Objects with dense diagonals

Walter Tholen
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Walter Tholen: York University, Department of Mathematics and Statistics

A chapter in Categorical Topology, 1996, pp 213-220 from Springer

Abstract: Abstract The purpose of this note is to show that, in a finitely complete category X with a proper ( ɛ, M)-factorization system for morphisms and a closure operator c w.r.t. the class $$M\, \subseteq \,mono\,(X)$$ in the sense of [DG], the full subcategory Δ( c) of those objects X ∈ X for which the diagonal δx: $${\delta _{\text{X}}}{\text{ : X }} \to {\text{ X}}$$ is c-dense, satisfies all the stability properties that one expects a category of “connected” objects to have. In fact, subject to suitable conditions on the given data, we show that ∇(c) is closed in X under ɛ -images, c-dense extensions, direct products, and under chained sinks. The first three closure properties appear essentially in [DT], Section 7.8, but not the crucial fourth property, which exhibits ∇(c) as a component subcategory in the sense of [Ti]; see also [T] and [C].

Keywords: 18A32; 18B30; 54B30; closure operator; dense diagonal; chained sink; Lawvere-Tierney topology. (search for similar items in EconPapers)
Date: 1996
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DOI: 10.1007/978-94-009-0263-3_19

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