More on action representability and normalizers
Abstract
We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category ℂ are equivalent: (a) The category of morphisms of ℂ admits generic split extensions; (b) Each functor category of ℂ with finite domain category admits generic split extensions; (c) ℂ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in ℂ to ℂ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular ℂ admits generic split extensions if and only if for each morphism f:X→ Z the functor sending each object B to the isomorphism class of split extensions in ℂ2 of (B,B,1B) with kernel (X,Z,f), is representable.
Related
- Normalizers in the category of topological groups
- On Intersections of System Normalizers
- Solvable Supplements to Normalizers of Cyclic 2-Subgroups
- V-Representability Beyond Singular Times in Time-Dependent Density Functional Theory: The Hubbard Dimer Case
- On tractability, complexity, and mixed-integer convex programming representability of distributionally favorable optimization
BibTeX
@misc{gray2026more,
title = {{More on action representability and normalizers}},
author = {James Richard Andrew Gray},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35122},
eprint = {2609.35122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35122},
}
RIS
TY - UNPB
TI - More on action representability and normalizers
AU - James Richard Andrew Gray
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35122
UR - https://arxiv.org/abs/2609.35122
AB - We show that under mild assumptions a functor is a prefibration if and only if it has terminal objects in its fibers and admits precartesian liftings of monomorphisms. We use this to show that the following conditions on a pointed category ℂ are equivalent: (a) The category of morphisms of ℂ admits generic split extensions; (b) Each functor category of ℂ with finite domain category admits generic split extensions; (c) ℂ admits normalizers and generic split extensions; (d) The kernel functor from the category split extensions in ℂ to ℂ is a prefibration. In addition we show that the category of morphisms of a pointed protomodular ℂ admits generic split extensions if and only if for each morphism f:X→ Z the functor sending each object B to the isomorphism class of split extensions in ℂ^2 of (B,B,1_B) with kernel (X,Z,f), is representable.
ER -