Algebraically Closed and Existentially Closed Substructures in Categorical Context

We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosický, Adámek and Borceux ([RAB, 02]), characterizing λ-injectivity classes in locally λ-presentable categories, is given. Introduction Algebraically closed embeddings are used in module theory and category theory (where they are called pure morphisms), as well as in model theory. The model theoretic definition allows seeing this concept as one of several related types of morphisms (like elementary embeddings and existentially closed embeddings) defined in terms of preservation of certain families of formulas. Adapting this line of thought, S. Fakir ([Fa, 75]) proposed a categorical version of the concepts of algebraically closed and existentially closed morphisms in the context of locally presentable categories. In this paper, we first revisit Fakir’s definitions, extending and simplifying them using two ideas: the first one is to use λ-presentable morphisms instead of λ-presentable (and λ-generated) objects, and the second one involves what we will call locally presentable factorization systems. The former idea was already exploited for the algebraically closed case in [H1, 98]. In a category C, a morphism f :A −→ B will be called λ-presentable if it is λ-presentable as an object of the comma category (A ↓ C). In [H1, 98], we characterized the classes closed under algebraically closed subobjects as the ones which are injective with respect to some class of cones formed by λ-presentable morphisms. As a corollary, we obtained a solution to a problem of L. Fuchs in the context of abelian groups ([Fu, 70]). In this paper, we use this characterization to obtain a different proof of the main result of [RAB, 02], characterizing the λ-injectivity classes as the ones closed under λ-algebraically closed subobjects and λ-reduced products (Theorem 3.5 below). As for the latter idea, we define a λ-presentable factorization system as a proper factorization system (E,M) in which the morphisms in E between λ-presentable objects are sufficient to determine the morphisms in M. In a locally λ-presentable category C, (Strong Received by the editors 2003-06-28 and, in revised form, 2004-04-16. Transmitted by Jiri Rosicky. Published on 2004-04-22. 2000 Mathematics Subject Classification: 18A20, 18C35, 03C60, 03C40.