Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 11-16 2005, Macquarie University, Sydney
Mon, 11 July: 9:00 - 10:00
The theory of quasi-categories (I)
Joyal, André (Université du Québec à Montréal)
We shall describe a few salient points of the theory of quasi-categories, stressing the similarities and the differences with category theory. For example, the theories of limits and Kan extensions are formally identical. The quasi-category HOT is the primary example of an $\infty$-topos.
We introduce the notion of an absolutely exact quasi-category, in which the equivalence relations are general groupoids. The notion may capture one of the main difference between HOT and the category of sets. The theorem of Kan on the equivalence between the homotopy category of pointed connected spaces and the category of group objects in TOP has the following generalisation: in any exact quasi-category the full quasi-category of pointed connected objects is equivalent to the quasi-category of group objects.
Absolutely exact quasi-categories abound. The quasi-category of models of any algebraic theory is absolutely exact. Here an algebraic theory is defined to be a quasi-category with finite products. Models of algebraic theories can be taken iteratively. A group object in the quasi-category of group objects is a braided group object or a 2-fold loop space, etc.
There is also a notion of category object in any left exact quasi-category. A category object is said to be reduced if its groupoid of isomorphisms is trivial (this is related to the notion of complete Segal space introduced by Charles Rezk). Every category object in HOT can be reduced (more generally, every category object of an absolutely exact quasi-category). The quasi-category of reduced categories in HOT is equivalent to QCAT, the quasi-category of small quasi-categories. More generally, there is a notion of reduced $n$-category object for every $n$. The quasi-category of reduced $n$-categories in HOT is equivalent to the quasi-category of quasi-$n$-categories properly defined. $\Theta$-sets can be used to model quasi-$n$-categories.