Kan extensions and the calculus of modules for $\mathrm{\infty }$-categories

Various models of $(\mathrm{\infty },1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\mathrm{\infty }$-cosmos. In a generic $\mathrm{\infty }$-cosmos, whose objects we call $\mathrm{\infty }$-categories, we introduce modules (also called profunctors or correspondences) between $\mathrm{\infty }$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\mathrm{\infty }$-category equipped with a left action of $A$ and a right action of $B$, 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 $\mathrm{\infty }$-cosmoi, to limits and colimits of diagrams valued in an $\mathrm{\infty }$-category, as introduced in previous work.