Abstract:
By a result of Gerstenhaber and Schack, the simplicial cohomology ring
$\operatorname{H}^{\bullet}({\mathcal C},k)$
of a poset
${\mathcal C}$
is isomorphic to
the Hochschild cohomology ring
$\operatorname{HH}^{\bullet}(k{\mathcal C})$
of the
category algebra
$k{\mathcal C}$,
where the poset is viewed as a category and
$k$
is a
field.
Extending results of Mishchenko, under certain assumptions on a category
${\mathcal
C}$,
we construct a category
${\mathcal D}$
and a graded
$k$-linear isomorphism
$\operatorname{HH}^{\bullet}(k{\mathcal C})\cong \operatorname{H}^{\bullet}({\mathcal
D},k)$.
Interpreting the degree one cohomology, we also show how the
$k$-space of
derivations on
$k{\mathcal C}$
graded by some semigroup corresponds to the
$k$-space of
characters on
${\mathcal D}$.
