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The construction of π_{0} in Axiomatic Cohesion

## Abstract

We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of π_{0} 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 π_{0} : Ɛ → Ɛ such that, for every X in E, π_{0}X = 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 π_{0}-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the π_{0}-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_{*}π_{0} : Ɛ → S and the category of π_{0}-algebras coincides with the subcategory p* : S → E.

###### Introduction. A Personal Tribute to Peter Freyd and Bill Lawvere

References [1] P. J. Freyd, Functor Theory, Dissertation, Princeton University, 1960. [2] P. J. Freyd, Abelian Categories, Harper and Row, New York, Evanston, and London, 1964. [3] P. J. Freyd, Stable Homotopy, in S. Eilenberg, D. K. Harrison, S. MacLane (Edts.), Proceedings of the Conference on Categorical Algebra. La Jolla 1965,121-172, Springer-Verag Berlin . Heidelberg. New York, 1966. [4] P. J. Freyd, Algebra Valued Functors In General and Tensor Products in Particular, Colloquium