On quasi-categories as a foundation for higher algebraic stacks

Abstract

We develop the basic theory of quasi-categories (a.k.a. weak Kan complexes or ([infinity], 1)- categories as in [BV73], [Joy], [Lur06]) from first principles, i.e. without reference to model categories or other ideas from algebraic topology. Starting from the definition of a quasi-category as a simplicial set satisfying the inner horn-filling condition, we define and prove various properties of quasi-categories which are direct generalizations of categorical analogues. In particular, we look at functor quasi-categories, Hom-spaces, isomorphisms, equivalences between quasi-categories, and limits. In doing so, we employ exclusively combinatorial methods, as well as adapting an idea of Makkai's ("very subjective morphisms," what turn out in this case to be simply trivial Kan fibrations) to get a handle on various notions of equivalence. We then begin to discuss a new approach to the theory of left (or right) fibrations, wherein the quasi-category of all left fibrations over a given base S is described simply as the large simplicial set whose n-simplices consist of all left fibrations over S x [delta]n.

(cont.) We conjecture that this large simplicial set is a quasi-category, and moreover that the case S = * gives an equivalent quasi-category to the commonly-held quasi-category of spaces; we offer some steps towards proving this. Finally, assuming the conjecture true, we apply it to give simple descriptions of limits in this quasi-category, as well as a straightforward construction of a Yoneda functor for quasi-categories which we then prove is fully faithful.