Contents

topos theory

Contents

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 as 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 $S$.

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

Here $LocTopos(S)$ is the 2-category whose

Then the 2-category $LocTopos$ is equivalent to the 2-category $Loc$ of locales (see C1.4.5 in the Elephant).

The 2-category $Loc$ is actually a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms. Thus this equivalence implies that $LocTopos$ 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 $Topos$ of all toposes is not locally essentially small.) Assuming sufficient separation axioms, the hom-posets of $Loc$, and hence $LocTopos$, become discrete.

Examples

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 $E = Set^G$ is not itself localic, but it has a localic slice $Set^G/G \simeq Set$ that covers it. Such a topos is called an étendue (cf. Lawvere’s 1975 monograph Variable Sets Etendu and Variable Structure in Topoi).1

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 called the Diaconescu cover of $E$.

• Then, using methods of descent theory, Joyal and Tierney deduce that every Grothendieck topos is equivalent to the category $B G$ 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 $Gal(\bar{k}/k)$, where $\bar{k}$ 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.

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

1. The ‘etendu’ in the title of Lawvere’s monograph might not be a misspelled noun, but an adjective as part of a back translation of a (hypothetical) French expression ‘ensembles étendus’. See this nForum thread for some discussion and speculation on this point.