Regular injectivity and exponentiability in the slice categories of actions of pomonoids on posets

TL;DR: In this paper, the authors consider the slice category Pos-S/B for an S-poset B, and study some categorical ingredients, including the necessary condition for a map (an ob-ject) here to be convex and present an example to show that the converse is not true, in general.

Abstract: . For a pomonoid S, let us denote Pos-S the category ofS-posets and S-poset maps. In this paper, we consider the slice cate-gory Pos-S/B for an S-poset B,and study some categorical ingredients.We first show that there is no non-trivial injective object in Pos-S/B.Then we investigate injective objects with respect to the class of regu-lar monomorphisms in this category and show that Pos-S/B has enoughregular injective objects. We also prove that regular injective objects areretracts of exponentiable objects in this category. One of the main aimsof the paper is to draw attention to characterizing injectivity in the cate-gory Pos-S/Bunder a particular case where Bhas trivial action. Amongother things, we also prove that the necessary condition for a map (an ob-ject) here to be regular injective is being convex and present an exampleto show that the converse is not true, in general. 1. Introduction and preliminariesA slice category is a construction in category theory which provides anotherway of looking at morphisms: instead of simply relating objects of a categoryto one another, morphisms become objects in their own right. This notion wasintroduced in 1963 by F. W. Lawvere, although the technique did not becomegenerally known until many years later.One of the very useful notions in many categories as well as in homologicalalgebra is the injectivity of objects with respect to a class M of morphisms. Forexample, in the category Pos of partially ordered sets and monotone mappings,injective objects, with respectto the classofall order-embeddings,coincidewiththe complete lattices (see [4]). Injective objects in slice categories have beeninvestigated in detail (see [1, 16]), especially in relation with weak factorizationsystems, a concept used in homotopy theory, in particular for model categories.