Homotopy Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it. On the other hand, those conditions are easy to check and provide a wide class of models some of which are listed in the paper.Comment: Corrected version of the published articl

Grothendieck quasitoposes

A full reflective subcategory E of a presheaf category [C*,Set] is the category of sheaves for a topology j on C if and only if the reflection preserves finite limits. Such an E is called a Grothendieck topos. More generally, one can consider two topologies, j contained in k, and the category of sheaves for j which are separated for k. The categories E of this form, for some C, j, and k, are the Grothendieck quasitoposes of the title, previously studied by Borceux and Pedicchio, and include many examples of categories of spaces. They also include the category of concrete sheaves for a concrete site. We show that a full reflective subcategory E of [C*,Set] arises in this way for some j and k if and only if the reflection preserves monomorphisms as well as pullbacks over elements of E.Comment: v2: 24 pages, several revisions based on suggestions of referee, especially the new theorem 5.2; to appear in the Journal of Algebr

Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory

Given a monad T on a suitable enriched category B equipped with a proper factorization system (E,M), we define notions of T-completion, T-closure, and T-density. We show that not only the familiar notions of completion, closure, and density in normed vector spaces, but also the notions of sheafification, closure, and density with respect to a Lawvere-Tierney topology, are instances of the given abstract notions. The process of T-completion is equally the enriched idempotent monad associated to T (which we call the idempotent core of T), and we show that it exists as soon as every morphism in B factors as a T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is satisfied as soon as B has certain pullbacks as well as wide intersections of M-embeddings. Hence the resulting theorem on the existence of the idempotent core of an enriched monad entails Fakir's existence result in the non-enriched case, as well as adjoint functor factorization results of Applegate-Tierney and Day