### Cohomology and Computation (Week 20)

#### Posted by John Baez

This week in our seminar on Cohomology and Computation, we began to see what’s so great about simplices:

- Week 20 (Apr. 12) - Cohomology and the category of simplices. Simplices as special categories: finite totally ordered sets, which are isomorphic to "ordinals". The algebraist’s category of simplices, $\Delta_{alg}$. Face and degeneracy maps. The functor from $\Delta_{alg}$ to Top sending the ordinal $n$ to the standard $(n-1)$-simplex. Simplicial sets. Preview of the cohomology of spaces.

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

A simplex is a special sort of space. A point is a 0-simplex, an interval is a 1-simplex, a triangle is a 2-simplex, a tetrahedron is a 3-simplex, and so on. Here’s a movie of a 4-simplex, rotating in the 4th dimension:

But beneath the level of topology, there’s a deeper level where a simplex is just a finite ordinal. And beneath that, a simplex is a special sort of category! We’ll learn more about the amazing algebraic properties of simplices later on.