Algebraic & Geometric Topology

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

Emily Riehl and Dominic Verity

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841313

Digital Object Identifier
doi:10.2140/agt.2017.17.189

Mathematical Reviews number (MathSciNet)
MR3604377

Zentralblatt MATH identifier
1362.18020

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

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

Citation

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. https://projecteuclid.org/euclid.agt/1510841313


Export citation

References

  • C Barwick, C Schommer-Pries, On the unicity of the homotopy theory of higher categories (2011)
  • G,S,H Cruttwell, M,A Shulman, A unified framework for generalized multicategories, Theory Appl. Categ. 24 (2010) No. 21, 580–655
  • R Haugseng, Bimodules and natural transformations for enriched $\infty$–categories, Homology Homotopy Appl. 18 (2016) 71–98
  • A Heller, Homotopy theories, Mem. Amer. Math. Soc. 383, Amer. Math. Soc., Providence, RI (1988)
  • A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207–222
  • T Leinster, fc–multicategories (1999) Notes for a talk at the 70th Peripatetic Seminar on Sheaves and Logic
  • T Leinster, Generalized enrichment for categories and multicategories (1999)
  • T Leinster, Generalized enrichment of categories, J. Pure Appl. Algebra 168 (2002) 391–406
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press, NJ (2009)
  • J Lurie, Higher algebra (2014) Available at \setbox0\makeatletter\@url http://www.math.harvard.edu/~lurie/papers/HA.pdf {\unhbox0
  • E Riehl, D Verity, The $2$–category theory of quasi-categories, Adv. Math. 280 (2015) 549–642
  • E Riehl, D Verity, Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions, Homology Homotopy Appl. 17 (2015) 1–33
  • E Riehl, D Verity, Homotopy coherent adjunctions and the formal theory of monads, Adv. Math. 286 (2016) 802–888
  • E Riehl, D Verity, Fibrations and Yoneda's lemma in an $\infty$–cosmos, J. Pure Appl. Algebra 221 (2017) 499–750
  • E Riehl, D Verity, Yoneda structures and the calculus of modules for quasi-categories In preparation
  • E Riehl, D Verity, On model independence and general change of base for $\infty$–category theories In preparation
  • M Shulman, Enriched indexed categories, Theory Appl. Categ. 28 (2013) 616–696
  • R Street, Elementary cosmoi, I, from “Category Seminar” (G,M Kelly, editor), Springer (1974) 134–180. Lecture Notes in Math., Vol. 420
  • D Verity, Enriched categories, internal categories and change of base, Repr. Theory Appl. Categ. (2011) 1–266
  • M Weber, Yoneda structures from $2$–toposes, Appl. Categ. Structures 15 (2007) 259–323
  • R,J Wood, Abstract proarrows, I, Cahiers Topologie Géom. Différentielle 23 (1982) 279–290
  • R,J Wood, Proarrows, II, Cahiers Topologie Géom. Différentielle Catég. 26 (1985) 135–168