Fibrational approach to Grandis exactness for 2-categories
Elena Caviglia, Zurab Janelidze, Luca Mesiti
TL;DR
The paper extends Grandis’ fibrational approach to exactness into the 2-categorical setting by introducing 2-dimensional notions of ideals, kernels, and cokernels and establishing a fibrational characterization via a $(1,1)$-proper factorization system. It proves a biequivalence between the weak 2-fibration of 2-quotients and the weak 2-opfibration of 2-subobjects relative to the factorization system, mediated by two normal pseudofunctors, thereby defining Grandis 2-exact and Puppe 2-exact 2-categories. A 2-dimensional theory of closed (and weakly closed) 2-ideals is developed, with three-piece factorizations and kernels/cokernels behaving coherently, generalizing classical 1-dimensional Grandis exactness. The work then compares the resulting framework to Dupont and Nakaoka’s pointed Gpd*-enriched notions, introducing weaker (weakly closed) variants that still yield biequivalences and capture many known examples, including abelian 2-categories and locally discrete cases. Overall, the paper provides a unified 2-dimensional homological algebra via fibrational and factorization approaches and clarifies how existing pointed-2-categorical theories fit into this broader context.
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 T.~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 M.~Dupont and E.~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 2-dimensional exactness, which we then compare with similar notions in the context of categories enriched in pointed groupoids arising in the work of M.~Dupont and H.~Nakaoka.