\title{Grothendieck Categories of Enriched Functors}

\author{Hassan Al Hwaeer}
\address{Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom}

\author{Grigory Garkusha}
\address{Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom}

\keywords{Grothendieck categories, enriched categories, model categories}

\subjclass[2010]{18E15, 18E30, 18G55}

\begin{abstract}
It is shown that the category of enriched functors $[\cc C,\cc V]$
is Grothendieck whenever $\cc V$ is a closed symmetric monoidal
Grothendieck category and $\cc C$ is a category enriched over $\cc
V$. Localizations in $[\cc C,\cc V]$ associated to collections of
objects of $\cc C$ are studied. Also, the category of chain
complexes of generalized modules $\Ch(\cc C_R)$ is shown to be
identified with the Grothendieck category of enriched functors
$[\modd R,\Ch(\Mod R)]$ over a commutative ring $R$, where the
category of finitely presented $R$-modules $\modd R$ is enriched
over the closed symmetric monoidal Grothendieck category $\Ch(\Mod
R)$ as complexes concentrated in zeroth degree. As an application,
it is proved that $\Ch(\cc C_R)$ is a closed symmetric monoidal
Grothendieck model category with explicit formulas for tensor
product and internal Hom-objects. Furthermore, the class of unital
algebraic almost stable homotopy categories generalizing unital
algebraic stable homotopy categories of
Hovey--Palmieri--Strickland~\cite{HPS} is introduced. It is shown
that the derived category of generalized modules $\cc D(\cc C_R)$
over commutative rings is a unital algebraic almost stable homotopy
category which is not an algebraic stable homotopy category.
\end{abstract}