Lax comma categories of ordered sets (Preprint)

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Type: Preprint
National /International: International
Title: Lax comma categories of ordered sets
Publication Date: 2022-12-27
Authors: - Maria Manuel Clementino
- Fernando Lucatelli Nunes
Abstract:

Let \( \mathsf{Ord} \) be the category of (pre)ordered sets. Unlike \( \mathsf{Ord}/X \), whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category \( \mathsf{Ord}//X \). In this paper we show that, when \( X \) is complete, the forgetful functor \( \mathsf{Ord}//X\to\mathsf{Ord} \) is topological. Moreover, \( \mathsf{Ord}//X \) is complete and cartesian closed if and only if \( X \) is. We end by analysing descent in this category. Namely, when \( X \) is complete and cartesian closed, we show that, for a morphism in \( \mathsf{Ord}//X \), being pointwise effective for descent in \( \mathsf{Ord} \) is sufficient, while being effective for descent in \( \mathsf{Ord} \) is necessary, to be effective for descent in \( \mathsf{Ord}//X \).

Institution: DMUC 22-49
Online version: http://www.mat.uc.pt...prints/eng_2022.html
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