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Kan extensions and the
calculus of modules for $\infty$–categories
Emily Riehl and Dominic Verity |
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Algebraic & Geometric Topology 17 (2017) 189–271
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Abstract |
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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. |
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Keywords
$\infty$–categories, modules, profunctors, virtual
equipment, pointwise Kan extension
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Mathematical Subject Classification 2010
Primary: 18G55, 55U35
Secondary: 55U40
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References |
Publication
Received: 25 October 2015
Revised: 15 May 2016
Accepted: 22 May 2016
Published: 26 January 2017
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Authors |
| Emily Riehl | |
| Department of Mathematics Johns Hopkins University 3400 N Charles Street Baltimore, MD 21218 United States |
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| http://www.math.jhu.edu/~eriehl | |
| Dominic Verity | |
| Department of Mathematics Macquarie University Sydney NSW 2109 Australia |
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