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Everything's a Kan extension

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If you have a question about this talk, please contact Ian Orton.

Kan extensions are a fundamental construction in category theory. As Mac Lane puts it, they “subsume all other fundamental concepts” in the field. I will give a brief introduction to Kan extensions: showing how the definition captures other important constructions, and how Kan extensions are hidden in many familiar examples. I will also give (largely without proof) some of the properties that make Kan extensions so useful.


  • definition of Kan extensions, their basic theory
  • examples of Kan extensions
  • other fundamental concepts as Kan extensions


  • basic category theory (natural transformation, functor)
  • some familiarity with the definition of adjunctions (I will cover this in the talk, but might be quite quick)


  • most of the material covered will be taken from Mac Lane’s ‘Categories for the Working Mathematician’ (chap. X); some will come from Awodey’s ‘Category Theory’ (esp. pp. 186-192 for the Yoneda Lemma, and pp. 208 – 234 for adjunctions and examples of Kan extensions)
  • for more background the Wikipedia page on Kan extensions is pretty good; the essay at provides a reasonably readable introduction as well.

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

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