Tbilisi Mathematical Journal

The construction of $\pi_0$ in Axiomatic Cohesion

Matías Menni

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

axiomatic cohesion topology


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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