The construction of π0 in Axiomatic Cohesion

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We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of π0 as the image of a natural map to a product of discrete spaces. A particular case of this construction produces, out of a local and hyperconnected geometric morphism p : Ɛ → S, an idempotent monad π0 : Ɛ → Ɛ such that, for every X in E, π0X = 1 if and only if (p*Ω)! : (p*Ω)1 → (p*Ω)X is an isomorphism. For instance, if E is the topological topos (over S = Set), the π0-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the π0-algebras in the category of compactly generated Hausdorff spaces are exactly the totally separated ones. Also, in order to relate the construction above with the axioms for Cohesion we prove that, for a local and hyperconnected p : Ɛ → S, p is pre-cohesive if and only if p* : S → Ɛ is cartesian closed. In this case, p! = p*π0 : Ɛ → S and the category of π0-algebras coincides with the subcategory p* : S → E.

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