Michael Shulman
2-pretopos

Definition

Let nn be 2, (2,1), (1,2), or 1. That is, 1n21\le n\le 2 and nn is directed; see n-prefix.

Definition

An nn-pretopos is an nn-exact nn-category which is also extensive. An infinitary nn-pretopos is an nn-pretopos which is infinitary-extensive.

As remarked here, regularity plus extensivity implies coherency. Thus an nn-pretopos is, in particular, a coherent nn-category. Conversely, we have:

Theorem

An nn-category is an nn-pretopos if and only if it is coherent and every (finitary) nn-polycongruence is a kernel.

Examples

  • CatCat is a 2-pretopos. Likewise, GpdGpd is a (2,1)-pretopos and PosPos is a (1,2)-pretopos.

  • A 1-category is a 1-pretopos precisely when it is a pretopos in the usual sense. Note that, as remarked for exactness, a 1-category is unlikely to be an nn-pretopos for any n>1n\gt 1.

  • Since no nontrivial (0,1)-categories are extensive, the definition as phrased above is not reasonable for n=(0,1)n=(0,1). However, for some purposes (such as the n-Giraud theorem), it is convenient to define an (infinitary) (0,1)-pretopos to simply be an (infinitary) coherent (0,1)-category (exactness being automatic).

Colimits

An nn-pretopos has coproducts and quotients of nn-congruences, which are an important class of colimits. However, it can fail to admit all finite colimits, for essentially the same reason as when n=1n=1: namely, some ostensibly “finite” colimits secretly involve infinitary processes. In a 1-category, this manifests in the construction of arbitrary coequalizers and pushouts, where we must first generate an equivalence relation by an infinitary process and then take its quotient.

For 2-categories it is even easier to find counterexamples: the 1-pretopos FinSetFinSet does in fact have all finite colimits, but the 2-pretopos FinCatFinCat of finite categories (that is, finitely many objects and finitely many morphisms) does not have coinserters, coinverters, or coequifiers. (The category FPCatFPCat of finitely presented categories does have finite colimits, but fails to have finite limits.)

However, I conjecture that just as in the case n=1n=1, once an nn-pretopos is also countably-coherent, it does become finitely cocomplete. See colimits in an n-pretopos.

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