Michael Shulman truncated 2-topos

Definition

Let $K$ be a Grothendieck 2-topos. We say that $K$ is $n$-truncated if it has a small eso-generator consisting of $(n-1)$-truncated objects. It is easy to see that if a coproduct of $(n-1)$-truncated objects is $(n-1)$-truncated (as is the case for all $n\ge 1$), then this is equivalent to saying that $K$ has enough $(n-1)$-truncated objects (i.e. every object admits an eso from an $(n-1)$-truncated one). In particular:

• $K$ is always 2-truncated.
• $K$ is (2,1)-truncated if it has enough groupoids.
• $K$ is 1-truncated if it has enough discretes.
• $K$ is (0,1)-truncated if the subterminal objects are eso-generating.
• $K$ is (-1)-truncated if the terminal object is an eso-generator.

$n$-sites and $n$-sheaves

By the 2-Giraud theorem, small eso-generating sets of objects correspond to small 2-sites of definition for $K$. Thus, if we define an $n$-site to be a 2-site which is an $n$-category (where $n\le 2$ as usual), we have:

Theorem

A Grothendieck 2-topos is $n$-truncated iff it is equivalent to the 2-category of 2-sheaves on some $n$-site.

Note that a 1-site is the same as the usual notion of site, and a $(0,1)$-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:

• $K$ 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.

• $K$ 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 $K$ localic.

• If $K$ 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 $X$ is a sublocale of the terminal locale $1$. Thus, just as (-1)-categories are subsets of $1$, (-1)-toposes are sublocales of $1$. If $Cat$ has classical logic, this implies that either $X\cong 0$ or $X\cong 1$; and hence that either $K\simeq 1$ or $K\simeq Cat$. However, constructively there may be many other sublocales of $1$.

• It would be nice if the only (-2)-truncated Grothendieck 2-topos were $Cat$. However, I don’t see a way to make this happen except by fiat.

Grothendieck $n$-toposes

Now, if $C$ is an $n$-site, it is also reasonable to consider $n$-sheaves on $C$, by which we mean 2-sheaves taking values in $(n-1)$-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 $n$-topos to be an $n$-category equivalent to the $n$-category of $n$-sheaves on an $n$-site. The case $n=1$ gives classical Grothendieck toposes; the case $n=(0,1)$ gives locales. Note the distinction between a Grothendieck $n$-topos and an $n$-truncated Grothendieck 2-topos. The relationship is that

1. The 2-category of 2-sheaves for the canonical coverage on a Grothendieck $n$-topos is an $n$-truncated Grothendieck 2-topos, and
2. Any $n$-truncated Grothendieck 2-topos arises in this way from the Grothendieck $n$-topos which is its full subcategory of $(n-1)$-truncated objects.

This relationship is completely analogous to the classical relationship between locales and localic toposes. In fact, if $Gr n Top$ denotes the $(n+1)$-category of Grothendieck $n$-toposes (that is, $n$-categories of $n$-sheaves on an $n$-site), we have inclusions

$$\array{ &&&&&& Gr(1,2)Top\\ &&&&& \nearrow && \searrow\\ Gr (-1) Top &\to& Gr(0,1)Top &\to & Gr1Top &&&& Gr2Top\\ &&&&& \searrow && \nearrow\\ &&&&&& Gr(2,1)Top }$$

where the inclusion from $Gr n Top$ to $Gr (n+1) Top$ is given by taking the $(n+1)$-category of $(n+1)$-sheaves for the canonical coverage. (See 2-geometric morphism for the morphisms in these categories.)