Classifying strict discrete opfibrations with lax morphisms

Abstract

We study discrete opfibration classifiers in enhanced 2-categories and show how, under suitable hypotheses, such classifiers can be endowed with the structure of a (lax or pseudo-)T-algebra and classify strict discrete opfibrations in 2-categories of (lax or pseudo-)T-algebras and lax morphisms. This leads to a notion of discrete opfibration classifier in the enhanced setting, in which `small' (e.g. strict) discrete opfibrations are classified by `loose' (e.g. lax) maps. We identify conditions on an enhanced 2-monad T and on a discrete opfibration classifier ensuring that this lifting to algebras is possible. These conditions hold in a broad range of examples, including double categories, monoidal and symmetric monoidal categories, orthogonal factorization systems, and, more generally, structures encoded by opfamilial 2-monads. In particular, this recovers and explains the role of Span(Set) as a classifier for strict double discrete opfibrations via lax double functors. We also characterize when representable copresheaves are pseudo rather than merely lax in terms of `cartesianness at the representing object', for an abstract notion of cartesianness we introduce.