#### Volume 17, issue 1 (2017)

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Kan extensions and the calculus of modules for $\infty$–categories

### Emily Riehl and Dominic Verity

Algebraic & Geometric Topology 17 (2017) 189–271
##### Abstract

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

##### Keywords
$\infty$–categories, modules, profunctors, virtual equipment, pointwise Kan extension
##### Mathematical Subject Classification 2010
Primary: 18G55, 55U35
Secondary: 55U40
##### Publication
Received: 25 October 2015
Revised: 15 May 2016
Accepted: 22 May 2016
Published: 26 January 2017
##### Authors
 Emily Riehl Department of Mathematics Johns Hopkins University 3400 N Charles Street Baltimore, MD 21218 United States http://www.math.jhu.edu/~eriehl Dominic Verity Department of Mathematics Macquarie University Sydney NSW 2109 Australia