther e exist : • a c ate gory 2 lim − → i ∈I b ( i ) and • a 2 -natu r al tr ansformation σ : b → 2 lim − → i ∈I b ( i ) such that for al l c ate gory C the functor : ( σ ◦ ) : Hom C (2 lim − → i ∈I b ( i ) , C ) → Hom ( b , C ) is an e quivalenc e of c ate gories. We say t hat a 2 - c olimit has t he str ong factorisation pr op erty if ( σ ◦ ) is an iso- morphism of c ate gories. More co ncretely , a 2-functor b : I → C AT admits a 2-colimit if a nd only if there exist : • a ca tegory 2 lim − → i ∈I b ( i ), • functor s σ i : b ( i ) → 2 lim − → i ∈I b ( i ) for any i ∈ I , • a nd a natura l equiv alence Θ σ s : σ i ∼ → σ j b ( s ) for any morphism s : i → j of I visualized by : 2 lim − → i ∈I b ( i ) σ i                   σ j   7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 θ s ∼ 3 ; o o o o o o o o o o o o o o b ( i ) b ( s ) / / b ( j ) such that σ : b → lim − → i ∈I b ( i ) is a 2-na tural tra nsformation. Moreov er these data satisfy condition assur ing tha t ( ◦ σ ) is an equiv alence. The first one transla te the fact tha t ( ◦ σ ) is e s sentially sur jectiv e a nd the second o ne that ( ◦ σ ) is fully faithfull. • F or a ny catego r y C , an y mo rphism of functors ρ : b → C , there exist a functor : F : 2 lim − → i ∈I b ( i ) − → C and an isomorphism ϕ F : ρ → F σ , which is a mo dification given, for all i ∈ I , by a natura l eq uiv a le nc e ϕ F i : ρ i → F ◦ σ i . This ma y be visualized by : 2 a ( i ) a ( s )   σ i ' ' N N N N N N N N N N N N N N N N N N N N N ρ i & & ∼ ϕ F i   θ s ∼                   2 lim − → i ∈I a ( i ) F / / C ϕ F j ∼ K S a ( j ) σ j 7 7 p p p p p p p p p p p p p p p p p p ρ j 8 8 The c ompatibility conditio n is given b y :  F • θ σ s  ◦ ϕ F i =  ϕ F j • a ( s )  ◦ θ ρ s The pa ir ( F, ϕ F ) is called a la x fac torization of the sys tem ρ . • Le t ρ and ρ ′ be t wo 2-natura l transformations and λ : ρ → ρ ′ a mo di- fication. Then for any lax factorization F : 2 lim − → i ∈I b ( i ) − → C of ρ and G : 2 lim − → i ∈I b ( i ) − → C o f ρ ′ there exists a unique natura l tra nsformation Λ : F → G suc h tha t : ϕ G i ◦ λ i =  Λ i • σ i  ◦ ϕ F i If the 2-colimit has the strong factor ization prop er t y then there exists a unique factorization such that ϕ F is the identit y . A contrav ariant 2-functor c : I → C AT is a 2-functor c : I op → C AT . A 2-limit is construction dual o f the 2 -colimit. Hence, we hav e the following definition : Definition 3. L et I b e a s m al l c ate gory and c : I → C AT a c ontr avariant 2 -functor. The system c admits a 2 -limit if and only if t her e exist : • a c ate gory 2 lim ← − i ∈I c ( i ) and • a 2 -natu r al tr ansformation σ : 2 lim ← − i ∈I b ( i ) → c such that for al l c ate gory C the functor : ( ◦ σ ) : Hom ( c , C ) → Hom C (2 lim ← − i ∈I b ( i ) , C ) is an e quivalenc e of c ate gories. We say t hat a 2 -limit has t he st ro ng factorisation pr op erty if ( ◦ σ ) is an isomor- phism of c ate gories. It is well known that the 2 -categor y C AT is complete and co-co mplete. The pro of of this result consist an explicit definition of the 2-limit a nd 2-colimit. W e are going to recall these de finitio ns , but morphisms of these catego ries won’t b e given in a cla ssical wa y . Usually they a re given as classe s of mor phisms b etw een 3 ob jects of b ( i ) which sa tisfy some conditions. Here we are going to express them in term of limit and colimit. Let b : I → C AT b e a 2-functor. Let us giv e so me us e ful notations. Let i, i ′ ∈ I and I ii ′ be the category de fined b y: • the ob jects o f I i,i ′ are : n ( i ′′ , s, s ′ ) | i ′′ ∈ O b I , s ∈ H om I ( i, i ′′ ) , s ′ ∈ H om I ( i ′ , i ′′ ) o visualized by : i s + + V V V V V V V i ′′ i ′ s ′ 3 3 h h h h h h h • the morphisms of I i,i ′ from ( i ′′ 1 , s 1 , s ′ 1 ) to ( i ′′ 2 , s 2 , s ′ 2 ) are : H om I ii ′  ( i ′′ 1 , s 1 , s ′ 1 ) , ( i ′ 2 , s 2 , s ′ 2 )  = { t ∈ H om I ( i ′′ 1 , i ′′ 2 ) | t ◦ s 1 = s 2 , t ◦ s ′ 1 = s ′ 2 } visualized by : i s 1 % % L L L L L L L L s 2 & & i ′′ 1 t / / i ′′ 2 i ′ s ′ 1 9 9 r r r r r r r s ′ 2 8 8 Lemma 4. If the c ate gory I is filter e d then t he c ate gory I ii ′ is filter e d. Pr o of. The pro of is s traight-fo ward. Prop ositio n 5. The c ate gory B defin e d b elow is a 2 -c olimit of b satisfying the str ong factorization pr op erty. • O bje cts of B ar e p airs ( i, X ) wher e i ∈ I and X ∈ b ( i ) , • let ( i, X ) and ( i ′ , Y ) b e two obje cts of B , the morphisms fr om ( i, X ) to ( i ′ , Y ) ar e the elements of the c olimit : H om B  X , Y  = lim − → ( i ′′ ,s,s ′ ) ∈I ii ′ H om b ( i ′′ )  b ( s ) X , b ( s ′ ) Y  wher e if t ∈ H om I ii ′  ( i ′′ 1 , s 1 , s ′ 1 ) , ( i ′′ 2 , s 2 , s ′ 2 )  , its image in the inductive system is the fol lowing morphism : H om b ( i ′′ 1 )  b ( s 1 )( X ) , b ( s ′ 1 )( Y )  − → H om b ( i ′′ 2 )  b ( s 2 )( X ) , b ( s ′ 2 )( Y )  h 7− → b − 1 t,s ′ 1 ◦ b ( t ) h ◦ b t,s 1 wher e b t,s 1 is the isomorphism given by the 2 - functor b : b t,s 1 : b ( t ◦ s 1 ) ∼ − → b ( t ) ◦ b ( s 1 ) Pr o of. See for ex ample [3]. 4 If I ii ′ is filtered, the morphisms from ( i, X ) to ( i ′ , Y ) are the elements of the quotient : a i ′′ ∈I ii ′ H om b ( i ′′ )  b ( s ) X , b ( s ′ ) Y  . ∼ Where, g iven h 1 ∈ H om b ( i ′′ 1 )  b ( s 1 ) X, b ( s ′ 1 ) Y  and h 2 ∈ H om b ( i ′′ 2 )  b ( s 2 ) X, b ( s ′ 2 ) Y  , we wr ite h 1 ∼ h 2 if and only if there exist ( i 3 , s 3 , s ′ 3 ) ∈ I ii ′ , t 13 ∈ H om  ( i 1 , s 1 , s ′ 1 ) , ( i 3 , s 3 , s ′ 3 )  and t 23 ∈ H om  ( i 2 , s 2 , s ′ 2 ) , ( i 3 , s 3 , s ′ 3 )  such that : b − 1 t 13 ,s ′ 1 ◦ b ( t 13 ) h 1 ◦ b t 13 ,s 1 = b − 1 t 23 ,s ′ 2 ◦ b ( t 23 ) h 2 ◦ b t 23 ,s 2 Similarly we are going to give an explicit constructio n of a 2-limit. As b efore morphisms of this categor y will be given by a limit. Let c b e a c ontra v ariant 2-functor : c : J op − → C AT Prop ositio n 6. The c ate gory C define b elow, is a 2 -limit of c satisfying the str ong factorization pr op erty. • The obje cts of C a re p airs ( X , ϑ X ) wher e : – X = { X j } j ∈J , for X j ∈ Ob ( c ( j )) , – ϑ X = { ϑ X t } t ∈ H om J wher e, for t ∈ H om J ( j, j ′ ) , ϑ X t is an isomor- phism : ϑ X t : X j ∼ − → c ( t ) X j ′ satisfying the fol lowing c onditions : A) for any j ∈ J we have I d j = c j ( X j ) ◦ ϑ X I d j , wher e c j is the morphism given by t he 2 -functor c : c j : c ( I d j ) ∼ − → I d c ( j ) B) for any two c omp osable morphisms t : j → j ′ and t ′ : j ′ → j ′′ the fol lowing e quation holds : c ( t )( ϑ X t ′ ) ◦ ϑ X t = c t,t ′ ( X j ′′ ) ◦ ϑ X t ◦ t ′ wher e c t,t ′ is the isomorphism given by the 2 -funct or c : c t,t ′ : c ( t ◦ t ′ ) ∼ − → c ( t ) ◦ c ( t ′ ) • L et ( X , ϑ X ) and ( Y , ϑ Y ) b e two obje cts of C , morphisms fr om ( X, ϑ X ) to ( Y , ϑ Y ) ar e elements of the limit : H om C ( X, Y ) = lim ← − j ∈J H om c ( j )  X j , Y j  wher e if t ∈ H om J ( j, j ′ ) , its image in the pr oje ctive system is the fol lowing morphism : H om c ( j ) ( X j , Y j ) − → H om c ( j ′ ) ( X j ′ , Y j ′ ) h 7− → ( ϑ Y t ) − 1 ◦ c ( t )( h ) ◦ ϑ X t This means that a mor phism b etw een tw o o b jects is the datum of { h j } j ∈J , where h j ∈ H om c ( j ) ( X j , Y j ) satisfies the equa lit y : h j = ( ϑ Y t ) − 1 ◦ c ( t )( h j ′ ) ◦ ϑ X t for all t : j ′ → j morphisms o f J . 5 3 In terc hange of filtered 2 -c olimits and finite 2 - limits W e are going to show that filtered 2-colimits co mm ute w ith finite 2 -limits. Let us give first a pr ecise meaning to this sentence. Let I b e a filtered category , J a finite category a nd a a 2-functor : a : I × J op − → C AT Let us de no te 2 F ( C ) the 2-category of 2-functor s g oing from the categ ory C to the 2-categ ory C AT . W e hav e the following prop o s ition, for a pro of see for example [3] : Prop ositio n 7. The c orr esp ondenc e : 2 F ( C ) − → C AT b 7− → 2 lim − → i ∈I b ( i ) c an b e extende d to a 2 -functor b etwe en 2 -c ate gories. A similar statement holds for 2 -limits. Now let us co nsider, the na tural 2-functor : J − → 2 F ( I ) j 7− → a ( · , j ) . The co mpo sition o f this 2-functor and the one defined in the propo sition g ives a functor fro m J to C AT . As C AT is c omplete we ca n cons ider its limit. Let us denote 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ) this limit. W e define in the same way the 2-co limit 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ). Remark The c omp o sition, for all i ∈ I a nd j ∈ J , of the functor define by the 2 -colimit and the 2-limit : i 2 lim ← − j ′ ∈J a ( i, j ′ ) − → a ( i, j ) − → 2 lim − → i ′ ∈I a ( i ′ , j ) i defines a functor : Ψ : 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ) − → 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ) Theoreme 8. The natur al functor Ψ : 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ) − → 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ) is an e quivalenc e of c ate gories. 6 Pr o of. The fact that Ψ is fully fa ithful comes dir ectly from the expression of the morphisms of these tw o ca tegories in terms of limits and co limits. In detail, using the tw o pro po sitions 5 a nd 6, we can give an ex plicit constr uction of the categ ories 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ) and 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ). The c ategory lim − → i ∈I lim ← − j ∈J a ( i, j ) is defined as follows : • its ob jects are the triples  i, X , ϑ X  where – i is an ob ject of I , – X = { X j } j ∈J for X j is an ob ject of a ( i, j ) – ϑ X = { ϑ X t } t ∈ H om J , where, for t : j ′ → j a n isomorphism of J , ϑ X t is a morphism : ϑ X t : X j − → a ( I d i , t ) X j ′ verifying the tw o following conditions : – for any j ∈ O bj J we hav e a ij ( X ij ) ◦ ϑ X I d i = I d X i , where a ij is the isomorphism a ij : a ( I d i , I d j ) ∼ → I d a ( i,j ) given by the 2- functor a , – for any tw o compo sable mo rphisms t : j → j ′ and t ′ : j ′ → j ′′ the following equatio n ho lds : a ( I d i , t )( ϑ X t ′ ) ◦ ϑ X t = a ( I d i ,t ) , ( I d i ,t ′ ) ( X ij ′′ ) ◦ ϑ t ′ ◦ t where a ( I d i ,t ) , ( I d i ,t ′ ) : a ( I d i , t ′ ◦ t ) ∼ → a ( I d i , t ′ ) a ( I d i , t ) is the morphism given by the 2- functor a . • the set of mo rphisms from  i, X , ϑ X  to  i ′ , Y , ϑ Y  is given by the limit : H om ( X , Y ) := lim − → i ′′ ∈I ii ′ lim ← − j ∈J H om a ( i ′′ ,j )  a ( s, I d j ) X j , a ( s ′ , I d j ) Y j  . The catego ry 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ) is defined as follows : • the ob jects a r e pair s ( X , θ X ), where – X = { ( i j , X i j ) } j ∈J with i j ∈ I and X i j ∈ a ( i j , j ), – θ X = { [ ϑ t ] } t ∈ H om J where, for t ∈ H om J ( j, j ′ ), [ ϑ X t ] belong s to the quotient : a i ′′ ∈I i j i j ′ H om a ( i ′′ ,j )  a ( s, I d j ) X i j , a ( s ′ , t ) X i j  . ∼ and where [ ϑ X t ] satisfies the following eq ualities : [ a ij ( X ij ) ◦ ϑ X I d i ] = [ I d X i ] (1) [ a ( I d i , t )( ϑ X t ′ ) ◦ ϑ X t ] = [ a ( I d i ,t ) , ( I d i ,t ′ ) ( X ij ′′ ) ◦ ϑ t ′ ◦ t ] (2) and for any tw o comp osa ble mo rphisms t : j → j ′ and t ′ : j ′ → j ′′ . 7 • the set of mo rphisms from ( X , θ X ) to ( Y , θ Y ) is given by the limit : lim ← − j ∈J lim − → i ′′ ∈I i j i ′ j H om a ( i ′′ ,j )  a ( s, I d j ) X j , a ( s ′ , I d j ) Y j  The natural functor b etw een the 2-limits is : ε : 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ) − → 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j )  i, { X j } , { ϑ X t }  7− →  { i, X j } , { [ ϑ X t ] }  Moreov er , if X = ( i, X , ϑ X ) and Y = ( i ′ , Y , ϑ Y ) are tw o ob jects of 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ), the morphism from H om 2 lim − → 2 lim ← − ( X , Y ) to H om 2 lim − → 2 lim ← − ( ε ( X ) , ε ( Y )) induce d by ε is the natura l morphism : lim − → i ∈I lim ← − j ∈J H om  a ( s, I d j ) X j , a ( s ′ , I d j ) Y j  − → lim ← − j ∈J lim − → i ∈I H om  a ( s, I d j ) X j , a ( s ′ , I d j ) Y j  As filtered co limits commute with finite limits, the morphism ab ov e is an is o - morphism and ε is fully faithful. The pr o of that Ψ is e ssentially surjective is similar to the pro of of the com- m uta tio n o f filtered limits a nd finite colimits. Let  { i j , X j } , { [ θ X t ] }  an ob ject of 2 lim ← − j ∈J 2 lim − → i ∈I a ( i, j ). Using the pro pe r ty ( ii ) of a filtered category inductiv ely , one prov es that there exist an ob ject k ′ ∈ I and, for a ny j ∈ J , a mor phis m s ′ j : i j → k ′ in I . Thu s , for all t mor phism o f J , [ θ X t ] can b e v iewed as the class o f an ob ject ϑ X t of H om  a ( s j , I d j ) X i j , a ( s j ′ , t ) X j ′  . Remark that even if the class [ θ X t ] s atisfies the equalities (1) and (2), the ob jects ϑ X t may not satisfy then. Using the prop- erty ( iii ) o f a filtered category inductively , one pr ov es that there exist an ob ject k of I and a morphism s k : k ′ → k such that a ll the equa lities hold also for the ob jects ϑ X t . Hence the triple :  k ,  a ( s k ◦ s j , I d j ) X i j  ,  a − 1 s k ,s j ◦ ϑ X t ◦ a s k ,s j ′   is an ob ject of 2 lim − → i ∈I 2 lim ← − j ∈J a ( i, j ). F or j ∈ J , the ob jects X i j and a ( s k ◦ s j , I d j ) X i j are isomor phic and, fo r all mor phis m t of J , we have [ ϑ X t ] = [ θ X t ] Hence we hav e shown that Ψ is an equiv alence of categories. References [1] S. Mac Lane. Cate gories for the working mathematician. S e c ond e dition. Graduate T exts in Mathematics, 5. Spring er-V erlag, 1 998. [2] R. Street. Categorical structur es. Handb o ok of algebr a , 1996. [3] I. W asc hkies . The stac k o f micro lo cal p erverse sheav es. Bul l. So c. Math. F r anc e 132 , 2 004. 8