Schreiber
Journal Club -- (infinity,1)-Categories

Some links for a journal club on (∞,1)-category theory. For more detailed seminar notes see Seminar on (∞,1)-Categories and ∞-Stacks.


Topic

The familiar collection of

  • topological spaces X,Y,X, Y, \cdots (resp. CW-Complexes)

  • continuous maps XYX \to Y between spaces;

  • homotopies X Y \array{& \nearrow\searrow\\X &\Downarrow& Y\\ & \searrow\nearrow} between continuous maps;

  • homotopies between homotopies;

  • and so on

naturally organizes itself into and is the archetypical example of a structure called an (infinity,1)-category: an infinity category in which all cells of degree k2k \geq 2 are invertible. A general (infinity,1)-category may be thought of as a generalized setup in which to do homotopy theory.

The idea of higher categories and the corresponding higher topos theory is an old one, going back to the ideas of Grothendieck who started to pursue it. For many years, though, it kept being pursued without finding a generally useful form.

This is changing now. For several years André Joyal amplified the fact that the weak Kan complexes introduced by Boardman and Vogt, which he started calling quasi-categories, are a model for (infinity,1)-categories for which a good comprehensive closed theory can be obtained, that completely parallels and generalizes ordinary category theory. Based on these ideas by Joyal and work by people like Carlos Simpson, more recently Jacob Lurie presented a comprehensive textbook on the subject

He then showed that using the (,1)(\infty,1)-categorical foundations developed in this book, an impressive collection of useful concepts and results are obtained in a wealth of areas, such as the homological algebra of stable (infinity,1)-categories and topological quantum field theory.

This considerable progress is currently the source of much interest in (infinity,1)-categories.

Idea of this Journal Club

The idea of this Journal Club would be to go section-wise through chapter 1 of Higher Topos Theory

The general patterns will be that concepts in quasi-categories are relatively easily understood as relatively straightforward generalizations of the corresponding familiar 1-categorical concepts, but that concrete computations in concrete models may tend to be a bit more demanding.

There are mainly four different concrete realizations of the notion (,1)(\infty,1)-category, which can all be related to each other and each have advantages and disadvantages for certain purposes:

The standard reference for the interrelation between these four models is

  • Julia Bergner, A survey of (,1)(\infty,1)-categories (arXiv).

Roughly

In Higher Topos Theory Lurie develops the two models quasi-category and simplicial category in parallel, passing back and forth between the two pictures.

Further Literature

Quasi-categories have originally been defined in

  • J.M. Boardman and R.M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.

They occured as weak Kan complexes in

  • R. Vogt, Homotopy limits and colimits, Math. Z., 134, (1973), 11–52.

Vogt’s main theorem involved a category of homotopy coherent diagrams defined on a topologically enriched category and showed it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.

Cordier in

  • J.-M. Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahiers de Top. Géom. Diff., 23, (1982), 93 –112,

defined a homotopy coherent nerve of any simplicially enriched category, which generalised the nerve of an ordinary category. In

  • J.-M. Cordier and T. Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65–90,

it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.

Their importance as a basis for category theory has been emphasized in the work by Joyal

  • A. Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra, 175 (2002), 207-222.

  • A. Joyal, Simplicial categories vs quasi-categories, in preparation.

Jacob Lurie developed the theory of quasi-categories in

in order to discuss infinity-stacks.

Meetings

a) Introduction

b) (,1)(\infty,1)-categories

c) model categories

d) models for \infty-stack (,1)(\infty,1)-toposes

Revised on May 29, 2012 22:04:00 by Andrew Stacey (129.241.15.200)