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The co-end justifies the co-means

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If you have a question about this talk, please contact Philip Saville.

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.

Covering:
  • 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
Prerequisites:
  • 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’)

This talk is part of the Logic & Semantics for Dummies series.

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