Some links for a journal club on (∞,1)-category theory. For more detailed seminar notes see Seminar on (∞,1)-Categories and ∞-Stacks.
Time and place:
The familiar collection of
topological spaces $X, Y, \cdots$ (resp. CW-Complexes)
continuous maps $X \to Y$ between spaces;
homotopies $\array{& \nearrow\searrow\\X &\Downarrow& Y\\ & \searrow\nearrow}$ 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 $k \geq 2$ 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 $(\infty,1)$-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.
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 $(\infty,1)$-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
Roughly
quasi-categories are the conceptually cleanest model; at least they are adequate in many situations.
simplicial categories are useful for many concrete constructions and a large repository of tools and machinery exists for them; in particular all homotopical categories in homotopy coherent category theory, such as model categories, Waldhausen categories, Brown categories present simplicial categories and hence $(\infty,1)$-categories; they are important in the construction of the Hammock-Localization by Dwyer and Kan.
the construction principle of complete Segal spaces lends itself best to iteration, which then yields models for (infinity,n)-categories for higher $n$: $\infty$-categories in which only the $k$-morphisms for $k \gt n$ are required to be invertible.
Segal categories are a weak form of simplicial categories. They have been frequently used by Carlos Simpson and Betrand Toën in their work.
In Higher Topos Theory Lurie develops the two models quasi-category and simplicial category in parallel, passing back and forth between the two pictures.
Quasi-categories have originally been defined in
They occured as weak Kan complexes in
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
defined a homotopy coherent nerve of any simplicially enriched category, which generalised the nerve of an ordinary category. In
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.