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localic topos

Definition

In intrinsic terms, a topos is localic if it is generated under colimits by the subobjects of its terminal object $1$.

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 $1$ (i.e., internal truth values). 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 internal truth values).

Properties

• A Grothendieck topos $E$ is a localic topos if and only if its unique global section geometric morphism to Set is a localic geometric morphism.

  Thus, in general we regard a localic geometric morphism $E\to S$ as exhibiting E as a “localic S-topos”.

• Moreover, just localic topoi can be identified with locales, for any base topos $S$ the 2-category of localic $S$-topoi is equivalent to the 2-category Loc$(S)$ of internal locales in $S$.

  $$\mathrm{LocTopos}(S)\simeq (\mathrm{Topos}/S{)}_{\mathrm{loc}}\simeq \mathrm{Loc}(S).$$LocTopos(S) \simeq (Topos/S)_{loc} \simeq Loc(S) \,.

  Here $\mathrm{LocTopos}(S)$ is the 2-category whose

  • objects are localic toposes over $S$;

  • 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-morphism 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

• Obviously, every Grothendieck topos that is a category of sheaves on (the category of open subsets of) a topological space is localic.

• Every sheaf topos over a posite is localic. (See there for details.)

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.

References

Localic toposes are discussed around proposition 1.4.5 of section C.1.4 of

• Peter Johnstone, Sketches of an Elephant