Reference Request: Ends and Coends

Question

I'm interested in learning about ends and coends, especially since I would like to develop categorical view of algebraic topology when I take a class in it this Fall. I would also like to learn about Kan Extensions. I have a solid background in the category theory necessary as a prerequisite. I am looking for a good reference on these subjects. Thanks for any help you can provide!

Answer 1

I'd recommend the appropriate chapters of Maclane or of Riehl's "Category Theory in Context" for Kan extensions, and the first chapter of Kelly's "Enriched Category Theory" for ends and coends.

If I remember, there's not a through consideration of extranatural transformations, which are what ends and coends represent, in Kelly's book (which also treats Kan extensions for enriched categories, which could be worth reading as in that context some concepts separate whose coincidence in ordinary categories is not really natural.) For this I'd send you back to the original source: Samuel Eilenberg and G. M. Kelly, A generalization of the functorial calculus, J. Algebra 3:3, 366–375 (1966)

Answer 2

There are these beautiful notes by Fosco Loregian.

Answer 3

The other answers make the recommendations I would make, but another somewhat unusual suggestion is Cáccamo's and Winskel's A Higher Order Calculus for Categories which formalizes a fragment of the internal language of locally small categories and ends play a significant role. The discussion of the justification for the rules they choose as well as the results proven with them are both interesting and aspects you should be familiar with. The rules themselves provide conservative but simple and compact descriptions of when and how to use ends with assumptions clearly stated.