Tbilisi Mathematical Journal

The construction of $\pi_0$ in Axiomatic Cohesion

Matías Menni

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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 $p^* : \mathcal{S} \rightarrow \mathcal{E}$.

Article information

Source
Tbilisi Math. J., Volume 10, Issue 3 (2017), 183-207.

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

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1524232080

Digital Object Identifier
doi:10.1515/tmj-2017-0108

Mathematical Reviews number (MathSciNet)
MR3731394

Zentralblatt MATH identifier
06816535

Subjects
Primary: 18B25: Topoi [See also 03G30]
Secondary: 18B30: Categories of topological spaces and continuous mappings [See also 54-XX]

Keywords
axiomatic cohesion topology

Citation

Menni, Matías. The construction of $\pi_0$ in Axiomatic Cohesion. Tbilisi Math. J. 10 (2017), no. 3, 183--207. doi:10.1515/tmj-2017-0108. https://projecteuclid.org/euclid.tbilisi/1524232080


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