july 2023 Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations
Fernando Lucatelli Nunes, Rui Prezado, Lurdes Sousa
Bull. Belg. Math. Soc. Simon Stevin 30(1): 130-139 (july 2023). DOI: 10.36045/j.bbms.221021

Abstract

For any suitable monoidal category $\mathcal{V}$, we find that $\mathcal{V}$-fully faithful lax epimorphisms in $\mathcal{V} \dash \mathcal{V}$ are precisely those $\mathcal{V}$-functors $F: \mathcal{A} \to \mathcal{B}$ whose induced $\mathcal{V}$-functors ${\mathfrak C} F: {\mathfrak C} \mathcal{A} \to {\mathfrak C} \mathcal{B}$ between the Cauchy completions are equivalences. For the case $\mathcal{V}= {\rm Set}$, this is equivalent to requiring that the induced functor $F^*:{\rm CAT}({\mathcal A},{\rm Cat}) \to {\rm CAT}({\mathcal B}, {\rm Cat})$ is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations ${\mathcal F}$ to the study of the codescent factorization, we find that the observations above on fully faithful lax epimorphisms provide us with a characterization of (effective) ${\mathcal F}$-descent morphisms in the category of small categories ${\rm Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of \textit{discrete} opfibrations,previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.

Citation

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Fernando Lucatelli Nunes. Rui Prezado. Lurdes Sousa. "Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations." Bull. Belg. Math. Soc. Simon Stevin 30 (1) 130 - 139, july 2023. https://doi.org/10.36045/j.bbms.221021

Information

Published: july 2023
First available in Project Euclid: 6 August 2023

Digital Object Identifier: 10.36045/j.bbms.221021

Subjects:
Primary: 18A20 , 18A22 , 18D20 , 18F20 , 18N10

Keywords: Cauchy completions , effective descent morphism , Enriched categories , fully faithful morphisms , lax epimorphisms , split fibrations

Rights: Copyright © 2023 The Belgian Mathematical Society

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Vol.30 • No. 1 • july 2023
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