Limits and colimits have very nice definitions in terms of Kan extensions, and therefore enjoy very nice adjointness properties. Mac Lane's Categories for the Working Mathematician gives a construction called the subdivision category of a category $C$, which allows one to reduce the theory of ends and coends to the theory of limits and colimits (and therefore the theory of Kan extensions). This construction feels a bit artificial and messy, although it is very useful for quick and dirty proofs of many of the details about ends and coends.

Can we give a definition of the end and coend as some sort of Kan extension without invoking the subdivision category construction?


Ends and coends should be thought of as very canonical constructions: as Finn said, they can be described as weighted limits and colimits, where the weights are hom-functors.

Recall that if $J$ is a (small) category, a weight on $J$ is a functor $W: J \to Set$. The limit of a functor $F: J \to C$ with respect to a weight $W$ is an object $lim_J F$ of $C$ that represents the functor

$$C^{op} \to Set: c \mapsto Nat(W, \hom_C(c, F-)).$$

Dually, given a weight $W: J^{op} \to Set$, the weighted colimit of $F: J \to C$ with respect to $W$ is an object $colim_J F$ that represents the functor

$$C \to Set: c \mapsto Nat(W, \hom_C(F-, c)).$$

Then, as Finn notes above, the end of a functor $F: J^{op} \times J \to C$ is the weighted limit of $F$ with respect to the weight $\hom_J: J^{op} \times J \to Set$, and the coend is the weighted colimit of $F$ with respect to $\hom_{J^{op}}: J \times J^{op} \to Set$.

The ordinary limit of $F$ is the weighted limit of $F$ with respect to the terminal functor $t: J \to Set$. Ordinary limits suffice for ordinary ($Set$-based) categories, but they are inadequate for enriched category theory. The concept of weight was introduced to give an adequate theory of enriched limits and colimits (replacing $Set$ by suitable $V$, and functors as above by enriched functors, etc.)

Weighted colimits and weighted limits (in particular coends and ends) can be expressed in terms of Kan extensions. For any weight $W$ in $Set^{J^{op}}$, the weighted colimit of $F: J \to C$ (if it exists) is the value of the left Kan extension of $F: J \to C$ along the Yoneda embedding $y: J \to Set^{J^{op}}$ when evaluated at $W$, in other words

$$(Lan_y F)(W)$$

A similar statement can be made for weighted limits, as values of a right Kan extension.

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  • $\begingroup$ A better answer than mine. I'll just point out, if only to remind myself, that the situation in your last paragraph comes from the diagram expressing y as the unit for the free cocompletion of J: given $F \colon J \to C$ with C cocomplete, the corresponding cocontinuous functor $[J^{op}, Set] \to C$ given by universality is $W \mapsto \operatorname{colim}^W F = (\operatorname{Lan}_y F) W$. $\endgroup$ – Finn Lawler Aug 31 '10 at 21:12
  • $\begingroup$ Precisely. The centrality of the free cocompletion in these studies is hard to overestimate. :-) (Not speaking to you Finn, but just generally: many people find ends and coends hard to grok, and Mac Lane is not really the best source to learn about them. First, extranatural transformations are more important than the more general dinatural transformations, so learn those first (and their "string diagram calculus"). Then, think of ends and coends as universal extranatural transformations. Finally, learn the canonical examples (such as tensor products of functors). Then they will seem easy.) $\endgroup$ – Todd Trimble Aug 31 '10 at 22:30
  • $\begingroup$ @Todd: The "universal extranatural transformation" definition is a good way to think about them, but it is very hard to do computations with them like that. $\endgroup$ – Harry Gindi Sep 1 '10 at 5:08
  • $\begingroup$ In my experience, far and away the most commonly arising examples are tensor products and homs of $V$-valued enriched functors (aka "modules"), and there is a kind of module calculus for dealing with computations of these. The point of my previous comment would then be to develop a clear mental model of this calculus, based on the typical ways of passing between extranaturality and naturality and their associated string diagrams (where one bends and unbends strings). With such clarity, one can then learn to compute with dispatch. $\endgroup$ – Todd Trimble Sep 1 '10 at 8:40
  • $\begingroup$ Note also that Mac Lane's "subdivision category" does not work in the enriched situation, so the weighted version is the only way to do it there. $\endgroup$ – Mike Shulman Sep 3 '10 at 6:00

I just happened across this old question and thought I would add another possible answer. Instead of the subdivision category it is possible to use the twisted arrow category $\mathrm{tw}(C)$ in exactly the same way: there is a functor $\mathrm{tw}(C) \to C^{\mathrm{op}}\times C$, and the end of a functor $C^{\mathrm{op}}\times C\to D$ is obtained by restricting it to $\mathrm{tw}(C)$ and taking the limit.

This may not be as satisfying as the other answers, since it is very similar to Mac Lane's version, but at least the twisted arrow category is somewhat less ad hoc than the subdivision category. It also has the advantage of being "homotopy correct", in that homotopy ends can be computed in the same way as homotopy limits over $\mathrm{tw}(C)$, which is not the case for the subdivision category (see e.g. this paper).

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I don't know of any definition involving Kan extensions, but (co)ends can be expressed as (co)limits weighted by a hom functor (see e.g. here), so that for $F \colon C^{op} \times C \to D$ the end $\int_c F(c,c)$ is $\lim^{\hom_{C}} F$, where hom is taken as a profunctor $C^{op} \times C \not\to 1$. Similarly for (pointwise) Kan extensions: $(\operatorname{Ran}_K G)(c) = \lim^{C(c,K-)} G$, where $C \overset{K}{\leftarrow} B \overset{G}{\to} D$, so that the weight is the 'representable' profunctor $C(1,K) \colon B \not\to C$. See e.g. Emily Riehl's notes referenced here.

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  • $\begingroup$ This really only works noncircularly in the $Set$-enriched case, since to define the enriched functor category, we must use coends, unless you're familiar with another way to define the enriched functor category. Can we define the enriched functor category without relying on coends? $\endgroup$ – Harry Gindi Aug 31 '10 at 20:14
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    $\begingroup$ No, the enriched functor category is defined by means of the usual end (not coend), and I don't think there's another natural definition. But it's not really important, because 'all concepts' are (pointwise) Kan extensions and therefore weighted limits. $\endgroup$ – Finn Lawler Aug 31 '10 at 21:03

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