# Internal object actions in homological categories

@article{Hartl2010InternalOA, title={Internal object actions in homological categories}, author={Manfred Hartl and Bruno Loiseau}, journal={arXiv: Category Theory}, year={2010} }

Let $G$ and $A$ be objects of a finitely cocomplete homological category $\mathbb C$. We define a notion of an (internal) action of $G$ of $A$ which is functorially equivalent with a point in $\mathbb C$ over $G$, i.e. a split extension in $\mathbb C$ with kernel $A$ and cokernel $G$. This notion and its study are based on a preliminary investigation of cross-effects of functors in a general categorical context. These also allow us to define higher categorical commutators. We show that any…

## 8 Citations

The ternary commutator obstruction for internal crossed modules

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- 2013

Abstract In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the…

Higher central extensions via commutators

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- 2012

We prove that all semi-abelian categories with the the Smith is Huq prop- erty satisfy the Commutator Condition (CC): higher central extensions may be charac- terised in terms of binary (Huq or…

Semidirect Products and Split Short Five Lemma in Normal Categories

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- 2014

This paper studies a generalization of the notion of categorical semidirect product to a non-protomodular context of categories where internal actions are induced by points, like in any pointed variety.

Further Remarks on the “Smith is Huq” Condition

- Mathematics, Computer ScienceAppl. Categorical Struct.
- 2015

The Smith is Huq condition (SH) is compared with three commutator conditions in semi-abelian categories to find the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight.

Normalities and commutators

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Abstract We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and…

A CHARACTERIZATION OF FINITE COCOMPLETE HOMOLOGICAL AND OF S EMI-ABELIAN CATEGORIES

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- 2011

Les categories semi-abeliennes et homologiques finiment co-completes sont definies en quatre (respectivement trois) axiomes simples exprimes en termes de notions categoriques de base introduites dans…

Hopf algebras in non-associative Lie theory

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We review the developments in the Lie theory for non-associative products from 2000 to date and describe the current understanding of the subject in view of the recent works, many of which use…

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- 2009

Ministerio de Educacion y Ciencia under grant
number MTM2006-15338-C02-02 (includes European FEDER support), by project Ingenio Mathematica
(i-MATH) under grant number CSD2006-00032 (Consolider…

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