1,071 research outputs found
Effective descent morphisms of regular epimorphisms
Let be a regular category with pushouts of regular epimorphisms by
regular epimorphism and the category of regular epimorphisms in .
We prove that every regular epimorphism in is an effective descent
morphism if, and only if, is a regular category. Then, moreover, every
regular epimorphism in is an effective descent morphism. This is the case,
for instance, when is either exact Goursat, or ideal determined, or is a
category of topological Mal'tsev algebras, or is the category of -fold
regular epimorphisms in any of the three previous cases, for any
Approximate Hagemann-Mitschke co-operations
We show that varietal techniques based on the existence of operations of a
certain arity can be extended to n-permutable categories with binary
coproducts. This is achieved via what we call approximate Hagemann-Mitschke
co-operations, a generalisation of the notion of approximate Mal'tsev
co-operation. In particular, we extend characterisation theorems for
n-permutable varieties due to J. Hagemann and A. Mitschke to regular categories
with binary coproducts.Comment: 11 pages. Dedicated to George Janelidze on the occasion of his
sixtieth birthda
Higher commutator conditions for extensions in Mal'tsev categories
We define a Galois structure on the category of pairs of equivalence
relations in an exact Mal'tsev category, and characterize central and double
central extensions in terms of higher commutator conditions. These results
generalize both the ones related to the abelianization functor in exact
Mal'tsev categories, and the ones corresponding to the reflection from the
category of internal reflexive graphs to the subcategory of internal groupoids.
Some examples and applications are given in the categories of groups,
precrossed modules, precrossed modules of Lie algebras, and compact groups.Comment: 32 page
Behavior of Quillen (co)homology with respect to adjunctions
This paper aims to answer the following question: Given an adjunction between
two categories, how is Quillen (co)homology in one category related to that in
the other? We identify the induced comparison diagram, giving necessary and
sufficient conditions for it to arise, and describe the various comparison
maps. Examples are given. Along the way, we clarify some categorical
assumptions underlying Quillen (co)homology: cocomplete categories with a set
of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application
The Auslander bijections and universal extensions
Universal extensions arise naturally in the Auslander bijections. For an
abelian category having Auslander-Reiten duality, we exploit a bijection
triangle, which involves the Auslander bijections, universal extensions and the
Auslander-Reiten duality. Some consequences are given, in particular, a
conjecture by Ringel is verified
Semi-localizations of semi-abelian categories
A semi-localization of a category is a full reflective subcategory with the
property that the reflector is semi-left-exact. In this article we first
determine an abstract characterization of the categories which are
semi-localizations of an exact Mal'tsev category, by specializing a result due
to S. Mantovani. We then turn our attention to semi-abelian categories, where a
special type of semi-localizations are known to coincide with torsion-free
subcategories. A new characterisation of protomodular categories in terms of
binary relations is obtained, inspired by the one discovered in the pointed
context by Z. Janelidze. This result is useful to obtain an abstract
characterization of the torsion-free and of the hereditarily-torsion-free
subcategories of semi-abelian categories. Some examples are considered in
detail in the categories of groups, crossed modules, commutative rings and
topological groups. We finally explain how these results extend similar ones
obtained by W. Rump in the abelian context.Comment: 30 pages. v2: introduction and references update
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