Algebraic & Geometric Topology

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

Emily Riehl and Dominic Verity

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 1 (2017), 189-271.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18G55: Homotopical algebra 55U35: Abstract and axiomatic homotopy theory
Secondary: 55U40: Topological categories, foundations of homotopy theory

$\infty$–categories modules profunctors virtual equipment pointwise Kan extension


Riehl, Emily; Verity, Dominic. Kan extensions and the calculus of modules for $\infty$–categories. Algebr. Geom. Topol. 17 (2017), no. 1, 189--271. doi:10.2140/agt.2017.17.189.

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