Topological categories with both epireflective and coreflective proper subcategories. (English) Zbl 0542.18004

Giuli, Eraldo

Commentat. Math. Univ. Carol. 24, 481-488 (1983).

For each nonempty set A, the author constructs the category \({\mathcal A}\) which has as objects pairs of sets (X,F) where F is simultaneously a subset of the sink of all functions from A to X and a superset of all constant functions from A to X. Morphisms are the usual ones between sinks. Each \({\mathcal A}\) is shown to be a Cartesian closed topological category for which final epi-sinks are hereditary; and is shown to contain a plethora of proper, nontrivial simultaneously epireflective and coreflective subcategories. In the case where A is a two-point set, \({\mathcal A}\) is the category of reflexive relations. The subcategory of symmetric reflexive relations is both reflective and coreflective in it.

Reviewer: G.E.Strecker

MSC:

18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
18B30	Categories of topological spaces and continuous mappings (MSC2010)
18B10	Categories of spans/cospans, relations, or partial maps

Keywords:

Cartesian closed topological category; reflective subcategories; reflexive relations