Michael Shulman
exact 2-category

A regular 1-category is exact when every congruence has a kernel. The definition for 2-categories is analogous.


Let 1<n2-1\lt n\le 2 be directed (see n-prefix). A 2-category is nn-exact if it is regular and every n-congruence is a kernel.

Recall that a 1-category is regular as a 1-category iff it is regular as a homwise-discrete 2-category. However, a regular 1-category is exact as a 1-category precisely when it is 1-exact as a 2-category. Since general 2-congruences in a 1-category are internal categories, they will obviously not all be kernels. This is why we add a prefix to “exact:” the best notion for 2-categories, which we call “2-exact,” is not a conservative extension of the established meaning of “exact” for 1-categories.

Likewise, it is unreasonable to expect a (2,1)-category to be any more than (2,1)-exact, or a (1,2)-category to be any more than (1,2)-exact.


  • Cat is 2-exact. Likewise, Gpd is (2,1)-exact, Pos is (1,2)-exact, and of course Set is 1-exact.

  • Every regular (0,1)-category (that is, every meet-semilattice) is (0,1)-exact, and in fact even 2-exact, since there are no nontrivial congruences of any sort in a poset.

  • If KK is 2-exact, then by the classification of congruences, gpd(K)gpd(K) is (2,1)-exact, pos(K)pos(K) is (1,2)-exact, disc(K)disc(K) is 1-exact, and Sub(1)Sub(1) is (0,1)-exact.

  • If KK is nn-exact, then so is K coK^{co}, by the remarks about opposite 2-congruences. For 1-categories and (2,1)-categories, of course, this is contentless, but for 2-categories and (1,2)-categories it is contentful.

Last revised on June 12, 2012 at 11:10:00. See the history of this page for a list of all contributions to it.