Search results for "Abelian category"

General Set-Up

2017

Abelian gradings on upper-triangular matrices

2003

Let G be an arbitrary finite abelian group. We describe all possible G-gradings on an upper-triangular matrix algebra over an algebraically closed field of characteristic zero.

Butterflies in a Semi-Abelian Context

2011

It is known that monoidal functors between internal groupoids in the category Grp of groups constitute the bicategory of fractions of the 2-category Grpd(Grp) of internal groupoids, internal functors and internal natural transformations in Grp, with respect to weak equivalences (that is, internal functors which are internally fully faithful and essentially surjective on objects). Monoidal functors can be equivalently described by a kind of weak morphisms introduced by B. Noohi under the name of butterflies. In order to internalize monoidal functors in a wide context, we introduce the notion of internal butterflies between internal crossed modules in a semi-abelian category C, and we show th…

A Push Forward Construction and the Comprehensive Factorization for Internal Crossed Modules

2014

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.

Weights and Pure Nori Motives

2017

In this chapter, we explain how Nori motives relate to other categories of motives. By the work of Harrer, the realisation functor from geometric motives to absolute Hodge motives factors via Nori motives. We then use this in order to establish the existence of a weight filtration on Nori motives with rational coefficients. The category of pure Nori motives turns out to be equivalent to Andre’s category of motives via motivated cycles.

Motivic Complexes and Relative Cycles

2019

This part is based on Suslin and Voevodsky’s theory of relative cycles that we develop in categorical terms, in the style of EGA. The climax of the theory is obtained in the study of a pullback operation for suitable relative cycles which is the incarnation of intersection theory in this language. Properties of this pullback operation, and on the conditions necessary to its definition, are made again inspired by intersection theory. We study the compatibility of this pullback operation with projective limits of schemes. In Section 9, the theory of relative cycles is exploited to introduce Voevodsky’s category of finite type schemes over an arbitrary base with morphisms finite correspondence…

Peiffer product and peiffer commutator for internal pre-crossed modules

2017

In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator 〈X, X〉 is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varieties, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.

Categories of (Mixed) Motives

2017

Nori’s Diagram Category

2017

We explain Nori’s construction of an abelian category attached to the representation of a diagram and establish some properties for it. The construction is completely formal. It mimics the standard construction of the Tannakian dual of a rigid tensor category with a fibre functor . Only, we do not have a tensor product or even a category but only what we should think of as the fibre functor.

INTERNAL CROSSED MODULES AND PEIFFER CONDITION

2010

In this paper we show that in a homological category in the sense of F. Borceux and D. Bourn, the notion of an internal precrossed module corresponding to a star-multiplicative graph, in the sense of G. Janelidze, can be obtained by directly internalizing the usual axioms of a crossed module, via equivariance. We then exhibit some sufficient conditions on a homological category under which this notion coincides with the notion of an internal crossed module due to G. Janelidze. We show that this is the case for any category of distributive Omega(2)-groups, in particular for the categories of groups with operations in the sense of G. Orzech.