REGULAR INJECTIVITY AND EXPONENTIABILITY IN THE SLICE CATEGORIES OF ACTIONS OF POMONOIDS ON POSETS
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For a pomonoid S, let us denote Pos-S the category of S-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 enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the cate- gory Pos-S/B under a particular case where B has trivial action. Among other 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 example to show that the converse is not true, in general.
Abstract
For a pomonoid S, let us denote Pos-S the category of S-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 enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the cate- gory Pos-S/B under a particular case where B has trivial action. Among other 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 example to show that the converse is not true, in general.
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