Year: 2022
Type: article
Abstract: Let $${\mathbb {A}}$$ A be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category $${\mathbb {A}}$$ A .
Source: Applied Categorical Structures
Author: Fernando Lucatelli Nunes
Institutions: Utrecht University, Centro de Matemática, University of Coimbra
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,780
OpenAlex ID: W2912349790