We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.

A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordant-dissonant and inseparable-separable.

. This paper is a first step in the study of symmetric cat-groups as the 2-dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2-categorical sense, in the 2-category of symmetric cat-groups. An application to exact sequences is given. Introduction A cat-group is a monoidal groupoid in which each object is invertible, up to isomorphisms, with respect to the tensor product [7, 10, 16, 24]. Cat-groups are a useful tool for ring theory, group cohomology and algebraic topology (for example, small and strict cat-groups correspond to crossed modules) [2, 5, 6, 13, 15, 21, 22, 25]. Symmetric cat-groups, together with symmetric monoidal functors and monoidal natural transformations, constitute a 2category which ca...

Abstract not found

Dedicated to Aurelio Carboni on the occasion of his sixtieth birthday

For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of so-called weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.

A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3–for–2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory. 1.

Abstract. It is known that factorisation systems in categories can be viewed as unitary pseudo algebras for the monad (–) 2, in Cat. We show in this note that an analogous fact holds for proper (i.e., epimono) factorisation systems and a suitable quotient of the former monad, deriving from a construct introduced by P. Freyd for stable homotopy. Structural similarities of the previous monad with the path endofunctor of topological spaces are considered.