The Limit-Colimit Coincidence for Categories

Paul Taylor 1994

Abstract Scott noticed in 1969 that in the category of complete lattices and maps preserving directed joins, limit of a sequence of projections (maps with preinverse left adjoints) is isomorphic to the colimit of those left adjoints (called embeddings). This result holds in any category of domains and is the basis of the solution of recursive domain equations. Limits of this kind (and continuity with respect to them) also occur in domain semantics of polymorphism, where we find that we need to generalise from sequences to directed or filtered diagrams and drop the “preinverse” condition. The result is also applicable to the proof of cartesian closure for stable domains. The purpose of this paper is to express and prove the most general form of the result, which is for filtered diagrams of adjoint pairs between categories with filtered colimits. The ideas involved in the applications are to be found elsewhere in the literature: here we are concerned solely with 2-categorical details. The result we obtain is what is expected, the only remarkable point being that it seems definitely to be about pseudo- and not lax limits and colimits. On the way to proving the main result, we find ourselves also performing the constructions needed for domain interpretations of dependent type polymorphism.

1 Filtered diagrams

The result concerns limits and colimits of filtered diagrams of categories, so we shall be interested in functors of the form FCCatcp I → where is a filtered category and FCCatcp is the 2-category of small categories with filtered colimits,I functors with right adjoints which preserve filtered colimits and natural transformations. Since we have to perform a lot of manipulation of filtered diagrams, we begin with a short discussion on them.

Definition 1.1 A category (diagram-type) is filtered if I (i) it is non-empty, (ii) it has the amalgamation property for objects, i.e. given i, j , there is some k and u : i k, v : j k in . ∈ I ∈ I → → I (iii) it has the amalgamation property for morphisms, i.e. given a parallel pair u, v : i ⇒ j in , there is some k and w : j k in with u ; w = v ; w. I ∈ I → I

Example 1.2 Any category with finite colimits is filtered, for example ordinals and finite power-set lattices. The definition says that a filtered category has “weak” finite colimits. Part of the data for the theorem is that the categories i “have filtered colimits.” We shall always take this to mean that there is a functor X

colim↑ : J X → X J

1 which is left adjoint to the “constant functor”; the counit is the colimiting cocone and the unit (since is connected) is an isomorphism. Often, however, we can avoid naming the colimit becauseJ we are given a cocone and set out to prove that it is colimiting, i.e. a colimit rather than the chosen colimit.

Definition 1.3 A functor F : is continuous if it preserves (commutes with) filtered colimits up to isomorphism φ: X → Y

X colim↑ φ X-j - colim↑ FXj J h i F colim↑ Xj J J j =  j  X X ∈J ∼ ∈J 6 6 φ σj F τj F J F ⇒ ? ? - FXj ======FXj J Y Y colim↑ Y J We shall make several uses of final subdiagrams, which generalise the idea of a cofinal subse- quence of an ordinal. They is dealt with in [Mac Lane], section 9.3.

Definition 1.4 A functor U : is final if for each i the comma category i U is J → I ∈ I ↓ connected. This means that there is some j and u : i Uj, and if there is another j0 ∈ J → ∈ J with u0 : i Uj0 then there is a “zig-zag” such that →

i ======i ======i ======i ======i

u = u0 u1 u2 u2n 1 u2n = u0 − ? ? ? ? ? Uv1- Uv2 Uv2n Uj = Uj0 Uj1 Uj2 Uj2n 1 Uj2n = Uj0 ··· − commutes.

Proposition 1.5 Let D : be a diagram in a category and U : be a final functor. I → X J → I Suppose that the diagram DU : has a colimit with cocone yj : DUj X. Then J → I → X → there is a unique cocone xi : Di X over the given diagram D such that xUj = yj, and this is colimiting. →

Proof For i , suppose j with u : i Uj, and put xi = Du ; yj. We have to check that this is well-defined,∈ I by induction∈ J on the length→ of a zig-zag between two candidates. It suffices to consider the case of a single “zig”, u0 = u;Uv, where v : j j0; then xi = Du;yj = Du;DUv;yj = → 0 Du0 ; yj . This also shows that xi is a cocone over . It is clearly also the only one with xUj = yj. 0 I Now let zi : Di Z be another cocone over . Then zUj : DUj Z is a cocone over → I → J and, since X is the colimit, there is a unique mediator z : X Z with yj ; z = zUj. Then → xi ; z = Du ; yj ; z = Du ; zUj = zi, so z mediates from also, and a fortiori it is unique. I  Definition 1.6 We shall need the following constructions on a fixed filtered category (with typical objects): I (i) itself I i

(ii) the coslice i0/ I u- i0 i

2 (iii) the double coslice (i0, j0)/ I (iv) the degenerate double coslice (i0, i0)/ I u- i i 0 - v

(v) [a construction needed in Lemma 8.2]...

Exercise 1.7 Let be a filtered category. Verify that the following functors are final: I (a) i0/ (i0, i0)/ by u (u, u); I → I 7→ (b) (i0, j0)/ (v); I → (c) i0/ i1/ by u p ; u; I → I 7→ (d) (i0, j0)/ (i1, j1)/ by (u, v) (p ; u, q ; v); I → I 7→ (e) by i (i, i); I → I × I 7→ where i0, i1, j0, j1 and p : i1 i0 and q : j1 j0 in . ∈ I → → I  Exercise 1.8 Suppose has binary coproducts. Show that the functors I (a) i0/ by i (i0 i0 + i), and I → I 7→ → (a) (i0, j0)/ by i (i0 i0 + j0 + i j0) I → I 7→ → ← (the maps being the coproduct inclusions) are final. Hence show that any colimit over i0/ , I (i0, j0)/ or may be calculated as a colimit over .  So theI aboveI × I assumptions about “categories with filteredI colimits” may be reduced to “cate- gories with colimits of type ”. Indeed, if we interpret the definition of filteredness constructively, any filtered category is equivalentI (for the purpose of finding colimits) to a category with finite colimits. Although we said that we wanted the categories to have all filtered colimits, it turns out that we only actually make use of colimits of diagrams of the above types. Clearly these are of essentially the same cardinality as the given diagram , which allows us to regard filtered colimits as an algebraic operation of fixed arity. We alsoI never consider the aggregate of all categories with filtered colimits, or perform constructions worse than products of the cardinality of . Consequently we have no problems of size, and may treat the categories as small (or locally small);I this is fortunate, considering the amount of 2-categorical work already cut out for us!

2 Pseudofunctors

( ) op In this section we shall examine the notion of pseudofunctor − : Cat. Although in the application to the limit-colimit coincidence (sections 5 andX 7 onwards)I → we shall need to be filtered, for the time being it may be any category whatever. I The problem is that in general for composable arrows u : i j and v : j k the functors → → u;v and u v X X X cannot be expected to be equal: at best isomorphic. This is because u is typically defined by some universal property, such as pullback along u. [More generally, oneX may study the case where

3 they are related by a non-invertible map one way or the other, and such constructions are called lax or oplax.] We shall find, however, that in the result which interests us we need an isomorphism

u,v u v α - u;v X X X Likewise, there is no a priori reason why id need be the identity, so we also provide an isomor- phism X γi idi id i - X X (We prefer for reasons of symmetry not to put γi = id, although this is possible and quite commonly done.) Since these maps are isomorphisms, we shall feel free to define compositions involving them 1 backwards or forwards, without writing α− .

Definition 2.1 A pseudofunctor op Cat is an assignment of categories i, functors u : j i and natural isomorphismsI αu,v→ and γi as above, subject to the laws X X X → X uαv,w u v w X - u v;w X X X X X

αu,v w A αu,v;w associativity X ? u;v,w ? u;v w α - u;v;w X X X and γi u uγj u X- idi u u idj X u X X X X X X

L id αidi,u αu,idj R u unit X u - idX ? ? u u X X Lemma 2.2 Any well-formed diagram consisting of α’s, γ’s and their inverses commutes. Proof By induction on the length of the compositions, using the associativity and unit laws. 

Corollary 2.3 If ~u and ~v are composable strings in with a common refinement, then there is a ~u def u I u ~v unique canonical isomorphism between = 1 ... n and .  This kind of lengthy presentation ofX data conflictsX X with theX spirit of category theory, and we have two alternatives: 2-functors and fibrations. (What about sheaves of categories?)

Construction 2.4 A 2-category I whose 0-cells are those of and whose 1-cells are composable strings of arrows of . There is a unique, invertible, 2-cell betweenI any two strings a common refinement. We thenI obtain a 2-functor (no longer pseudo)

op u1 un d : I Cat by (u1, ..., un) ... → 7→ X X whose effect on 2-cells is well-defined by the corollary. 

Proposition 2.5 There is an isomorphism between pseudofunctors op Cat and 2-functors Iop Cat. I → → 

4 Notation 2.6 Besides the equations A, L, R, etc., we shall denote the naturality square of φ : F G with respect to f : X Y by → → φX F f FX - GX FX - FY

F f N Gf or φX Z φY

? φY ? ? Gf ? FY - GY GX - GY

3 Fibrations

The other approach, sometimes called the Grothendieck construction, is to form the total category Σi : .Xi and display it as a fibration over . I I Construction 3.1 The total category Σi : . i. I X (i) The objects of the total category are the pairs (i, X) with i and X i; ∈ I ∈ X (ii) the morphisms from (i, X) to (j, Y ) are the pairs (u, f) where u : i j in and f : X uY ; → I → X i (iii) the identity on (i, X) is (idi, γ X); (iv) the composite of (u, f):(i, X) (j, Y ) and (v, g):(j, Y ) (k, Z) is (u ; v, f ; ug ; αu,v). → → X The display functor P :Σi : . i I X → I (v) takes (i, X) to i and (u, f) to u.

i0 i The inclusion functor Ii : Σi : . 0 X → I X i0 (vi) takes X to (i0,X) and f : X Y to (idi , f ; γ Y ). → 0 u The coherence υu : Ii Ij X → (vii) at Y j is (u, id uY ) and satisfies the equations ∈ X X i v Iiγ- id u v υu - v Ii Ii Ii X Ij X X X X

U u,v id υid and Iiα M υv - ? ? υ ? u;v u;v - Ii Ii Ik X Proof First, Σi : . i is a category. For the identities: I X u j u,id (u, f); id = (u ; idj, f ; γ Y ; α Y ) (j,Y ) X = (u ; idj, f ; idY ) = (u, f) using R and

i id id,u id ;(u, f) = (idi ; u, γ X ; f ; α Y ) (i,X) X i u id,u = (idi ; u, f ; γ Y ; α Y ) X = (u, g ; id)

5 using naturality and L. Associativity follows from:

u( vh ; αv,wT ) = u vh ; αu,v wT ; αu;v,wT A X X X X X = αu,vZ ; u;vh ; αu;v,wT N X

Functoriality of P is obvious, as is the fact that Ii preserves identities; for composition:

i i Iif ; Iig = (idi, f ; γ Y );(idi, g ; γ Z) i id id i id,id = idi, f ;(γ Y ; g);( γ Z ; α Z) X X
i = (idi, f ; g ; γ Z ; id idZ ) Z, R X

Naturality of υ: id uY - uY X X

uy uy X X ? ? u j u u γ Y-0 u id Y 0 Y 0 X Y 0 X X X X id i u L R 0 γ Y 0 id id Y X u, - α ? ? id u - u Y 0 Y 0 X X αid,uY X The U equation is essentially just L, and for M:

u v id - u v

X X γ i X X X u u,v α X v L id - ? αid,u v ? u;v N id u v X- u v X X X X X X

γi u;v idαu,v A αu,v X X ? ? id,u;v ? id u;v id- id u;v α - u;v X X X X X



Definition 3.2 Let P : be a functor. A morphism g : Y Z in is S → I → S (i) vertical if PY = PZ and P g = idPY , and (ii) horizontal over (i.e. mapping to) v : j k (where j = PY , etc.) if for every h : X Z over u ; v there is a unique f : X Y over→u with h = f ; g. → → The functor P is called a fibration if for every v : j k and Z with PZ = k there is some horizontal g : Y Z over v. → → Lemma 3.3 P :Σi : .Xi is a fibration, in which (v, g) is vertical iff v = id and horizontal I → I iff g is invertible, the horizontal lifting of v at Z being υvZ.

6 Proof The only part which isn’t obvious is that if (v, g) is horizontal then g is invertible (for u,v the converse recall that α is also invertible); for this we use horizontality twice with u = idi. v j 1 First let X1 = Z, h1id and derive f1 such that [f1 ;(γ Y )− ]; g = id. Then put X2 = Y , h2g; X j 1 now id and g ;[f1 ;(γ Y )− ] both serve for f2.  Lemma 3.4 If P : is a fibration then there is a factorisation system in consisting of the (*pseudo!)vertical andS → horizontal I maps. S Proof Let h : X Z over u : i j. Let g : Y Z be a horizontal lifting of u at Z and f : X Y the unique→ vertical map with→ h = f ; g. Now→ if we are given v ; h = k ; g with v vertical and g →horizontal, then h is over P h = P k ; P g and so there is a unique f over P k with f ; g = h; likewise v ; h is over P h = P k ; P g and v ; f and k both serve as the mediator, so v ; f = k.  Exercise 3.5 Conversely, if P is any functor satisfying the definition of fibration with the word “horizontal” deleted, and the vertical and horizontal maps in form a factorisation system, then P is a fibration. S

i j Proposition 3.6 Σi : . is the lax colimit of the diagram, i.e. if Nj : are functors and u I X X → Y nu : Ni Nj natural transformations such that X → i v u,v id = Niγ ; nid and nu ; nv = Niα ; nu;v X then there is a unique functor N :Σi : . i such that I X → Y NIi = Ni and Nυu = nu

Proof On objects, we are forced to put N(i, X) = NiX. For morphisms we use the vertical- horizontal factorisation: (u, f) = Iif ; υuY , so we must put N(u, f) = Nif ; nuY . Universality of the factorisation makes this *functorial.  Now for the converse; clearly the category i has objects those X with PX = i and morphisms the vertical maps over i. However,X since horizontal maps are∈ defined S by a universal property, the lifting of v : j k at Z with PZ = k is defined only up to isomorphism. We therefore have to make a choice→of liftings in order to determine a functor v : k j. X X → X Lemma 3.7 v is *pseudofunctorial in v. X  Proposition 3.8 There is a weak equivalence from pseudofunctors op Cat to fibrations . I → − → I  4 Oplax and pseudo limits

Having established what we mean by a (pseudo)diagram of categories, we can now calculate its pseudolimit. Although we have chosen not to consider lax diagrams, we can still form a lax or oplax limit of the diagram, which corresponds to a Π-type. This has application to the construction of filtered colimits in Σ-types.

( ) op i Construction 4.1 Let − : Cat be a pseudofunctor. The categories Lim i : . and Πi : . i have X I → I X I X (i) objects the families Xi, xu with Xi i for i and xu : Xi uXj for u : i j such that h i ∈ X ∈ I → X → γiXi αu,vXk Xi - idi Xi u vXk - u;vXk X 6 X X6 X 6 U id xidi and uxv M xu;v - X xu Xi uXj  Xi X

7 commute, where xu is respectively an isomorphism and arbitrary; (ii) morphisms from Xi, xu to Y i, yu the families f i : Xi Y i for i such that h i h i h → i ∈ I uf j uXj X - uY j X 6 X 6 xu H yu

Xi - Y i f i commutes; (iii) componentwise identities and composition. (iv) There are projection functors to i given by the extraction of i-components, for which we also write i, and X X (v) coherences ξu : i u j given by extraction of u-components, which satisfy the equations U and M X → X X γi i = ξidi and ξu;v = ξu ; uξv ; αu,v k X X X analogous to part (i) above. Proof Obvious. 

Lemma 4.2 Πi : . i is isomorphic to the category of sections of the display functor Σi : . i . I X I X → I Proof A section is a functor S : Σi : . i such that PS = id. This means that I → I X → I Si = (i, Xi) and Su = (u, xu) where Xi i and xu : Xi uXj. Functoriality corresponds exactly to the equations U and M. Likewise,∈ X a natural transformation→ X φ : S T between sections satisfies P φ = id, so → i φi = (idi, f ) where naturality makes f i : Xi Y i satisfy the equation H. →  Proposition 4.3 Πi : . i is the oplax limit and Lim i : . i is the pseudolimit. I X I X Proof Given functors F i : i and natural transformations σu : F i uF j such that the equations U and M hold: A → X → X

γiF i = σidi and σu;v = σu ; uσv ; αu,vF k X then there is a unique functor F : Πi : . i such that A → I X iF = F i and ξuF = σu X On objects we are forced to put FA = F iA, σuA , and this is an object of the oplax limit by equations U and M. Likewise on morphismsh we havei F a = F ia , with H given by N. Then F is obviously a functor and unique. Moreover σ are isomorphisms,h i then F factors through the inclusion Lim i : . i Πi : . i. I X ⊂ I X 

8 Proposition 4.4 The limits are Cat-enriched: if φi : F i Gi is a compatible family of natural transformations, i.e. the square → φi F i - Gi

σu H τ u ? ? uφj uF j X - uGj X X commutes, then there is a unique natural transformation φ : F G such that φi = iφ. → X Proof Obvious. 

5 Continuous fibrations

In this section we assume that each j has (filtered) colimits of types and j/ and that these are preserved by u. X I I X Proposition 5.1 The pseudo and oplax limits have filtered colimits, which are computed com- ponentwise and preserved by the projection functors i. X Proof Let be a filtered diagram for which each i has and u preserves colimits. For j let Xi, xu beJ an object of the oplax or pseudo limit,X and for v X: j j let f i : Xi Xi be∈ J a j j 0 v j j0 morphism,h i so that we have a diagram of type in the limit. Then we→ also haveh diagrams→ ini each i i i i I , and we may let fj : Xj X be a colimiting cocone. X Define xu as the unique→ map making xu Xi - uXi0 6 X 6

f i uf i0 j X j u xj Xi - uXi0 j X j commute for each j . If each xu is invertible then so is xu, so the result specialises from ∈ J j the oplax to the pseudo limit. Once we have shown that Xi, xu is an object of the limit, this h i i diagram immediately makes fj a morphism, and it belongs to a cocone because each fj does. The *equations U and M for the object Xi, xu are simply the colimits of the same equations for Xi, xu . h i h j j i If gi : Xi Zi is another cocone, then by the componentwise colimits we have a unique h j j → i mediator gi : Xi Z, and this is a *morphism g = gi by uniqueness of mediators Xi uZi0 . → h i → X  Construction 5.2 Let Xi, xu Πi : . i. A diagram of type i/ in i which takes h i ∈ I X I X u v u;v uxv αu,v Xk (i j) - (i k) to uXj X - u vXk - u;vXk → → X X X X Proof We have to check functoriality: uxid uXi X - u idXi X 6 X-X i γ u,id k id U R α X ? id uXi - uXi X X

9 and uxv αu,vXk uXj X - u vXk - u;vXk X X X X

MNu vxw u;vxw X X X ? ? αu,v wXl uxv;w u v wXl X - u;v wXl X X X X X X

l X u;v,w l v,w A α X u α ?  u,v;w l ? X α X u v;wXl - u;v;wXl X X X 

Definition 5.3 Let Y i be the colimit with cocone zu : uXj Y i, i.e. X → zu uXj - Y i X 6 uxv E zu;v X ? αu,vXk u vXk - u;vXk X X X commutes for all u i/ . ∈ I

t i0 t i Construction 5.4 An isomorphism y : Y ∼= Y . X t Proof Let t : i0 i in . There is a diagram of type i/ given by applying to the previous construction,→ and sinceI this functor is continuous, tzIu is a colimiting cocone.X Using ZA, αt,uXj ; zt;u : t uXj Y i0 is a cocone, so there is aX unique mediator tY i Y i0 . Now t,u j 1 t u X Xt;u j → i X → (α X )− ; z : X Y is a cocone over a final subdiagram of the diagram defining Y i0 and so extendsX toX a cocone→ over the whole diagram, whose mediator yt : Y i0 tY i is the inverse. This is the unique map such that → X

yt Y i0 - tY i 6 X 6 zt;u F tzu over j/ X I αt,uXj t;uXj  t uXj X X X commutes for all u : i j, the vertical maps forming colimiting cocones. → Lemma 5.5 Y i, yu is an object of Lim i : . i Πi : . i. h i I X ⊂ I X Proof For the U equation: yid =? Y i - idY i i i 6 γ Y X 6 zu N idzu id,u j X α X uXj id uXj X - X X γi uXj X 10 which is the colimit of U, and the associative law is obtained in the same way:

u,v k u v k α Y - u;v k Y Y X X X- 6 X u w 6 v ;v z X u z w N X αu,v wXl u v wXl X - u;v wXl X X X X X

uyv FAFuαu,vXl αu;v,wXl yu;v X X ? ? αu,v;wXl u v;wXl - u;v;wXl X X X

;w z u v F ;v u z ;w -  X yu uY j  Y i X 

Construction 5.6 Morphism ζ : Xi, xu Y i, yu by ζi = γiXi ; zidi . h i → h i Proof It is a morphism:

xu Xi - uXj ======uXj ======uXj X X X

γiXi ZLRγi uXj id uγiXj X X ? ? ? ? idxu αid,uXj αu,idXj idXi X - id uXj - uXj  u idXj X X X X X X

zidi E zu uzidj X ? ? ? yu Y i ======Y i - uY j X



Lemma 5.7 ζ is universal into the pseudolimit. Proof Suppose bi : Xi, xu Ai, au is a morphism with au invertible, so the square h i h i → h i bi Xi - Ai

xu H au ? ? ubj uXj X - uAj X X

11 u j u 1 i commutes. I claim that b ;(a )− is a cocone for the diagram defining Y : X ubj au uXj X - uAj  Ai X X w uxu H uav w X X w w ? u v k ? w u v k b u v k w X X X - A M w X X X X w w w αu,vXk Z αu,vAk w w ? ? w u;vbk au;v w u;vXk X - u;vAk  Awi X X

Hence there is a unique mediator ci : Y i Ai such that → ci Y i - Ai 6 zu G au ? ubj uXj X - uAj X X commutes. Finally the following diagram shows that c : Y A is a morphism: → i i c - i Y  A

z u G au ? ubj F uXj X - uAj X6 X yu αu,idXj U uγjXj Z uγjAj X X ? ? u idbj u idXj X X - u idAj X X X X id 6 u z G uaid uγjAj X X X ?  ucj uY j X - uAj X X 

Proposition 5.8 The full inclusion functor Lim i : . i Πi : . i is continuous and has a reflection (left adjoint post-inverse). I X ⊂ I X Proof The first part is immediate from Proposition 5.1; note also that the inclusion is full because the two categories have the same definition of morphisms. We have just found the unit of the adjunction; observe that if ~x are already isomorphisms then ζ is also an isomorphism.  There is an application of these results to finding filtered colimits in Σ-types.

Definition 5.9 A functor P : is a continuous fibration if S → C

12 (i) it is a fibration, (ii) the fibres i have, and the functors u preserve, filtered colimits (of types ), X X J (iii) has filtered colimits (of types ), and C I ci (iv) if ci : Ci C is a colimiting cocone for a diagram cu of type then is a pseudolimiting cone. → I X

Lemma 5.10 An object over a colimit is the colimit of its liftings. C i Proof Let ci : Ci C be a colimiting cocone in and X . Let υci X : Ci,X C,X → i ci C ∈ X u h i → h i be the horizontal lifting of ci, where X = X; then with x = υc we have an object of X u Πi : . Ci . By (iv) this corresponds to an object Y C with Xi = ci Y ... I X ∈ X ∼ X  Proposition 5.11 If P : is a continuous fibration then has and P preserves filtered colimits (of type ). S → C S I Proof We are given , a diagram of type , and suppose that the fibration P corresponds to a pseudofunctor opI →Cat S . Write I C → ( ) ( ) op op op C − : and − : Cat I → S → C X I → S → C → i u so that the diagram has vertices Xi . The arrows of the diagram are of the form x : Xi uXj and satisfy the *equations,∈ X and so we have an object of Πi : . i. Applying the reflector→ X yields an object Y i, yu of Lim i : . i, which corresponds to a uniqueI X object Y (up to h i I X isomorphism) of the fibre over C = colim↑ Ci. By the lemma, Y is the colimit of Y~ , and from the ~ reflection it is also the colimit of the X. (explain why)  Every object is the colimit of its liftings; the colimiting cocone of a diagram of horizontal maps is horizontal.

6 Adjunctions

Recall that we call two functors h : , c : adjoint, and write c h, if there is a bijection A → B B → A a cB A → B hA which is natural in A and B. In particular, corresponding→ to id : cB cB there is the unit ηB : B hcB and to id : hA hA the counit A : chA A. These are→ natural in A and satisfy the triangular→ identities: → → cη ηh c - chc hch  h

X id c c Y id - ? ? c h [Mac Lane 1971], 4.1 shows that (c, h, η, ) suffice to characterise the adjunction. Since we no longer have any mention§ of the objects of the categories and , the latter may now be objects of an abstract 2-category C, and we have an equational AdefinitionB of an adjunction between two opposite arrows (1-cells) between two objects (0-cells) of a 2-category. The above natural bijection is a well-known and easy to remember way of presenting the adjunction, but it has a less well known dual. Instead of applying c and h to objects of (or functors into) and , we can apply functors out of these categories: A B Bhˆ Aˆ → Bˆ Acˆ → 13 naturally in Aˆ : and Bˆ : . We could say “h is co-left adjoint to c.” A → X B → X

Construction 6.1 A natural bijection between natural transformations θ : h1 h2 and φ : c2 → → c1 which identifies identities. Using both correspondences together we obtain

η2h1 ιh-1 h2c2h1 h1 h2c1h1

c h h θ h c 2 2 2  1 1 1 κ 2 6 - h 6  ? 2 1 c2θ h2 θc η1 -  κ -  ι c2h1 id h2c1 id -

φh1 c1 h η2  1  2 φ - 6 ?  ? 1 2 c c1h1 κc φ 1 h2c2

c2η1 c2-ι c2h1c1 c2 c2h2c1

(commutation of one of which suffices) where we can verify θ ι φ κ θ are bijections 7→ 7→ 7→ 7→ using the triangle laws.  We shall write η1 - id h1c1

η2 D θc1 ? ? h2φ- h2c2 h2c1

Lemma 6.2 The bijection respects composition and identites.

Proof It suffices to show that θ ; θ0 and φ0 ; φ satisfy the equation.

η1 - id h1c1

η 2 D θc1 - ? h2φ- η3 D0 h2c2 h2c1

θ0c2 Z θ0c1 ? ? ? h3φ-0 h3φ- h3c3 h3c2 h3c1

The result for identities is obvious. 

Definition 6.3 The 2-category Ca has the same objects as C; its 1-cells are adjunctions (c, h, η, ) and its 2-cells are pairs of natural transformations (φ, θ) satisfying the equations of the construc- tion. The composition is in the obvious way.

14 Proposition 6.4 The forgetful 2-functor Ca C which extracts the left (or right) part of the adjunction and natural transformations is full→ and faithful at the 2-level. Loosely speaking, the 2-categories of left and right adjoints are dual.  Note carefully that we shall find it convenient to write composition in both right-handedly as u ; v and left-handedly v u: these notations are completely synonymousI. We shall also adopt the convention of writing u◦ for the functors in the limit diagram (which are contravariant in u) X and u for their left adjoints in the colimit diagram (which are covariant in u). X Definition 6.5 Let P : be a functor. A morphism f : X Y is op-horizontal over P f = u : i j if for any hS: X → C Z over u ; v : i j k there is a unique→ g : Y Z over v with h = f ; g. We→ call P an op-fibration→ if each u has→ an op-horizontal→ lifting at each →X with PX = i. If P is both a fibration and an op-fibration we call it a bifibration.

Lemma 6.6 P : is an op-fibration iff P : op op is a fibration. S → C S → C Proof Obvious. 

( ) op i Proposition 6.7 Let − : Cat be a pseudofunctor. Then P :Σi : .X is a bifibration u X I → I iff each has a left adjoint, written u. X X

Note 6.8 Since the left adjoints (op-substitutions) u are covariant in u : i j, but we still write composition of functors from right to left, it is convenientX to use the notations→

u ; v and v u ◦ synonymously for the composite i u j v k. → → Proof [ ] Any morphism (u, f):(i, X) (j, Y ) can be factorised as a vertical followed by a horizontal, ⇒ or as an op-horizontal followed→ by a vertical:

- (i, X) (j, uX) X (u, f ) ? - ? (i, uY ) - (j, Y ) X

where uX is introduced as the codomain of the op-horizontal lifting. Making such a choice X i i j for all X , we can extend this uniquely to a functor u : . I claim this is the ∈ X X X →u X left adjoint: f corresponds bijectively to either vertical part X Y or uX Y . The unit and counit are given by → X X →

u X u Y X X ophoriz - ηuX and  Y ophoriz u ? - ? u horiz- u horiz - uX uX Y Y X X X X

u u u u and the equations η ; u = id and uη ; u u = id may be verified using the uni- X X uX X versal properties of the horizontal map Y Y and the op-horizontal map X uX respectively. X → → X

15 u [ ] I claim that (u, η X):(i, X) (j, uX) is op-horizontal. Let (u ; v, f):(i, X) (k, Z), ⇐ u,v 1 → X u v i v → so that f 0 = f ;(α Z)− : X Z in . Let f 00 : uX Z be its adjoint → X X u X u X → X transpose; this is the unique solution of f 0 = η X ; f 00. Then (v, f 00):(j, uX) (k, Z) X X → is the required factorisation. 

Lemma 6.9 The functors u have coherences δi : id id and βv,u : v u v u which satisfy equations dual to those in definitionX 2.1: X → X ◦ → X X

wβv,u w v u X w v u X X X X X ◦ 6 6

βw,v u A βw,v u associativity X ◦ βw v,u w v u ◦ w v u X ◦ X X ◦ ◦ and δj u uδi- u X id u u id X u X  X j X X X i X 6 6 - id L βidj ,u βu,idi R u unit X u idX

u u X X Proof Immediate from lemmas 3.7 and 6.8. 

u,v i Lemma 6.10 The coherences α , βv,u, γ and δi are related to the units by the *equations

ηid ηw wηw0 - id - w -w w w0 id id id w X X w w X X X X X X X 0 X

id w;w0 w,w0 id C δk and η B α w w X X 0 X ? ? ? k ? w;w0 γ βw ,w - id w;w0 0 - w;w0 id w w X w w X X X 0◦ X X 0 X and to the counits by the equations

   w0  id id  w0 w0  w0 w w w0 id id id w X X w w X X X 0 X X 0 X X X 6 6 6 6

k w,w0 id C idγ and w w B w w α X 0◦ X X 0

w;w0 δ βw ,w  k w;w0  0 w;w0 id id w w X w w X X 0◦ X X 0 X X



7 Transfer functors

Now we shall begin the construction of the equivalence between the pseudolimit and pseudocolimit. Imagining for the moment that we already have this equivalence, we can form

j i i X- Colim k Lim k X- j X k X ∼ k X X ∈I ∈I

16 using the colimiting cocone and limiting cone. It is our purpose now to construct this composite j functor, Mi . When we have it we can use the universal property of the pseudolimit to construct functors i Lim k : . k; these will be left adjoints to i and will form the colimiting cocone X → I X X i. X Construction 7.1 A Diagram of type (i, j)/ of functors i j. On objects, (i u k v j) v I X → X → ← 7→ u. The image of w : k k0 is X X → v w v,w v η -u v w α w -u v;w u X X w u X X w u X X X X X X X X X 6 6 v w v;w βw,u N βw,u X X X αv,w v w w u- v;w v0 w u X ◦ w u = u X X X ◦ X X ◦ X X 0

Proof We have to show that we have defined a functor. The effect on the identity idk is the v v long anticlockwise route from u on the top left to u on the top right. We use the k X X X X correspondence between γ and δk, together with one unit and one counit law.

v id- v id- v u u u X X X X -X X

id k η C γ R id v, ? ? α v id δ-k v id id u u X X X X X X X 6 -

βid,u R id

v id u X X X For the composite w ; w0,

v v u ======u X X X X

ηw B ηw;w0

? w w,w ? η 0 α 0 βw ,w v w - v w w0 - v w;w0  0 v w;w0 w u w w u w w u w w u X X X X X X X X 0 X X X X X 0 X X X X X 0◦ X

αv,w ZAZαv,w αv,w;w0 αv,w;w0

? w ? v;w,w ? ? η 0 α 0 βw ,w v;w - v;w w0 - v;w;w0  0 v;w;w0 w u w w u w w u w w u X X X X X X 0 X X X X 0 X X X X 0◦ X 6 6 6 6

βw,u NNAβw,u βw,u βw w,u 0◦ w v;w,w η 0 α 0 βw ,w u v;w - v;w w0 - v;w;w0  0 v;w;w0 w u w w u w w u ◦ w v u X X ◦ X X X 0 X ◦ X X 0 X ◦ X X 0◦ ◦ where the left-hand side is the effect on w, the bottom that on w0 and the right-hand side that on w ; w0. 

j i j v v Construction 7.2 Functor M : is the (pointwise) colimit, with cocone ν : u i X → X u X X →

17 j Mi , i.e. v v νu - j u M X X i 6 v w v;w η u K νw u X X ◦ ? v,w v;wβ v w α w -u v;w  w,u v;w w u X X w u X w u X X X X X X X X X ◦ i Proof We know that (i, j)/ is a filtered category, so it is possible to form the colimit Mj X for each X; it is standard that thisI extends to a functor which is the colimit in the functor category. v v v i w In detail, for f : X Y we have a cocone uf ; νuY : uX Mj Y by naturality of η , v,w →i X X X X → v α and βw,u, so Mj f is the unique mediator and it is automatic that νu is natural. 

t t j j0 Construction 7.3 Coherences µs : Mi s = M for s : i0 i and t : j0 j such that X X ∼ i0 → → µt t j s - j0 Mi s M X X i0 6 6 t v t;v ν s P ν X uX u;s t,v t;vβ t v α u -s t,v  u,s t;v u s X X u s X u s X X X X X X X X X ◦

t v j0 Proof The lower composite u s M is a cocone by naturality of α, β and η and i0 associativity of α and β (the proofX X isX veryX similar→ to the diagram in 7.1). Since t is continuous, t v t X ν s is colimiting and µ is then defined as the mediator. It is invertible because (i, j)/ X uX s I → (i0, j0)/ is final, and natural by the universal property of the colimit. I 

t t j t t j j0 Lemma 7.4 The coherences µi = Mi δi ; µ : Mi = M for s : i0 i satisfy U, M and X idi X ∼ i0 → µt t j i - j0 Mi M X i0 6 6 t v t;v ν s P ν X uX u;s t,v t v α -u t,v u X u X X X X X

Proof These are just colim↑ U u and colim↑ M u. X X  i k j j u u Definition 7.5 i : Lim k : . such that i = Mi and ξ i = µi . We use the t X X → I X X X X coherences µi and the universal property of the pseudolimit. 

8 Unit and counit

i Now we shall show that i and that i is a colimiting cocone. X a X X Lemma 8.1 The maps

v u v k v v i u-ξ v u k u - v k ξ j u X X u X X X X X ∼= X X X X X X ∼= X form a cocone for the diagram defining M j i. i X

18 a,b,c def a b c Proof Recall that d,e,f = d e f , and we also omit the functors applied to natural transformations. ThenX the diagramX X X X X X

u v v i ξ - v u,k u - v,k  ξ j Xu X Xu X X X

w w w w η w η ZZη ξ w w w - w ? u ?  w v,w i ξ- v,w u,k u - v,w k v,w,l w w,u w,u w N w X X X X X X X w 6 w w w ξ u η w ; MNξ w ξ w w w w - - - w u,w  w v,w u;w,l α v,w u,w,l -u v,w w,l w βw,u YMid w Xw,u X  Xw,u X Xw X w w w ZBβ  w w,u w w - w ? w ξu;w  w v,w i - v,w u;w,l w u - v,w,l w w u w u ◦ w X ◦ X X ◦ X X w w w αv,w ZZαv,w αv,w w w ? ? ? w ξu;w  ξv;w w v;w i - v;w u;w,l w u - v;w,l  wj w u w u ◦ X ◦ X X ◦ X X X commutes. 

j j i j Definition 8.2 Natural transformation i : Mi is the mediator, the unique map such that X → X j M j i i - j i X X 6 νv i Q ξv uX v u v k ? v i uξ- v u k u - v k u X X u X X X X X X X X X X X commutes.

i j j Construction 8.3 i : i idLim such that  = i. X X → i X Proof The coherence equation H is obtained by varying the above diagram along t : j0 j, which is the colimit of three commutative squares which amount to naturality of αt,v and M→for ξ. 

19 Lemma 8.4 The diagram

j j i i - j Mi Mi Mi 6 6 v v µ i νu Q

µu   vM i -i v uM k u- vM k Xu i Xu X i X i 6 6 6 id id v PNPνk νu νu

 νu v u,id u - v,id u Xu Xu Xu 6 6 id α v, u, id k k R α R γ N γ -  v u id- v u u - v id - v Xu Xu Xu Xu Xu Xu commutes. 

i Lemma 8.5 i : i 1L is a colimiting cocone. X X → Proof *Cocone (over what?) Colimiting: use finality.

j i - colim↑ i  j ======j i X X X X ∈I 6 6 j j i ξ ? j i j i v k i ======i X X X X X X X X 6 6 v i v k ν id uX X X u = id v id v i k v k idξ- v id k u ======id X X id X X X X X X X X X X

j The details are in the diagram defining i . *Cocones, *coherences. 

u u i i Lemma 8.6 η ; νu : id i Mi = i is independent of the choice of u : i j. X → X X → Proof It suffices to show that ηu ; νu = ηidi ; νidi . This follows from definition 7.2 (of ν) with u idi i w = id, together with the U and C equations for γ and δi. 

i u u Definition 8.7 η = η ; νu .

i i i Lemma 8.8 η ; i = id i . X X X

20 Proof i i  = i M i i i X - i i X X 6

νidi i Q ξidi idi X

? idi idi v i id ξ id idi i i- idi idi i i - v k id X X id X X X X i X X X i X X X X 6 6 -

ηidi i Zηidi idi i Y X X X id

id i ξ - u i X X X 

i Lemma 8.9 iη = i i. X X j i j Proof By enrichment of the pseudo limit, it suffices to verify componentwise that M η ;  i = i i X id j . The diagram Mi M jηi j j i - j i i i - j Mi Mi Mi X Mi 6 6 6 v i Z νuMi

νv vM i νv u Xu i u - 6 η vνu Xu u v η u u v - v u X X- v Xu Xu Xu Xu commutes for all (u, v) (i, j)/ ; the bottom row is the identity by X. The left-hand side is part of a colimiting cocone,∈ and theI diagram and provides another cocone for which the top row and the identity both serve as the mediator. 

i Lemma 8.10 Lim i : . is the pseudo-colimit, with cocone i. i I X X Proof Let Gi : be another cocone for this diagram with coherences ... Consider the X → A i i diagram of type of functors Lim i : . with i Ai which takes u : i j to I I X → A 7→ X →

Functoriality... Let A be the colimit. Then

j v A i = colim↑ AjMi = colim↑ colim↑ Aj u = colim↑ Ak u = Ai X j j (u,v) /(i,j) X X ∼ u:k i X ∈I → *Uniqueness. 

21 9 Pseudolimit of adjunctions

Finally we shall show that we have the limit in the category of homomorphisms. Let Gi : X i i →i X be a cone of homomorphisms and Fi G be the corresponding comparisons, with units υ and a i i counits δi. Since L is the limit we have a unique mediating functor G : X L with G = G , → X and similarly, since it is also the colimit, F : L X with Fi = F i. → X

Construction 9.1 Natural transformations υ : idL GF and δ : FG idX , by the diagrams → → υ - δ - idL GF sF G idX 6 6 6 -

k kGF k F kG δ k k k k kυ - k k k k X X kG Fk FkG X X X X

Lemma 9.2 F G with unit υ and counit δ. a Proof The two triangle laws follow from these diagrams:

F υ δF F - F GF - F 6 6 6

k k k k kυ - k k δkFk - k Fk X X FkG Fk X Fk X X X and jυG Gjδ Gj X - GjFG - Gj 6 6 6

j i - j i i - j i Mi G Mi G FiG Mi G 6 6 6

v i i v i v j uυ G- v i i uG -δi v i uG X X uG FiG X X uG X X X X X X 

Theorem 9.3 For any cofiltered diagram of homomorphisms, the limit of the homomorphisms quˆa continuous functors, the colimit of the comparisons quˆa continuous functors and the limit of the homomorphisms quˆa homomorphisms exist and are naturally equivalent. Proof It only remains to formulate and prove naturality. One way of stating this is that the hm hm op forgetful functors FCCat FCCat and FCCat FCCat create limits.  We call L the bilimit of the→ diagram. →

22 10 Notes

Definition 8.2:

v k ξ v l -  X6 X X v X w k 0 l i Q X

v j v u νw;u ξ k j  v j w u - v w0 l Mj X w u X X ◦ w X X X ◦ X X X 0 X X 6

k u v u Mj ξ NMw uξ α X X ◦ ? νv u i ? k u i  w;u v u i ξ - v u w l Mj X X w u w u X. X X X ◦ X X X X ◦ X X X - . 6 i .6 u . X . k . P β µ uX . . . id v . ν . α idM k u u . - id v v u - v u i N β B i u  i X . w u i w u X X X X . X X X X X X X X X X X . 6 . . . k . Z id γ . γ . . . v u v uw w ? k u i νu u i - v u i wu ξ- v u w l M u X X X w u X X w u i X X X X X X X X X X X X X X 6 6 6

k i v i v w l M u NZwu wu i X X X X X X X

v i v w k i  νu v i wξ - v l M X w X X w i X X X X X X X l v w k Q X i X

? v  k ξ - v l X X X Construction 9.1:

i X δ A id  FG i 6 6 -

δk F kG u σu X ⇓- A

k k - k FkG = Fk G F kX G j X X A ? j X

23 i - i Lim Ai X - A6 X i i τ X A u i σu ξ · - ? - i u j Aj u A X6 X X X - ξu σ u u j ==== u ==== Ai Aju ⇒ X X ⇒ X τ j ? ? - j j Aj X X A i X 6

τi i X

σ σi i i - Ai i A i Ai X X X  = - 6 ∼ - i η A i u A A i Aiνu i  i  τ i i i A -- u Aiη - u i Ai Ai u Ai Ai i X X X X 6 6 6 u σu σu u σu X X A ηu A  u j u - u j u - Aj u X Aj u u X Aj u X X X X X Aιu- A j u A i A  X X X - A i τ i σj u σi X ? ? i σu - i  σi i Aj u Ai Ai X A i X X X X A  6 A i τi

i i  σi i Ai X A i X X X 6 6 i i Aii N A ii X X X u i u i σi u u i Ai u X X X A i u X X X X X X X 6 6 u u σuξ pseudo colim Aιuξ − σ j j  j j Aj X A j X X X i i i i i i υ F ; F δi = idF υ : id i F Gi δi : GiFi idA θi : Gi G i X → → → X

24 i i i i υ F - i i F δi - i F F GiF F  6 6A F i i i i i j η F defF θi F def F δ G i X X X a- ? i i ? i i i υ i -F i i F Gi-F i i id adj iF X X X FG i F X F GF G F X X X X X X- a -X

i i i iF N F GiF i X id X a ? ? X iυF ? - iF X - iF GF Lim X X i υ - i id i FGi X X

i i i δi - η def F θi GiF id X .A .6 ? i ? . i ν -i i i i . i X X FG i θiF = θi F def . δ X X X X X . . 6 ? . v i i GiF - ν G iF = G i F GF u X X X colimiting

v u X X i Giυ - i δiGi - Gi GiF Gi Gi 6 i θi Z θiF Gi def δGi

? i ? G iυ - i GiFGi- G i X G i FGi GF Gi X X X

i i G iη def G i Fiθi NNGF θi θi X X X ? i ? ? i G i υ -i i GiFG-i G i i X X X G i FG i X GF G i X X X X X X X

Gi i Z GiFG i δG i X X X ? ? ? - Gυ i - δG i -  G i X GF G i X G i X X X

25