Let be 2, (2,1), (1,2), or 1.
An -category is an -pretopos if and only if it is coherent and every (finitary) -polycongruence? is a kernel.
Since no nontrivial (0,1)-categories are extensive, the definition as phrased above is not reasonable for . 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).
An -pretopos has 2-coproducts? and 2-quotients? of -congruences, which are an important class of colimits. However, it can fail to admit all finite colimits, for essentially the same reason as when : 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 does in fact have all finite colimits, but the 2-pretopos of finite categories (that is, finitely many objects and finitely many morphisms) does not have coinserters, coinverters, or coequifiers. (The category of finitely presented categories does have finite colimits, but fails to have finite limits.)
However, it is natural to conjecture that just as in the case , once an -pretopos is also countably-coherent, it does become finitely cocomplete. See colimits in an n-pretopos?.
This is due to