The n-Category Café

Cohomology and Computation (Week 22)

Posted by John Baez

This week in our seminar on Cohomology and Computation, we explained how homology ‘counts holes’:

• Week 22 (May 3) - Cohomology and chain complexes. The functor from simplicial sets to simplicial abelian groups. The functor from simplicial abelian groups to chain complexes. The homology of a chain complex as a general method of ‘counting holes’. Some examples: the hollow triangle has $H_1 = \mathbb{Z}$ because it has a ‘1-dimensional hole’. The twice filled triangle (a triangulated 2-sphere) has $H_1 = \{0\}$ but $H_2 = \mathbb{Z}$ because it has a ‘2-dimensional hole’.

Last week’s notes are here; next week’s notes are here.

Next time we’ll start describing how to ‘count holes’ in any algebraic gadget described using a monad!