Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Functor categories and cohomology of small categories

Let $C$ be a small category, $k$ a field and $Vect_k$ the category of $k$-vector spaces. We are interested in the functor category of covariant functors $Fct(C, Vect_k)$. By a theorem of B. Mitchell, there exists a fully faithful functor from $Fct(C, Vect_k)$ to $kC$-mod, where $kC$ is the category algebra of $C$ which we will define. Moreover, when $C$ has finitely many objects, the above functor induces an equivalence between the two abelian categories. We consider the tensor structure on $kC$-mod, and show how one can define the ordinary cohomology ring of $kC$, in a way similar to what one can do for a Hopf algebra. If time permits, we will continue to study the Hochschild cohomology ring of $kC$ and then establish a split surjection from the Hochschild cohomology ring to the ordinary cohomology ring.