## Tbilisi Mathematical Journal

### The construction of $\pi_0$ in Axiomatic Cohesion

Matías Menni

#### 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
Revised: 30 October 2017
First available in Project Euclid: 20 April 2018

https://projecteuclid.org/euclid.tbilisi/1524232080

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

Mathematical Reviews number (MathSciNet)
MR3731394

Zentralblatt MATH identifier
06816535

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

#### References

• M. Barr and R. Paré. Molecular toposes. J. Pure Appl. Algebra, 17(2):127–152, 1980.
• F. Cagliari, S. Mantovani, and E.M. Vitale. Regularity of the category of Kelley spaces. Appl. Categ. Struct., 3(4):357–361, 1995.
• A. Carboni and S. Mantovani. An elementary characterization of categories of separated objects. Journal of pure and applied algebra, 89:63–92, 1993.
• S. P. Franklin. Spaces in which sequences suffice. Fundamenta Mathematicae, 57:107–115, 1965.
• J. M. E. Hyland. Filter spaces and continuous functionals. Annals of mathematical logic, 16:101–143, 1979.
• P. T. Johnstone. On a topological topos. Proceedings of the London mathematical society, 38:237–271, 1979.
• P. T. Johnstone. Stone spaces. Cambridge University Press, 1982.
• P. T. Johnstone. Sketches of an elephant: a topos theory compendium, volume 43-44 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York, 2002.
• P. T. Johnstone. Remarks on punctual local connectedness. Theory Appl. Categ., 25:51–63, 2011.
• F. W. Lawvere. Cohesive toposes and Cantor's “lauter Einsen”. Philos. Math. (3), 2(1):5–15, 1994. Categories in the foundations of mathematics and language.
• F. W. Lawvere. Categories of spaces may not be generalized spaces as exemplified by directed graphs. Repr. Theory Appl. Categ., 9:1–7, 2005. Reprinted from Rev. Colombiana Mat. 20 (1986), no. 3-4, 179–185.
• F. W. Lawvere. Axiomatic cohesion. Theory Appl. Categ., 19:41–49, 2007.
• F. W. Lawvere and M. Menni. Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness. Theory Appl. Categ., 30:909–932, 2015.
• S. Mac Lane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer Verlag, 1971.
• S. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Universitext. Springer Verlag, 1992.
• F. Marmolejo and M. Menni. On the relation between continuous and combinatorial. J. Homotopy Relat. Struct., 12(2):379–412, 2017.
• M. Menni. Continuous cohesion over sets. Theory Appl. Categ., 29:542–568, 2014.
• M. Menni. Sufficient cohesion over atomic toposes. Cah. Topol. Géom. Différ. Catég., 55(2):113–149, 2014.