The colimits in the generalized algebraic categories
Jiřı́ Adámek, Vácłav Koubek, Věra Pohlová · Czech digital mathematics library · 1972
The aim of this paper is to discuss the cocompletness of a certain class of categories, the generalized algebraic categories.These categories are a natural generalization of the categories of universal algebras -they were first defined in [1] by Trnkova and Goralcik in connection with Wyler's paper [2].This class of categories contains not only all the categories of universal algebras, but also some categories of topological and convergent spaces and other well-known categories.A generalized algebraic category, denoted by A(F, G), where F and G are set functors (i.e.functors from the category of sets into itself), is the category, the objects of which are pairs (X, co), X a set, co a mapping from FX to GX; the morphisms from (X, co) to (X', co') are all the mappings / : X -> X' such that the diagram, consisting of Ff, co, Gf, co' is commutative.In particular in the covariant case (both F and G covariant) we have FX 0) GX Ff FX' to Gf -> GX' in the contravariant case (both functors contravariant) FX co GX Ff FX' co Gf -> GX' We shall deal only with these A(F, G) which have a common variance of F and G; the case of different variances is the subject of another paper [6].Several papers study the question of the existence of limits and colimits in A(F, G) in connection with the choice of the two set functors ([1], [3] -[6]).In the