Let be a Grothendieck 2-topos. We say that is -truncated if it has a small eso-generator consisting of -truncated objects. It is easy to see that if a coproduct of -truncated objects is -truncated (as is the case for all ), then this is equivalent to saying that has enough -truncated objects (i.e. every object admits an eso from an -truncated one). In particular:
By the 2-Giraud theorem, small eso-generating sets of objects correspond to small 2-sites of definition for . Thus, if we define an -site to be a 2-site which is an -category (where as usual), we have:
A Grothendieck 2-topos is -truncated iff it is equivalent to the 2-category of 2-sheaves on some -site.
Note that a 1-site is the same as the usual notion of site, and a -site is sometimes called a posite. In particular, any frame is a (0,1)-site with its canonical coverage (the covering families are given by unions).
Particular cases include:
is 1-truncated iff it is equivalent to the 2-category of 2-sheaves (stacks) on an ordinary small (1-)site, and therefore to the 2-category of stacks for the canonical coverage on some Grothendieck 1-topos.
is (0,1)-truncated iff it is equivalent to the 2-category of stacks on a posite, and therefore also to the 2-category of stacks on some locale. We call such a localic.
If is (-1)-truncated, then it is in particular localic, and its terminal object is a (strong) generator. It is not hard to see that this is equivalent to saying that the corresponding locale is a sublocale of the terminal locale . Thus, just as (-1)-categories are subsets of , (-1)-toposes are sublocales of . If has classical logic, this implies that either or ; and hence that either or . However, constructively there may be many other sublocales of .
It would be nice if the only (-2)-truncated Grothendieck 2-topos were . However, I don’t see a way to make this happen except by fiat.
Now, if is an -site, it is also reasonable to consider -sheaves on , by which we mean 2-sheaves taking values in -categories. Thus, a 1-sheaf on a 1-site is precisely the usual notion of sheaf on a site. And a (0,1)-sheaf on a (0,1)-site is easily seen to be a lower set that is an “ideal” for the coverage.
We define a Grothendieck -topos to be an -category equivalent to the -category of -sheaves on an -site. The case gives classical Grothendieck toposes; the case gives locales. Note the distinction between a Grothendieck -topos and an -truncated Grothendieck 2-topos. The relationship is that
This relationship is completely analogous to the classical relationship between locales and localic toposes. In fact, if denotes the -category of Grothendieck -toposes (that is, -categories of -sheaves on an -site), we have inclusions
where the inclusion from to is given by taking the -category of -sheaves for the canonical coverage. (See 2-geometric morphism for the morphisms in these categories.)