Jason Schuchardt - Miscellaneous Math

Miscellaneous Math

This page collects miscellaneous calculations, ideas that came up during reading, and mathematical exposition that I decided to write up.

Understanding the Moore complex and simplicial groups

Jason Schuchardt

If GG is a simplicial group, then there is a complex of (not necessarily) groups, NGNG, called the Moore complex with NGq=i=1qkerdi.NG_q = \bigcap_{i=1}^q \ker d_i. The key result we’d like to understand is the following. f:GHf:G\to H is injective (resp. surjective) if and only if Nf:NGNHNf:NG\to NH is injective (resp. surjective).

Which triple functors to Set\Set give an enrichment?

Jason Schuchardt

If V\calV is a monoidal category, C\underline{\calC} is a V\calV-category, then we get a functor Vop×Cop×CSet,\calV^\op\times \calC^\op\times \calC\to \Set, where C\calC is the underlying ordinary category of C\underline{\calC} defined by V(,C(,))\calV(-,\underline{\calC}(-,-)). We’d like to understand which such functors describe enriched categories.

Extending 2-variable adjunctions to diagram categories

Jason Schuchardt

We’d like to understand and generalize the result that the category of simplicial C\calC-objects is enriched, tensored and cotensored over sSet.\sSet.