Fibrational approach to Grandis exactness for 2‐categories

Abstract In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures substantial information about the exactness structure of a category. Indeed, as it was shown by the second author and Weighill, categories equipped with a proper factorization system such that the opfibration of subobjects relative to the factorization system is isomorphic to the fibration of relative quotients are precisely the Grandis exact categories. In this paper we characterize those (1,1)‐proper factorization systems on a 2‐category in the sense of Dupont and Vitale, for which the weak 2‐opfibration of relative 2‐subobjects is biequivalent to the weak 2‐fibration of relative 2‐quotients. This results in a new notion of two‐dimensional exactness, which we then compare with similar notions in the context of categories enriched in pointed groupoids arising in the work of Dupont and Nakaoka.