Enrico M. Vitale

Regular and exact completions

The regular and exact completions of categories with weak limits are proved to exist and to be determined by an appropriate universal property. Several examples are discussed, and in particular the class of examples given by categories monadic over a power of Set: any such a category is in fact the exact completion of the full subcategory of free algebras. Applications to Grothendieck toposes and geometric morphisms, and to epireflective hulls are also discussed

A Picard-Brauer exact sequence of categorical groups

A categorical group is a monoidal groupoid in which each object has a tensorial inverse. Two main examples are the Picard categorical group of a monoidal category and the Brauer categorical group of a braided monoidal category with stable coequalizers. After discussing the notions of kernel, cokemel and exact sequence for categorical groups, we show that, given a suitable monoidal functor between two symmetric monoidal categories with stable coequalizers, it is possible to build up a five-term Picard-Brauer exact sequence of categorical groups. The usual Units-Picard and Picard-Brauer exact sequences of abelian groups follow from this exact sequence of categorical groups. We also discuss the direct sum decomposition of the Brauer-Long group

Beck's theorem for pseudo-monads

In this work we establish a 2-categorical analogue of Becks theorem characterizing monadic functors. We show that a 2-functor (a pseudo-functor) U is monadic iff it is a right pseudo-adjoint, it reflects adjoint equivalences and it creates U-absolute pseudo-coequalizers of codescent objects

Localizations of algebraic categories - II

Using the exact completion of a weakly left exact category, we specialize previous results on monadic categories over LET to obtain a characterization of localizations of finitary algebraic categories. As a corollary, we have the Popescu-Gabriel representation theorem for Grothendieck categories

When do completion processes give rise to extensive categories

We consider various (free) completion processes: the exact completion and the regular completion of a category with weak finite limits, the pre-regular completion of a category with finite products and weak finite limits, the exact completion of a regular category, the regular reflection of a pre-regular category, and the filtered-colimit completion of a small category. In each case we give necessary and sufficient conditions for the completion to be extensive; or, in the case of the pre-regular completion, for the completion to satisfy a weakened notion of extensivity which we call pre-extensivity

Proper factorization systems in 2-categories

Starting from known examples of factorization systems in 2-categories, we discuss possible definitions of proper factorization system in a 2-category. We focus our attention on the construction of the free proper factorization system on a given 2-category

On the notion of bimodel for functorial semantics

We discuss the notion of bimodel in order to obtain a classification of the equivalences between categories of models in the sense of functorial semantics.

On the categorical structure of H-2

The categorical structure of H-2 is shown to be a particular instance of the cokernel of a morphism between symmetric categorical groups

Regularity of the category of Kelley spaces

We show that the cartesian closed category of compactly generated Hausdorff spaces is regular, but is neither exact, nor locally cartesian closed. In fact we find a coequalizer of an equivalence relation which is not stable under pullback.

On Fibrations Between Internal Groupoids and Their Normalizations

We characterize fibrations and