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June 2017 The construction of $\pi_0$ in Axiomatic Cohesion
Matías Menni
Tbilisi Math. J. 10(3): 183-207 (June 2017). DOI: 10.1515/tmj-2017-0108

Abstract

We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of $\pi_0$ 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 $p : \mathcal{E} \rightarrow \mathcal{S}$, an idempotent monad $\pi_0 : \mathcal{E} \rightarrow \mathcal{E}$ such that, for every $X$ in $\mathcal{E}$, $\pi_{0}X = 1$ if and only if $(p^*\Omega)^! : (p^*\Omega)^1 \rightarrow (p^*\Omega)^X$ is an isomorphism. For instance, if $\mathcal{E}$ is the topological topos (over $\mathcal{S} = Set$), the $\pi_0$-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the $\pi_0$-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 $p : \mathcal{E} \rightarrow \mathcal{S}$, $p$ is pre-cohesive if and only if $p^* : \mathcal{S} \rightarrow \mathcal{E}$ is cartesian closed. In this case, $p_! = p_* \pi_0 : \mathcal{E} \rightarrow \mathcal{S}$ and the category of $\pi_0$-algebras coincides with the subcategory $p^* : \mathcal{S} \rightarrow \mathcal{E}$.

Citation

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Matías Menni. "The construction of $\pi_0$ in Axiomatic Cohesion." Tbilisi Math. J. 10 (3) 183 - 207, June 2017. https://doi.org/10.1515/tmj-2017-0108

Information

Received: 22 August 2017; Revised: 30 October 2017; Published: June 2017
First available in Project Euclid: 20 April 2018

zbMATH: 06816535
MathSciNet: MR3731394
Digital Object Identifier: 10.1515/tmj-2017-0108

Subjects:
Primary: 18B25
Secondary: 18B30

Rights: Copyright © 2017 Tbilisi Centre for Mathematical Sciences

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Vol.10 • No. 3 • June 2017
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