nLab
localic topos

Definition

In intrinsic terms, a topos is localic if it is generated under colimits by the subobjects of its terminal object. In equivalent but extrinsic terms, a category is a localic topos if it is equivalent to the category of sheaves on a locale with respect to the topology of jointly epimorphic families (accordingly, every localic topos is a Grothendieck topos). The frame of opens? specifying the locale may indeed be taken as the poset of subobjects of 1. From the perspective of logic, localic toposes are those categories which are equivalent to the category of partial equivalence relations of the tripos given by a complete Heyting algebra (as before, the complete Heyting algebra may be taken as the poset of subobjects of 1 (i.e., internal truth values)).

Properties

Let $\mathrm{LocTopos}$ be the 2-category whose

• objects are localic toposes;

• morphisms are geometric morphisms, i.e. adjunctions in which the left adjoint preserves finite limits, considered as pointing in the direction of their right adjoint; and

• 2-morphisms are mate?-pairs of natural transformations.

Then the 2-category $\mathrm{LocTopos}$ is equivalent to the 2-category $\mathrm{Loc}$ of locales (see C1.4.5 in the Elephant). The 2-category $\mathrm{Loc}$ is actually a (1,2)-category; its 2-cells are the pointwise ordering of frame homomorphisms. Thus this equivalence implies that $\mathrm{LocTopos}$ is also a (1,2)-category, and moreover that it is locally essentially small, in the sense that its hom-categories are essentially small. (The 2-category $\mathrm{Topos}$ of all toposes is not locally essentially small.) Assuming sufficient separation axioms, the hom-posets of $\mathrm{Loc}$, and hence $\mathrm{LocTopos}$, become discrete.

Examples

Every Grothendieck topos that is a category of sheaves on (the category of open subsets of) a topological space is localic.

Many familiar toposes $E$, even when they are not localic, can be covered by a localic slice $E/X$ (“covered” means the unique map $X\to 1$ is an epi). For example, if $G$ is a group, then $E={\mathrm{Set}}^G$ is not itself localic, but it has a localic slice ${\mathrm{Set}}^G/G\simeq \mathrm{Set}$ that covers it. Such toposes have been dubbed etendu (see Lawvere’s 1976 monograph Variable Sets, Etendu, and Variable Structure in Topoi).

A significant result due to Joyal and Tierney is that for any Grothendieck topos $E$, there exists an open surjection $F\to E$ where $F$ is localic. This fact is reproduced in Mac Lane and Moerdijk’s text Sheaves in Geometry and Logic (section IX.9), where the localic cover taken is the Diaconescu cover of $E$.

• Then, using methods of descent theory, Joyal and Tierney deduce that every Grothendieck topos is equivalent to the category $BG$ of continuous discrete representations of a localic groupoid $G$. (Their result is relativized so as to hold internally over any Grothendieck topos $S$ as base.) This should be regarded as a major extrapolation of Grothendieck’s Galois theory (as in SGA 1), where it is shown that the etale topos of a field $k$ is equivalent to the category of continuous discrete representations of the fundamental pro-group $\mathrm{Gal}(\overline{k}/k)$, where $\overline{k}$ denotes the separable closure of $k$. It was a watershed event for the penetration of localic methods in topos theory.

Generalizations

In the context of (∞,1)-topos theory there is a notion of n-localic (∞,1)-topos.

Notice that a locale is itself a (Grothendieck) (0,1)-topos. Hence a localic topos is a 1-topos that behaves essentially like a (0,1)-topos. In the wider context this would be called a 1-localic (1,1)-topos.