  In this paper we propose a categorical theory of intensionality. We first
revisit the notion of intensionality, and discuss we its relevance to logic and
computer science. It turns out that 1-category theory is not the most
appropriate vehicle for studying the interplay of extension and intension. We
are thus led to consider the P-categories of \v{C}ubri\'{c}, Dybjer and Scott,
which are categories only up to a partial equivalence relation (PER). In this
setting, we introduce a new P-categorical construct, that of exposures.
Exposures are very nearly functors, except that they do not preserve the PERs
of the P-category. Inspired by the categorical semantics of modal logic, we
begin to develop their theory. Our leading examples demonstrate that an
exposure is an abstraction of well-behaved intensional devices, such as G\"odel
numberings. The outcome is a unifying framework in which classic results of
Kleene, G\"odel, Tarski and Rice find concise, clear formulations, and where
each logical device or assumption involved in their proofs can be expressed in
the same algebraic manner.
