This thesis investigates relations over a category C relative to an (E;M)-factorization system of C. In order to establish the 2-categoryRel(C) of relations over C in the rst part we discuss su cient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as de ...

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory of spans. The main theorem states that Van Kampen cocones are precisely those diagrams in a category that induce bicolimit diagrams in its associated bicatego...

Abstract We apply the theory of weighted bicategorical colimits to study problem existence and computation such birepresentations finitary bicategories. The main application our results is complete classification simple transitive a bicategory studied previously by Zimmermann. confirms conjecture he has made.

In SGA 4 [1], one of the principal building blocks of the theory of topoi is Giraud’s theorem, which says that the condition of a 1-category A being the category of sheaves on a site, may be characterized by intrinsic, internal conditions in A. The intrinsic conditions are basically existence of certain limits and colimits, plus a condition about generation by a small set of objects. In this pa...

Let $S$ be a monoid. In this paper, we prove every class of $S$-acts having a flatness property is closed underdirected colimits, it extends some known results. Furthermore thisresult implies that every $S$-act has a flatness cover if and only if it has a flatness precover.