The points and localisations of the topos of \(M\)-sets (Q6594011)

A \textit{point} of a Grothendieck topos \(\mathcal{E}\) is a geometric morphism p = (p\(_*\), p\(^*\)): \(\mathcal{S}\mathrm{ets}\to \mathcal{E}\) from the topos of sets to \(\mathcal{E}\). A \textit{localisation} \(\mathcal{T}\) of a Grothendieck topos \(\mathcal{E}\) is a full subcategory of \(\mathcal{E}\) such that the inclusion functor \(\iota\colon \mathcal{T}\to \mathcal{E}\) has a left adjoint which preserves finite limits and if \(y\in \mathcal{E}\) is isomorphic to some \(x\in \mathcal{T}\), then \(y\in \mathcal{T}\). The author studies localisations and points of the topos \(\mathcal{S}\mathrm{ets}_M\) (\(_M\mathcal{S}\mathrm{ets}\)) of right (resp. left) \(M\)-sets, where \(M\) is a non-commutative monoid. In a special case of a left zero monoid \(S_+=S\cup\{1\}\) (\(\vert S\vert\geq 2\)), both \(\mathcal{S}\mathrm{ets}_{S_+}\) and \(_{S_+}\mathcal{S}\mathrm{ets}\) have exactly two non-isomorphic points (because \(S_+\) and \(S\) are the only filtered \(S_+\)-sets in the first case and \(S_+\) and singletons in the second case). Localisations of topoi \(\mathcal{S}\mathrm{ets}_{S_+}\) and \(_{S_+}\mathcal{S}\mathrm{ets}\) are also described. It is proved that for any finite monoid \(M\), any filtered \(M\)-set is of the form \(Me\) for an idempotent \(e\in M\), and, therefore, the category \(\mathcal{I}(M)\) of idempotents of \(M\) and the category \textbf{Pts}(\(\mathcal{S}\mathrm{ets}_M\)) of points of \(\mathcal{S}\mathrm{ets}_M\) are contravariantly equivalent. It is also proved that there is a one-to-one correspondence between the localisations of \(\mathcal{S}\mathrm{ets}_M\) and idempotet ideals of \(M\).