The construction of $\pi_0$ in Axiomatic Cohesion

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