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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

Various models of (,1)–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 A to B is an –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 –cosmoi, to limits and colimits of diagrams valued in an –category, as introduced in previous work.

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