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 (n1)-truncated objects. It is easy to see that if a coproduct of (n1)-truncated objects is (n1)-truncated (as is the case for all n1), then this is equivalent to saying that K has enough (n1)-truncated objects (i.e. every object admits an eso from an (n1)-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 n2 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 X0 or X1; and hence that either K1 or KCat. 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 (n1)-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 (n1)-truncated objects.

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

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