Abstract

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.

Covering:

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

Prerequisites:

• 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)

Material

• 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 http://www.math.harvard.edu/theses/senior/lehner/lehner.pdf provides a reasonably readable introduction as well.