Normalizers in the category of topological groups

Maria Manuel Clementino; James Richard Andrew Gray

Home/Archive/arXiv

arXiv · 2611.03253

Normalizers in the category of topological groups

Maria Manuel Clementino·James Richard Andrew Gray

View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that the category of morphisms of topological groups admits split extension classifiers and hence the category of topological groups admits normalizers. We give explicit descriptions of both normalizers and of split extension classifiers of morphisms of topological groups in terms of split extension classifiers of topological groups.

Explore related subjects

math.CT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Maria Manuel Clementino, James Richard Andrew Gray. 2026-11-02. Normalizers in the category of topological groups. https://arxiv.org/abs/2611.03253

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Contextful Arrows via Wreath Products

We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows---comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span construction. Contextads are defined in terms of Lack--Street wreaths, suitably categorified for pseudomonads in a tricategory of spans in a 2-category with display maps. The associated wreath product provides the Ctx construction, and by its universal property we conclude trifunctoriality. This abstract approach lets us work up to structure, and thus swiftly prove that, under very mild assumptions, a contextad equipped colaxly with a 2-algebraic structure produces a similarly structured double category of contextful arrows. We also explore the role contextads might play qua dependently graded comonads in organizing contextful computation in functional programming. We show that many side-effects monads can be dually captured by dependently graded comonads, and gesture towards a general result on the `transposability' of parametric right adjoint monads to dependently graded comonads.

math.CT↗

arXiv2026

On the representability of actions of unital algebras

Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the category of all unital algebras is not even action accessible. We then consider this problem in the context of operadic, action accessible, unit-closed varieties. Using the construction of the external weak actor, we prove that for any algebra $X$ in such a variety $\mathcal{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathcal{V}_1$ of unital algebras in $\mathcal{V}$ is action representable, and the actor of a unital algebra $X$ is $X$ itself. The action accessibility of $\mathcal{V}$ is not necessary for this conclusion: in characteristic zero, unital Jordan algebras form an action representable category, although the variety of all Jordan algebras is not action accessible. Finally, we prove action representability for the categories of unital Poisson algebras and unital commutative Poisson algebras.

math.CT↗

arXiv2026

Higher zeroes in pointed infinity-categories

We construct an endofunctor on (small) simplicial sets which freely adjoins a zero object to a (small) infinity-category, as the co-cone construction freely adjoins a terminal object. We then prove that this endofunctor induces a left adjoint to the inclusion of the infinity-category of (small) pointed infinity-categories and morphisms that preserve zero objects into the infinity-category of (small) infinity-categories. This adjunction, in turn, leads to the definition of a coherent choice of nullhomotopies on a pointed infinity-category. A coherent choice of nullhomotopies is a non-canonical, coherent choice of "higher zeroes" on a pointed infinity-category, mirroring the obvious definition of higher zeroes on the differential graded nerve of a differential graded category of chain complexes over an additive category. We then show that coherent choices of nullhomotopies give an alternate characterization of fiber sequences in pointed infinity-categories, generalizing the 1-categorical definition of kernels in terms of morphisms that compose to zero.

math.CT↗