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The stable category of preorders in a pretopos I: General theory
Abstract
In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category PreOrd(C) of internal preorders in any coherent category C, that enlightens the categorical nature of this notion. When C is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all σ-pretoposes and all elementary toposes, with the property that this functor sends any short Z-exact sequences in PreOrd(C) (where Z is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.
... If the category C is a pretopos, it is possible to construct a pointed quotient category, called the stable category, with an interesting property: the pretorsion theory (Eq, ParOrd) in PreOrd(C) becomes a "classical" torsion theory in the stable category. This construction was presented in [2,3] as a natural generalization of the one proposed in [11] for the category PreOrd of preordered sets. It is worth noting that in the case C = Set, the stable category provides an example of a pointed category arising as a quotient category of a topological category (the category of Aleksandrov-discrete spaces), where short exact sequences naturally occur, as in the case of abelian categories. ...
... The pretorsion theory (Eq, ParOrd) gives rise to an interesting Galois theory, whose main properties have been explored in [13]. The stable category presented in [2,3] has an important categorical interpretation: the canonical functor Σ: PreOrd(C) → Stab (where Stab is the stable category) is universal among all (finite coproduct preserving) torsion theory functors G : PreOrd(C) → X, where X is a pointed category with coproducts equipped with a torsion theory (see [3,Section 2] for more details). Informally speaking, the quotient functor Σ: PreOrd(C) → Stab is the best possible finite coproduct preserving functor transforming the pretorsion theory (Equiv(C), ParOrd(C)) into a torsion theory. ...
... In this paper we show that this is indeed possible whenever the base category is merely a lextensive category [6], hence naturally including the example mentioned above and many others (see [11,12,13,24] and the last section of this article for more details). Most of techniques contained in this paper are based on those of [2,3]. The main reason for this is that the category PreOrd(C) of internal preorders in a pretopos C is lextensive, and the proofs in [2,3] essentially used the "good" properties relating coproducts and limits in the category PreOrd(C). ...
We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders in a pretopos. Some examples are provided in the categories of topological spaces and of (small) categories.
... We prove a general theorem on the existence and the universality of the stable category, which applies in particular to the case of preordered objects in an exact coherent category (see [11,3]), and to the case of internal categories in a Grothendieck topos (see [6,7]). But the flexibility of our approach allows also choosing only trivial subobjects (i.e. ...
... Their union is of course the whole sheaf. This shows that our notion of saturated (clopen) sub-preordered object differs from that in [3], where complementarity of the subobject is forced. Of course, both notions coincide in the Set case. ...
... These two notions are cartesian ones (see [14]), thus make sense in every category with finite limits. They yield a pretorsion theory in every Barr exact category (see [1,3]). ...
In a coherent category, the posets of subobjects have very strong properties. We emphasize the validity of these properties, in general categories, for well-behaved classes of subobjects. As an example of application, we investigate the problem of the various torsion theories which can be universally associated with a pretorsion one.
... This article is meant as the sequel of [2] and it deals with the study of the universal property of the stable category Stab(C) of the category PreOrd(C) of internal preorders in a pretopos C. It reveals the categorical feature of a natural construction due to A. Facchini and C. Finocchiaro in the category PreOrd of preordered sets [6], that we first briefly recall. ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
We prove that the stable category associated with the category PreOrd(C) of internal preorders in a pretopos C satisfies a universal property. The canonical functor from PreOrd(C) to the stable category Stab(C) universally transforms a pretorsion theory in PreOrd(C) into a classical torsion theory in the pointed category Stab(C). This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when C is the category of sets.
... If the category C is a pretopos, it is possible to construct a pointed quotient category, called the "stable category", with an interesting property: the pretorsion theory (Eq, ParOrd) in PreOrd(C) becomes a "classical" torsion theory in the stable category. This construction was presented in [2,3] as a natural generalization of the one proposed in [10] for the category PreOrd of preordered sets. It is worth noting that in the case C = Set, the stable category provides an example of a pointed category arising as a quotient category of a topological category (the 2020 Mathematics Subject Classification. ...
... The pretorsion theory (Eq, ParOrd) gives rise to an interesting Galois theory, whose main properties have been explored in [12]. The stable category presented in [2,3] has an important categorical interpretation: the canonical functor Σ : PreOrd(C) → Stab (where Stab is the stable category) is universal among all (finite coproduct preserving) torsion theory functors G : PreOrd(C) → X, where X is a pointed category with coproducts equipped with a torsion theory (see [3,Section 2] for more details). Informally speaking, the quotient functor Σ : PreOrd(C) → Stab is the best possible finite coproduct preserving functor transforming the pretorsion theory (Equiv(C), ParOrd(C)) into a torsion theory. ...
... In this paper we show that this is indeed possible whenever the base category is merely a lextensive category [6], hence naturally including the example mentioned above and many others (see [10,11,12,20] and the last section of this article for more details). Most of techniques contained in this paper are based on those of [2,3]. The main reason for this is that the category PreOrd(C) of internal preorders in a pretopos C is lextensive, and the proofs in [2,3] essentially used the "good" properties relating coproducts and limits in the category PreOrd(C). ...
We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders in a pretopos. Some examples are provided in the categories of topological spaces and of (small) categories.
... This article is meant as the sequel of [2] and it deals with the study of the universal property of the stable category Stab(C) of the category PreOrd(C) of internal preorders in a pretopos C. It reveals the categorical feature of a natural construction due to A. Facchini and C. Finocchiaro in the category PreOrd of preordered sets [6], that we first briefly recall. ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
We prove that the stable category associated with the category of internal preorders in a pretopos satisfies a universal property. The canonical functor from to the stable category universally transforms a pretorsion theory in into a classical torsion theory in the pointed category . This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when is the category of sets.
... This article is meant as the sequel of [2] and it deals with the study of the universal property of the stable category Stab(C) of the category PreOrd(C) of internal preorders in a pretopos C. It reveals the categorical feature of a natural construction due to A. Facchini and C. Finocchiaro in the category PreOrd of preordered sets [6], that we first briefly recall. ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
... In the first article [2] of this series we proved that, whenever C is a coherent category [11], it is possible to give a purely categorical construction of the stable category Stab(C) of the category PreOrd(C) of internal preorders in C (we recall this construction in the first section of this article). Moreover, when C is a pretopos, the functor Σ : PreOrd(C) → Stab(C) preserves coproducts and sends short Z-exact sequences in PreOrd(C) to short exact sequences in the pointed category Stab(C) (Theorem 7.14 in [2]). ...
In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category of internal preorders in any coherent category , that enlightens the categorical nature of this notion. When is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all -pretoposes and all elementary toposes, with the property that this functor sends any short -exact sequences in (where is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.
... Choosing the equivalence relations as torsion objects and the partial orders as torsion-free objects, one obtains a pretorsion theory on PreOrd (see [14]) with the discrete objects as trivial ones. This fact has been generalized to the category PreOrd(C) of preordered objects in a Barr-exact category C (see [16,5,6]). Other examples of pretorsion theories have been studied in [15,7,17,33,19]. ...
... Choosing the equivalence relations as torsion objects and the partial orders as torsion-free objects, one obtains a pretorsion theory on PreOrd (see [14]) with the discrete objects as trivial ones. This fact has been generalized to the category PreOrd(C) of preordered objects in a Barr-exact category C (see [16,5,6]). Other examples of pretorsion theories have been studied in [15,7,17,33,20]. ...
We describe a pretorsion theory in the category of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in that identify pairs of objects: they are faithful and reflect isomorphisms.
Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general (non-pointed) categories. Many examples have been investigated in several different contexts, such as topological spaces and topological groups, internal preorders, preordered groups, toposes, V-groups, crossed modules, etc. In this paper, we show that pretorsion theories naturally appear also in the “classical” framework, namely in abelian categories. We propose two ways of obtaining pretorsion theories starting from torsion theories. The first one uses “comparable” torsion theories, while the second one extends a torsion theory with a Serre subcategory. We also give a universal way of obtaining a torsion theory from a given pretorsion theory in additive categories. We conclude by providing several examples in module categories, internal groupoids, recollements and representation theory.
This is a review of
Borceux, Francis; Clementino, Maria Manuel
On coherent systems of subobjects with application to torsion theories. (English) Zbl 07844807
This is a review of
Borceux and Federico
The stable category of preorders II:the universal property
This is a review of
Borceux, Francis; Campanini, Federico; Gran, Marino
The stable category of preorders in a pretopos. II: The universal property. (English) Zbl 07605314
Ann. Mat. Pura Appl. (4) 201, No. 6, 2847-2869 (2022).
We prove that the stable category associated with the category PreOrd(C) of internal preorders in a pretopos C satisfies a universal property. The canonical functor from PreOrd(C) to the stable category Stab(C) universally transforms a pretorsion theory in PreOrd(C) into a classical torsion theory in the pointed category Stab(C). This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when C is the category of sets.
We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair (T, F) of full replete subcategories in a category C, the corresponding full subcategory Z=T∩F of trivial objects in C. The morphisms which factor through Z are called Z-trivial, and these form an ideal of morphisms, with respect to which one can define Z-prekernels, Z-precokernels, and short Z-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when Z is reduced to the 0-object of C. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets.
Several elementary properties of the symmetric group Sn extend in a nice way to the full transformation monoid Mn of all maps of the set X := {1, 2, 3,…,n} into itself. The group Sn turns out to be the torsion part of the monoid Mn. That is, there is a pretorsion theory in the category of all maps f:X → X, X an arbitrary finite set, in which bijections are exactly the torsion objects.
In a pointed category with kernels and cokernels, we characterize torsion–free classes in terms of their closure under extensions. They are also described as indexed reflections. We obtain a corresponding characterization of torsion classes by formal categorical dualization.
We provide a characterisation of the category KH of compact Hausdorff spaces and continuous maps by means of categorical properties only. To this aim we introduce a notion of filtrality for coherent categories, relating certain lattices of subobjects to their Boolean centers. Our main result reads as follows: Up to equivalence, KH is the unique non-trivial well-pointed pretopos which is filtral and admits all set-indexed copowers of its terminal object.
We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free.
We propose here a study of ‘semiexact’ and ‘homological’ categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied. This is a sequel of a book on ‘Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups’, published by the same Editor, but can be read independently of the latter. The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell — a moderate generalisation of abelian categories that is nevertheless crucial for a theory of ‘coherence’ and ‘universal models’ of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework. According to the present definitions, a semiexact category is a category equipped with an ideal of ‘null’ morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple. Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of ‘pairs’ of topological spaces or groups; they also include their codomains, since the sequences of homotopy ‘objects’ for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets. © 2013 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
By a multi-pointed category we mean a category CC equipped with an ideal of null morphisms, i.e. a class NN of morphisms satisfying f∈N∨g∈N⇒fg∈Nf∈N∨g∈N⇒fg∈N for any composable pair f,gf,g of morphisms. Such categories are precisely the categories enriched in the category of pairs X=(X,N)X=(X,N) where XX is a set and NN is a subset of XX, whereas a pointed category has the same enrichment, but restricted to those pairs X=(X,N)X=(X,N) where NN is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories.
We propose a theory of central extensions for universal algebras, and more generally for objects in an exact category , centrality being defined relatively to an “admissible” full subcategory of . This includes not only the classical notions of central extensions for groups and for algebras, but also their generalization by Fröhlich to a pair consisting of a variety of ω-groups and a subvariety . Our notion of central extension is adapted to the generalized Galois theory developed by the first author, the use of which enables us to classify completely the central extensions of a given object B, in terms of the actions of an “internal Galois pregroupoid”.
We introduce and study the notion of torsion theory in the non-abelian context of homological categories, and we investigate the properties of the corresponding closure operator. We then consider several new examples of torsion theories in the category of topological groups and, more generally, in any category of topological semi-abelian algebras. We finally characterize the hereditary torsion theories, and we analyse a new example in the homological category of crossed modules. (c) 2006 Elsevier Inc. All rights reserved.
We introduce a relativized notion of (semi)normalcy for categories that come equipped with a proper stable factorization system, and we use radicals (in the sense of module theory) and normal closure operators in order to study torsion theories in such categories. Our results generalize and com-plement recent studies in the realm of semi-abelian and, in part, homological categories. In particular, we characterize both, torsion-free and torsion classes, in terms of their closure under extensions. We pay particular attention to the homological and, for our purposes more importantly, normal categories of topological algebra, such as the category of topological groups. But our applications go far beyond the realm of these types of categories, as they include, for example, the normal, but non-homological category of pointed topological spaces, which is in fact a rich supplier for radicals of topological groups.
It is shown that the descent constructions of finite preorders provide a simple motivation for those of topological spaces, and new counter-examples to open problems in Topological descent theory are constructed.
In recent years, there has been considerable discussion as to the appropriate definition of distributive categories. Three definitions which have had some support are: (1) A category with finite sums and products such that the canonical map δ:A×B+ A×C→A×(B+C) is an isomorphism (Walters). (2) A category with finite sums and products such that the canonical functor +:A/A× A/B→A/(A+B) is an equivalence (Monro). (3) A category with finite sums and finite limits such that the canonical functor + of (2) is an equivalence (Lawvere and Schanuel).There has been some confusion as to which of these was the natural notion to consider. This resulted from the fact that there are actually two elementary notions being combined in the above three definitions. The first, to which we give the name distributivity, is exactly that of (1). The second notion, which we shall call extensivity, is that of a category with finite sums for which the canonical functor + of definitions (2) and (3) is an equivalence. Extensivity, although it implies the existence of certain pullbacks, is essentially a property of having well-behaved sums. It is the existence of these pullbacks which has caused the confusion. The connections between definition (1) and definitions (2) and (3) are that any extensive category with products is distributive in the first sense, and that any category satisfying (3) satisfies (1) locally.The purpose of this paper is to present some basic facts about extensive and distributive categories, and to discuss the relationships between the two notions.
A pretorsion theory for the category of all categories
- J Xarez
J. Xarez, A pretorsion theory for the category of all categories, preprint, arXiv:2011.13448
(2020), to appear in Cahiers Top. Géom. Diff. Catég.
Rosen's no-go theorem for regular categories
- F Lorengian
F. Lorengian, Rosen's no-go theorem for regular categories, arXiv:2012.11648 (2020).
A new Galois structure in the category of internal preorders
- Facchini
- A Facchini
- C A Finocchiaro
- M Gran
A. Facchini, C.A. Finocchiaro and M. Gran, A new Galois structure in the category of internal
preorders, Theory Appl. Categories 35 (2020) pp. 326-349.
Italy Email address: federico
- Padova
Padova, Italy
Email address: federico.campanini@unipd.it