### Cohomology and Computation (Week 23)

#### Posted by John Baez

In this week’s seminar on Cohomology and Computation, we finally came within sight of the promised land. We began sketching how to get simplicial sets from quite arbitrary algebraic gadgets, using the mind-bogglingly beautiful ‘bar construction’:

- Week 23 (May 10) - Simplicial sets from algebraic gadgets. Algebraic gadgets and adjoint functors. The unit and counit of an adjunction: the unit ‘includes the generators’, while the counit ‘evaluates formal expressions’. The canonical presentation of an algebraic gadget. Simplicial objects from adjunctions: the bar construction. 1-simplices as proofs.

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

To understand adjunctions, it’s really good to keep concrete examples firmly in mind, like the adjunction between

$L: Set \to AbGp$

(the functor sending each set to the free abelian group on that set) and

$R: AbGp \to Set$

(the functor sending each abelian group to its underlying set).

One of the students was quite impressed by the example of $LR\mathbb{Z}$, the group of *formal integer linear combinations of integers*.

But, this was just a warmup for understanding abelian groups like $LRLR\mathbb{Z}$, which consists of *formal integer linear combinations of formal integer linear combinations of integers*, and $LRLRLR\mathbb{Z}$, which consists of *formal integer linear combinations of formal integer linear combinations of formal integer linear combinations of integers!*

Applying the bar construction to $\mathbb{Z}$, we take all the above groups and lump them into a simplicial abelian group, which has:

- formal integer linear combinations of integers as vertices,
- formal integer linear combinations of formal integer linear combinations of integers as edges,
- formal integer linear combinations of formal integer linear combinations of formal integer linear combinations of integers as triangles,

and so on! The $(-1)$-dimensional simplices are just the integers themselves.

Amazing what the human mind will dream up — and eventually understand — and eventually find a necessary tool for further thought!

## Re: Cohomology and Computation (Week 22)

Is it possible to discuss simplicial sets, homotopies, topologies, … related to formal theories? I can see one direct way: if our theory defines some category of (algebraic) models, and for each model we can build corresponding simplicial set, then I think it’s possible to do that for whole category or some interesting “small” fragment of this category… But maybe someone could suggest more direct way for calculating topological parameters of formal theories (using n-categories for example).

If we can (theoretically) do such things then we can (also theoretically) analyze all the humans knowledge with this qualitative approach, calculating topological invariants for each theory and comparing it?