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

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

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

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