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2017 Kan extensions and the calculus of modules for $\infty$–categories
Emily Riehl, Dominic Verity
Algebr. Geom. Topol. 17(1): 189-271 (2017). DOI: 10.2140/agt.2017.17.189


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.


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Emily Riehl. Dominic Verity. "Kan extensions and the calculus of modules for $\infty$–categories." Algebr. Geom. Topol. 17 (1) 189 - 271, 2017.


Received: 25 October 2015; Revised: 15 May 2016; Accepted: 22 May 2016; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1362.18020
MathSciNet: MR3604377
Digital Object Identifier: 10.2140/agt.2017.17.189

Primary: 18G55, 55U35
Secondary: 55U40

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.17 • No. 1 • 2017
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