Normalizers and Split Extensions

Bourn, Dominique; Gray, James Richard Andrew

doi:10.1007/s10485-014-9382-7

Cited by 2 publications

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“…Proof. Recall that: (i) a unital category is algebraically cartesian closed if and only if it admits centralizers (in the sense above), Proposition 1.2 of [4]; (ii) a pointed exact protomodular category is action accessible if and only if for each normal monomorphism n : S → C the normalizer of n, n : S → C × C exists, Theorem 3.1 of [10], (see also [5]). The claim now follows from the previous theorem applied to the pre-cover relations in Examples 1.4 and 1.5.…”

Section: The Main Resultsmentioning

confidence: 99%

A note on images of cover relations

Gray¹

2020

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For a category C, a small category I, and a pre-cover relation ⊏ on C we prove, under certain completeness assumptions on C, that a morphism g : B → C in the functor category C I admits an image with respect to the pre-cover relation on C I induced by ⊏ as soon as each component of g admits an image with respect to ⊏. We then apply this to show that if a pointed category C is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii) admits normalizers, then the same is true of each functor category C I with I finite. In addition, our results give explicit constructions of images in functor categories using limits and images in the underlying category. In particular, they can be used to give explicit constructions of both centralizers and normalizers in functor categories using limits and centralizers or normalizers (respectively) in the underlying category.

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Section: The Main Resultsmentioning

confidence: 99%