Various models of
–categories,
including quasi-categories, complete Segal spaces, Segal categories, and
naturally marked simplicial sets can be considered as the objects of an
–cosmos. In a generic
–cosmos, whose objects
we call
–categories,
we introduce
modules (also called
profunctors or
correspondences) between
–categories,
incarnated as spans of suitably defined fibrations with groupoidal fibers. As the name suggests,
a module from
to
is an
–category equipped
with a left action of
and a right action of
,
in a suitable sense. Applying the fibrational form of the Yoneda lemma, we
develop a general calculus of modules, proving that they naturally assemble
into a multicategory-like structure called a
virtual equipment, which is
known to be a robust setting in which to develop formal category theory.
Using the calculus of modules, it is straightforward to define and study
pointwise Kan extensions, which we relate, in the case of cartesian closed
–cosmoi,
to limits and colimits of diagrams valued in an
–category,
as introduced in previous work.
Keywords
$\infty$–categories, modules, profunctors, virtual
equipment, pointwise Kan extension
