We introduce and study relative Rickart objects and dual relative Rickart objects in abelian categories. We show how our theory may be employed in order to study relative regular objects and (dual) ...relative Baer objects in abelian categories. We also give applications to module and comodule categories.
Let
Gls
denote the category of (possibly large) ordered sets with Galois connections as morphisms between ordered sets. The aim of the present paper is to characterize semi-abelian and regular ...protomodular categories among all regular categories ℂ, via the
form of subobjects
of ℂ, i.e. the functor ℂ →
Gls
which assigns to each object
X
in ℂ the ordered set Sub(
X
) of subobjects of
X
, and carries a morphism
f
:
X
→
Y
to the induced Galois connection Sub(
X
) → Sub(
Y
) (where the left adjoint maps a subobject
m
of
X
to the regular image of
fm
, and the right adjoint is given by pulling back a subobject of
Y
along
f
). Such functor amounts to a Grothendieck bifibration over ℂ. The conditions which we use to characterize semi-abelian and regular protomodular categories can be stated as self-dual conditions on the bifibration corresponding to the form of subobjects. This development is closely related to the work of Grandis on “categorical foundations of homological and homotopical algebra”. In his work, forms appear as the so-called “transfer functors” which associate to an object the lattice of “normal subobjects” of an object, where “normal” is defined relative to an ideal of null morphism admitting kernels and cokernels.
3 X 3 lemma for star-exact sequences Gran, Marino; Janelidze, Zurab; Rodelo, Diana
Homology, homotopy, and applications,
2012, Volume:
14, Issue:
no. 2
Journal Article
Peer reviewed
Open access
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 \times 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 \times 3
lemma is equivalent, in the universal algebraic terminology, to
congruence 3-permutability; equivalently, regular categories satisfying
such 3 \times 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.
Concordant-dissonant and monotone-light factorisation systems on categories, ways to construct them, and conditions for them to coincide, as well as their examples are studied in this article. These ...factorisation systems are constructed from a reflection induced from a ground adjunction and a specified prefactorisation system. Furthermore, we give additional conditions, under which the monotone-light and the concordant-dissonant factorisations coincide for sub-reflections of the induced reflection. The adjunctions given by right Kan extensions, from the category of presheaves on sets, turn out to be very well-behaved examples, provided they satisfy the cogenerating set condition, which allows to describe the four classes of morphisms in the reflective and concordant-dissonant (= monotone-light) factorisations. It is also noticed that the faithfulness of the composite of the left-adjoint with the Yoneda embedding can be seen as a generalisation of the cogenerating set condition. Using this generalisation it is possible to present a convenient simplified version of the sufficient conditions above for the case of an adjunction from the category of presheaves on sets into a cocomplete category, satisfying the faithfulness of the abovementioned composite. Then, the same is done for induced sub-reflections from categories of models of (limit) sketches; in particular this explains why the monotone-light factorisation for categories via preordered sets is just the restriction of the same factorisation for simplicial sets via ordered simplicial complexes.
We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an ...effective descent morphism.
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