### Cohomology and Computation (Week 26)

#### Posted by John Baez

This week in our seminar on Cohomology and Computation we continued discussing the bar construction, and drew some pictures of a classic example:

- Week 26 (May 31) - The bar construction, continued. Comonads as comonoids. Given adjoint functors $L: C \to D$ and $R: D \to C$, the bar construction turns an object $d$ in $D$ into a simplicial object $\overline{d}$. Example: the cohomology of groups. Given a group $G$, the adjunction $L: Set \to Grp$, $R: Grp \to Set$ lets us turn any $G$-set $X$ into a simplicial $G$-set $\overline{X}$. This is a "puffed-up" version of $X$ in which all equations $g x = y$ have been replaced by edges, all equations between equations (syzygies) have been replaced by triangles, and so on. When $X$ is a single point, $\overline{X}$ is called $E G$. It’s a contractible space on which $G$ acts freely. The "group cohomology" of $G$ is the cohomology of the space $B G = E G/G$.

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

I’m running behind on putting up these course notes… but maybe that’s good: you have a bit more to chew on even though classes are actually over here at UCR!

I seem to be vacillating a lot between denoting the result of the bar construction with an underline and denoting it with an overline, as here:

$the bar construction turns objects d \in D into simplicial objects \overline{d}: \Delta^{op} \to D$

In this particular environment I’m having more trouble drawing underlines than overlines! But don’t let this confuse you: there’s no difference.

## Re: Cohomology and Computation (Week 26)

You’re saying this bar construction works for any adjunction?

If so, then we get a cohomology for the Giry monad.