Michael Shulman
truncated 2-topos

Definition

Let KK be a Grothendieck 2-topos. We say that KK is nn-truncated if it has a small eso-generator consisting of (n1)(n-1)-truncated objects. It is easy to see that if a coproduct of (n1)(n-1)-truncated objects is (n1)(n-1)-truncated (as is the case for all n1n\ge 1), then this is equivalent to saying that KK has enough (n1)(n-1)-truncated objects (i.e. every object admits an eso from an (n1)(n-1)-truncated one). In particular:

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

nn-sites and nn-sheaves

By the 2-Giraud theorem, small eso-generating sets of objects correspond to small 2-sites of definition for KK. Thus, if we define an nn-site to be a 2-site which is an nn-category (where n2n\le 2 as usual), we have:

Theorem

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

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

  • KK 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.

  • KK 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 KK localic.

  • If KK 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 XX is a sublocale of the terminal locale 11. Thus, just as (-1)-categories are subsets of 11, (-1)-toposes are sublocales of 11. If CatCat has classical logic, this implies that either X0X\cong 0 or X1X\cong 1; and hence that either K1K\simeq 1 or KCatK\simeq Cat. However, constructively there may be many other sublocales of 11.

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

Grothendieck nn-toposes

Now, if CC is an nn-site, it is also reasonable to consider nn-sheaves on CC, by which we mean 2-sheaves taking values in (n1)(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 nn-topos to be an nn-category equivalent to the nn-category of nn-sheaves on an nn-site. The case n=1n=1 gives classical Grothendieck toposes; the case n=(0,1)n=(0,1) gives locales. Note the distinction between a Grothendieck nn-topos and an nn-truncated Grothendieck 2-topos. The relationship is that

  1. The 2-category of 2-sheaves for the canonical coverage on a Grothendieck nn-topos is an nn-truncated Grothendieck 2-topos, and
  2. Any nn-truncated Grothendieck 2-topos arises in this way from the Grothendieck nn-topos which is its full subcategory of (n1)(n-1)-truncated objects.

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

Gr(1,2)Top Gr(1)Top Gr(0,1)Top Gr1Top Gr2Top Gr(2,1)Top\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 GrnTopGr n Top to Gr(n+1)TopGr (n+1) Top is given by taking the (n+1)(n+1)-category of (n+1)(n+1)-sheaves for the canonical coverage. (See 2-geometric morphism for the morphisms in these categories.)

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)