Normalization equivalence, kernel equivalence and affine categories

Bourn, Dominique

doi:10.1007/bfb0084212

Cited by 140 publications

(210 citation statements)

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“…The whole diagram and the square on the left are pullbacks, so that by the pullback cancelation property of protomodular categories (see [5]) also the square on the right is a pullback, thus showing that ϕ is the kernel of γ in Pt A (C).…”

Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 95%

“…Following [5], we say that g is the cokernel of f , and we write g = coker(f ), when (2) is a pushout. Let us notice that this definition of cokernel is not dual to that of kernel given above, unless the category is pointed.…”

Section: Action Of Quotientsmentioning

confidence: 99%

“…When both conditions above are satisfied, i.e. when (2) is at the same time both a pullback and a pushout, we call the pair (f, g) short exact sequence (see [5]), and we describe it by the diagram:…”

Section: Action Of Quotientsmentioning

confidence: 99%

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A note on strong protomodularity, actions and quotients

Metere

2017

Journal of Pure and Applied Algebra

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In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions. IntroductionThe present work originates from the investigation of the categorical properties related to two well-known features of group actions. Actions on quotientsSuppose we are given a pair (ξ, g):where ξ is a left-action of groups, and g is a surjective homomorphism. We discuss the following problem: under what conditions does the action ξ induces an action on the quotient Z? Indeed, it is not difficult to see that ξ is well-defined on the cosets of Y mod X = Ker(g), precisely when it is well-defined on the 0-coset X, i.e. when it restricts to X. We shall state this property as follows:(KC) An action passes to the quotient if, and only if, it restricts to the kernel. Action of quotientsSuppose now that we are given a group action ξ as before, and a surjective group homomorphism q : A → Q. A natural question arises: when does the given A-action induce a Q-action? In this case, the restriction of the action ξ to the kernel K of q always exists, and the condition under which the action of the quotient is well defined, amounts to the fact that the kernel of q acts trivially. 1These issues can be addressed in any category where a notion of internal object action is available, e.g. in any semi-abelian category (see [10]). Indeed, we will show that the property (KC) characterizes strongly protomodular categories among semi-abelian categories, and that, in such contexts, actions of quotients behave substantially in the same way as in the case of groups.On the other hand these issues can be dealt with also in more general contexts. Indeed, when an object A acts on object X, just like in the case of group, one can consider the split epimorphism X ⋊ A → A given by the semidirect product projection together with its canonical section. Vice-versa, any split epimorphism with codomain A gives rise to the conjugation A-action on the kernel of the split epimorphism.This allows to formulate our issues in terms of split epimorphisms, or points, even in contexts where the machinery of internal actions is not at all available. This line of investigation will lead us to the study of some new classifying aspects of the fibration of points. In particular, with Proposition 3.3, we will give a characterization of protomodular categories among quasi-pointed regular ones as those with kernel functors that reflect short exact sequences. Then, we will show that the problem of extending actions along quotients translates (in term of points) in a property closely connected with strong protomodularity, i.e. the fact that kernel functors reflect kernels In fact, th...

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Section: Exactness Properties Of Kernel Functorsmentioning

confidence: 95%

Section: Action Of Quotientsmentioning

confidence: 99%

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A note on strong protomodularity, actions and quotients

Metere

2017

Journal of Pure and Applied Algebra

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“…We recall that a pointed and regular category is Bourn protomodular [5] if and only if the (Regular) Short Five Lemma holds: this means that for any commutative 2010 Mathematics Subject Classification. 17A32, 18B99, 18E99, 18G50, 20J05.…”

Section: Semi-abelian Categoriesmentioning

confidence: 99%

Universal Central Extensions in Semi-Abelian Categories

Casas

Linden

2013

Appl Categor Struct

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“…It was recently proved by the second author [18] that, in any normal category (i.e., a pointed regular category where every regular epimorphism is a normal epimorphism), the upper and the lower 3 × 3 lemmas are equivalent, and they hold precisely when the normal category is subtractive [17]. The middle 3 × 3 lemma turns out to be stronger than the other two, and it is equivalent to protomodularity [3]. The "denormalized 3 × 3 lemma", studied by D. Bourn in [5], replaces short exact sequences with exact forks, i.e., kernel pairs of regular epimorphisms.…”

Section: Introductionmentioning

confidence: 99%

3 × 3 lemma for star-exact sequences

Gran

Janelidze

Rodelo

2012

Homology, Homotopy and Applications

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A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 × 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with "exact forks" and then, the corresponding 3 × 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 × 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.

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Normalization equivalence, kernel equivalence and affine categories

Cited by 140 publications

References 7 publications

A note on strong protomodularity, actions and quotients

A note on strong protomodularity, actions and quotients

Universal Central Extensions in Semi-Abelian Categories

3 × 3 lemma for star-exact sequences

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