Abstract

Ends and coends are generalisations of limits and colimits: if you think of a colimit as a generalised `sum’, then the corresponding coend is a kind of `integral’. They also have close connections to the Yoneda lemma and Kan extensions. I will recap the definitions of (co)limits, then introduce (co)ends and their basic theory. In particular, I will show how every limit is an end, and every end is a limit. Finally I will try to present examples of ends and coends at work, and show how they can be extremely useful tools for reasoning in category theory.

• recap of definition of (co)limits,
• definition of (co)ends,
• basic theory of (co)ends: relationship to limits, Fubini theorem
• (co)ends in the wild: Kan extensions, Yoneda, other examples

• basic category theory: functors, natural transformations
• I will cover the definition of limits and colimits, having some intuition already will be useful (eg. as covered in Awodey’s `Category Theory’)