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Dinatural Transformations Dinatural transformations can be viewed as an attempt to generalize the notion of a to that needn’t be of the same variance. Here is the general concept:

Definition. Given functors S,T : Cop × C → D, a dinatural transforma- tion u : S −→.. T is a family of in D,

uc : S(c, c) → T (c, c), c ∈ C, such that for every f : c → d in C we have a commutative hexagon

ud S(d, d) / T (d, d) S(1,f) : T (f,1)

% S(d, c) T (c, d). 9

S(f,1) T (1,f) % S(c, c) / T (c, c) uc

Clearly, the restriction to the diagonal of a natural transformation u : S ⇒ T is dinatural. In line with the introduction, one considers functors C → D and Cop → D and considers dinatural transformations of the functors obtained by pre- composing with projections from Cop × C. For example, given two functors F,G : C → D, a natural transformation F ⇒ G is just a dinatural transfor- .. mation F ◦ pr2 −→ G ◦ pr2. Similarly for contravariant functors, using pr1. One also obtains a notion of “transformation” from a covariant F to .. a contravariant functor G: it is a dinatural transformation F ◦pr2 −→ G◦pr1, which means we have

uc : F (c) → G(c), c ∈ C, so that for every f : c → d in C we have a commutative diagram

F (f) F (c) / F (d)

uc ud  G(f)  G(c) o G(d).

1 Wedges and Coends We will be interested in dinatural transformations from or to a constant func- tor e ∈ D, also called wedges (or extranatural transformations).

Explicitly, u : S −→.. e consists of arrows

uc : S(c, c) → e, c ∈ C so that for every f : c → d in C we have

S(1,f) S(d, c) / S(d, d)

S(f,1) ud   S(c, c) / e. uc

Definition. A coend of a functor S : Cop × C → D is an object, written R c S(c, c) ∈ D, together with an initial wedge u : S −→.. R c S(c, c).

R c That is, u is a wedge uc : S(c, c) → S(c, c) so that for any other wedge R c wc : S(c, c) → f there exists a unique arrow φ : S(c, c) → f in D with φ ◦ uc = wc for all c ∈ C.

In other words, a coend of S : Cop × C → D is just a

S(1,f) S(d, c) / S(d, d)

inf ind `  `  R c S(d, c) // S(c, c) / S(c, c). f:c→d in C c∈C O O inf inc

S(d, c) / S(c, c) S(f,1)

In particular, in a co-complete all coends exist. Conversely, if F :

I → C is a diagram, then a wedge from F pr2 is just a co-, so Z i F (i) = F pr2(i, i) → F pr2(i, i), i ∈ I,

2 is a colimit of F ; thus the existence of all coends implies that C is co-complete. The coequalizer description of the coend shows also that a co-continuous functor preserves all coends.

Let F : Cop → M and G : C → M be functors into a (M, ⊗). Then the coend R c F (c) ⊗ G(c) of ⊗ ◦ (F × G) is called the of F and G and written F ⊗ G.

Example. (Top, ×) is monoidal. Let X ∈ S = Cat(∆op, Set) be a simpli- cial set and consider also the functor ∆ → Top, n 7→ |∆n|. A coend of the functor op m ∆ × ∆ → Top, (n, m) 7→ Xn × |∆ | is called the geometric realization of X: Z n • n |X| = X × |∆ | = Xn × |∆ |.

Example. Let M be a simplicial . In particular, M is ten- sored over S, given by a functor ⊗ : M × S → M. If F : I → M is a diagram, then the homotopy colimit is by definition Z i op hocolimIF = F (i) ⊗ B(i ↓ I) ∈ M

Here B(− ↓ I)op is viewed as the functor Iop → S which associates to each i ∈ I the classifying of the (i ↓ I)op. Thus op B(i ↓ I)n = {i0 ← i1 ← · · · ← in ← i in I} .

Ends A dinatural transformation u : e −→.. S, also called a wedge, consists of arrows

uc : e → S(c, c), c ∈ C, so that for every f : c → d we have a commutative diagram

uc e / S(c, c)

ud S(1,f)   S(d, d) / S(c, d). S(f,1)

3 Definition. An of a functor S : Cop × C → D is an object R S(c, c) ∈ D R .. c together with a final wedge u : c S(c, c) −→ S. R That is, u is a wedge uc : S(c, c) → S(c, c) so that for any other wedge c R wc : f → S(c, c) there exists a unique arrow φ : f → c S(c, c) in D with uc ◦ φ = wc for all c ∈ C.

The homotopy is an example of an end.

Properties of Coends We will say that a functor U : D → E creates coends, if for every functor S : Cop × C → D and final wedge u : US −→.. e

1. there exists a unique wedge v : S −→.. d with Uv = u to some d ∈ D

2. this unique wedge is already final.

The next proposition asserts that a “parameter-dependent” coend R c S(c, c, p) of a functor S : Cop × C × P → D is functorial in p.

Proposition. Suppose S(−, −, p) has a coend for each p ∈ P. Then the coend of S : Cop × C → DP may be computed pointwise. More precisely, the forgetful functor DP → DOb(P) creates all coends, viewing Ob(P) as a discrete category. In particular, the coend may be viewed as a functor Z : Fun(Cop × C, D) → D.

Proof. Suppose we are given wedges uc(p): S(c, c, p) → e(p) for each p ∈ P. .. P We wish to find a unique extension to a wedge uc : S(c, c, −) −→ e in D for an extension e of the object e(p). Necessarily, for f : p → q in P,

uc(p) S(c, c, p) / e(p)

S(1c,1c,f) e(f)

 uc(q)  S(c, c, q) / e(q),

4 P so if uc is to be a wedge in D , there is just one way to define e on morphisms, namely as the map induced on the coend e(p) by the wedge uc(q)◦S(1c, 1c, f). It is now easy to check that the forgetful functor creates coends. The last assertion follows by taking P = [1], viewing a natural transformation as a functor Cop × C → D[1]. Next is a Fubini-type property of coends.

Proposition. Let S : Cop × C × Dop × D → E be a functor and suppose each S(−, −, p, q) has a coend R c S(c, c, p, q). Then

Z p Z c Z (c,p) S(c, c, p, p) = S(c, c, p, p), meaning that S has a coend iff R c S(c, c, −, −) has a coend and that they coincide. If in addition each S(c, d, −, −) has a coend,

Z p Z c Z c Z p S(c, c, p, p) = S(c, c, p, p).

Proof. A wedge S(c, c, p, p) −→.. e is dinatural in c (and in p) and thus yields a wedge R c S(c, c, p, p) −→.. e. Conversely, a wedge R c S(c, c, p, p) −→.. e (dinatural in p) may be precomposed with S(c, c, p, p) → R c S(c, c, p, p) to make a wedge on S. The proof is an easy diagram chase, where one uses to existence of the coend of S(−, −, p, q) for all p, q to insert terms into the diagram so one is able to use the functoriality of S.

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