Weak Factorisation Systems in the Effective Topos
Daniil Frumin
Abstract:
In this thesis we present a new model of Martin-Löf type theory with identity types in the effective topos. Using the homotopical approach to type theory, the model is induced from a Quillen model category structure on a full subcategory of the effective topos. To aid the construction we introduce a general method of obtaining model category structures on a full subcategory of an elementary topos, by starting from an interval object I and restricting our attention to fibrant objects, utilizing the notion of fibrancy similar to the one that Cisinksi employed for constructing a model category structure on a Grothendieck topos with an interval object.
We apply this general method to the effective topos Eff . Following Van Oosten we take the interval object to be I = ∇(2), and derive a model structure on the subcategory Eff_f of fibrant objects. This Quillen model category structure gives rise to a model of type theory in which the identity type for a type X is represented by X^I. It follows that the resulting model supports functional extensionality.