A Note on Topological Hom-functors

This note establishes internal criteria on a category C and a separator 2 in C which characterize the condition that the 2-induced covariant hom-functor /i2: C -> Set is (epi, mono-source)-topological. Introduction. Hoffmann [4] showed how topological functors (of Herrlich [2]) may be recovered from factorizations of sources and sinks in the domain category. This process was extended to (F, M)-topological functors in our paper [1], where we claimed, without elaborating details, that the results could be used to internally characterize the condition that a covariant hom-functor «2: C -> Set is (epi, monosource)-topological. In this note, we establish such a characterization which, in fact, depends on the supporting results of that paper. Our references are only intended to be immediately relevant rather than exhaustive, and our terminology is generally that of [1, 2, 3]. We note the following concepts before stating our main result: Definitions. Let C be a category, 2 an object in C and e: X -» y a morphism in C. Further, let «2 : = C(2, — ), the covariant hom-functor induced by 2. (1) C is said to admit ~2-disjoint coproducts iff for every coproduct sink («,: A, -* II A,), and morphism y: 2 -> 11 A,, there exist /" G / and x: 2 -» A, such that v = w, ° x, i.e. iff («2M/)/ is an epi-sink. (2) The morphism e: X -* y is said to be 1,-coextendible (cf. [3]) iff for every v: 2 -> y, there exists x: 2 -* A such that y = e ° x, i.e. iff «2e is onto. We have the following characterization: Theorem. Let C be a category with an object 2 such that there exists at most a set of nonisomorphic objects C with «2C = 0. Then the following conditions are equivalent: (1) «2: C -» Set is an (epi, mono-source)-topologicalfunctor. (2) C and 2 satisfy the following four conditions: (a) C is. a co-complete category; (b) 2 is a separator; (c)for any set S, every morphism 2 ->s 2. is a copower injection; (d) C admits ^.-disjoint coproducts and every regular epi in C is ~2.-coextendible. Proof. (1) implies (2): (2)(a) follows from the fact that Set is co-complete and «2 lifts colimits; Received by the editors May 6, 1981. 1980 Mathematics Subject Classification. Primary 18A20, 18A22, 18A30, 18A32, 18A40, 18B99, 18D30.