  We present in this paper a reformulation of the usual set-theoretical
semantics of the description logic $\mathcal{ALC}$ with general TBoxes by using
categorical language. In this setting, $\mathcal{ALC}$ concepts are represented
as objects, concept subsumptions as arrows, and memberships as logical
quantifiers over objects and arrows of categories. Such a category-based
semantics provides a more modular representation of the semantics of
$\mathcal{ALC}$. This feature allows us to define a sublogic of $\mathcal{ALC}$
by dropping the interaction between existential and universal restrictions,
which would be responsible for an exponential complexity in space. Such a
sublogic is undefinable in the usual set-theoretical semantics, We show that
this sublogic is {\sc{PSPACE}} by proposing a deterministic algorithm for
checking concept satisfiability which runs in polynomial space.
