### Cohomology and Computation (Week 27)

#### Posted by John Baez

In the last of this year’s classes on Cohomology and Computation, we sketched a few of the simplest consequences of the bar construction:

- Week 27 (June 7) - Cohomology of algebraic gadgets. The bar construction "puffs up" any algebraic gadget, replacing equations by edges, syzygies by triangles and so on, with the result being a simplicial object with one contractible component for each element of the original gadget. Examples: Ext and Tor, group cohomology and homology, Lie algebra cohomology and homology. How Ext and Tor arise from the adjoint functors between the category of abelian groups and the category of modules of a ring. Free resolutions. Group cohomology as a special case of Ext. Group cohomology as the cohomology of the the classifying space $B G = E G/G$.

Last week’s notes are here.

We never reached what I was really aiming for, namely an automatic weakening procedure that turns $\lambda$-calculi into “simplicial $\lambda$-calculi”, in which proofs become 1-simplices, and so on. But, this should be easy for anyone who followed the whole year’s course. So, I leave it as an exercise for the reader.

I just saw Derek Wise graduate — in fact, I just “hooded” him. I thank him for taking wonderful notes on these seminars for the last 4 years. He’s going on to a postdoc at U. C. Davis.

I’m now taking a week-long vacation at a secret location far from the internet.

Have a fun summer! Next year’s seminar will be on groupoidification.

## Re: Cohomology and Computation (Week 27)

Congratulations to Derek!!