In [2] we constructed homological localizations of spaces, groups, and 17"modules; here we generalize those constructions to give "factorization systems" and "homotopy factorization systems" for maps in categories. In Section 2 we recall the definition and basic properties of factorization systems, and in Section 3 we give our first existence theorem (3.1)for such systems. It can be viewed as a...

We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups uq(sl2).

We define a notion of flexible functor from a category X to the category of topological spaces (or other similar categories). This is a generalization of the notion of functor, where the condition of compatibility with composition is replaced by the data of homotopies, homotopies between the homotopies, and so on. When X is a site, we define a sheaf condition and show that a flexible functor (t...

namely 90 a M0 a M! a M1 a Yset : set !Mset: We recall from [8] or [9] that a locally small category B is said to be total (abbreviating totally cocomplete) if Y : B !MB has a left adjoint, X: Considerable motivation for the terminology is given in either reference. Examples include categories of algebras, categories of spaces and categories of sheaves on a Grothendieck site. The reader is advi...

There are various doctrines wherein theories are small categories and the models of a theory constitute a full subcategory of the category of functors from the theory to sets. We define the notion of bimodel at the nondoctrinaire level where those specified subcategories are arbitrary; as a first application we look for a classification of the equivalences between categories of models. We obtai...

For a given point set S (in general position), two pointed pseudo-triangulations are compatible if their union is plane. We show that for any set S there exist two maximally disjoint compatible pointed pseudotriangulations, that is, their union is a triangulation of S. In contrast, we show that there are point sets S and pointed pseudo-triangulations T such that there exists no pointed pseudo-t...

Let I be a small indexing category, G : I → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (re...

Let I be a small indexing category, G : I → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (re...

In this thesis we construct a new model of intensional type theory in the category of cubical sets with connections. To facilitate this we introduce the notion of a nice path object category, a simplification of the path object category axioms of [vdBG12] that nonetheless yields the full path object category structure. By defining cubical n-paths and contraction operators upon them we exhibit t...