Abstract
In my three-part paper "Generalized sketches as a framework for completeness theorems" (JPAA 1997), I construct, for each of a number of categorical doctrines, call it D, a presheaf category C such that D is the full subcategory of C with objects that are injective relative to a small, usually finite, number of arrows, the "sketch-axioms", in C ; the set of the sketch-axioms I denote by A . For example, if D is the category of small finite-limit categories (with arrows the functors preserving finite limits in the non-strict sense), than C is the category (with suitable arrows!) of finite-limit sketches. In each of the examples of D , one has two weak factorization systems (the factoring diagonal is not required to be unique), one of them giving rise, using the above set A of "sketch-axioms" to the objects of D as the Kan complexes arise as the fibrant objects, from the horn-extensions in the Quillen model structure on simplicial sets. I am interested in the question for which of my examples of sketch-categories C the two factorization systems determine a Quillen model structure; in some simple cases, I already know that this is case. In the sketch-categories, the strict anodyne maps play a distinguished role. These are the ones that, in the classical case of simplicial sets, are obtained from the Gabriel-Zisman definition of anodyne map by omitting reference to retracts. In the sketch-category C, the strict anodyne maps are the transfinite composites of pushouts of the sketch-axioms, the arrows in the set A . (In the classical case also, the strict anodynes are the transfinite composites of the pushouts of the horn-extensions.) In the talk, I will start with discussing the classical case of the category of simplicial sets with a special emphasis on the strict anodyne maps, and a variant of the latter related to Andre Joyal's model structure on simplicial sets where the fibrant object are the quasi-categories.