Dinatural Transformations Dinatural transformations can be viewed as an attempt to generalize the notion of a natural transformation to functors 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 morphisms in D, uc : S(c; c) ! T (c; c); c 2 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 functor 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 2 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 2 D, also called wedges (or extranatural transformations). Explicitly, u : S −!:: e consists of arrows uc : S(c; c) ! e; c 2 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) 2 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 2 C. In other words, a coend of S : Cop × C ! D is just a coequalizer 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 c2C O O inf inc S(d; c) / S(c; c) S(f;1) In particular, in a co-complete category all coends exist. Conversely, if F : I ! C is a diagram, then a wedge from F pr2 is just a co-cone, so Z i F (i) = F pr2(i; i) ! F pr2(i; i); i 2 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 monoidal category (M; ⊗). Then the coend R c F (c) ⊗ G(c) of ⊗ ◦ (F × G) is called the tensor product of F and G and written F ⊗ G. Example. (Top; ×) is monoidal. Let X 2 S = Cat(∆op; Set) be a simpli- cial set and consider also the functor ∆ ! Top; n 7! j∆nj: A coend of the functor op m ∆ × ∆ ! Top; (n; m) 7! Xn × j∆ j is called the geometric realization of X: Z n • n jXj = X × j∆ j = Xn × j∆ j: Example. Let M be a simplicial model category. 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) 2 M Here B(− # I)op is viewed as the functor Iop ! S which associates to each i 2 I the classifying space of the comma category (i # I)op. Thus op B(i # I)n = fi0 i1 · · · in i in Ig : Ends A dinatural transformation u : e −!:: S, also called a wedge, consists of arrows uc : e ! S(c; c); c 2 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 end of a functor S : Cop × C ! D is an object R S(c; c) 2 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 2 C. The homotopy limit 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 2 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 2 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 2 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 function 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. 5.