Abelian Categories

A comprehensive study of generators in terms of epic families as well as of their induced (generalized) hom-functors is given, with special emphasis on the subtle differences between the notions of regular and dense generator. Applications concern subcategories which contain a generator of their supercategory, and a characterization of categories with the property that every object is a coproduct of objects whose representables preserve coproducts. AMS Subject Classification: 18A40, 18A30, 18A35 * This paper is based on a preliminary report (cf. Seminarberichte 28, Fernuniversitit Hagen 1988) which was prepared while the second author was visiting Fernuniversitat and the University of Sydney in 1987-88; work was completed while the first author was visiting York University in 1989. Partial financial support under the Australian Research Grant Scheme and by NSERC (Canada) is gratefully acknowledged.

A category is adhesive if it has all pullbacks, all pushouts along monomorphisms, and all exactness conditions between pullbacks and pushouts along monomorphisms which hold in a topos. This condition can be modified by considering only pushouts along regular monomorphisms, or by asking only for the exactness conditions which hold in a quasitopos. We prove four characterization theorems dealing with adhesive categories and their variants.

We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where the operad is on an abelian category, and the internal structures in question are those of internal category, internal n-category, or internal (cubical) n-tuple category. This allows an operadic treatment of crossed modules and other crossed structures.

Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks. We give a new proof of the fact that every topos is adhesive. We also prove a converse: every small adhesive category has a fully faithful functor in a topos, with the functor preserving the all the structure. Combining these two results, we see that the exactness conditions in the definition of adhesive category are exactly the relationship between pushouts along monomorphisms and pullbacks which hold in any topos.

We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where the operad is on an abelian category, and the internal structures in question are those of internal category, internal n-category, or internal (cubical) n-tuple category. This allows an operadic treatment of crossed modules and other crossed structures.

Let C be a small category with weak finite limits, and let Flat(C) be the category of flat functors from C to the category of small sets. We prove that the free exact completion of C is the category of set-valued functors of Flat (C) which preserve small products and filtered colimits. In case C has finite limits, this gives A. Carboni and R. C. Magno's result on the free exact completion of a small category with finite limits.

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been a empted over time, some topological in avor, based on the vector spacevalued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category so that functors to that category induce persistence functions. We port the interleaving and bo leneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we de ne colorable ranks, persistence diagrams and prove the equality between multicolored bo leneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on ltered topological spaces with a group action and labeled point cloud data.

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S n+p in characteristic p, where 0 ≤ n ≤ p − 1, and of the Deligne category Rep ab S t , where t ∈ N. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl 2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and P GL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

In this paper, some categorical properties of the category Pre-Dcpo of all predcpos; pre-ordered sets which are also pre-directed complete, with pre-continuous maps between them is considered. In particular, we characterize products and coproducts in this category. Furthermore, we show that this category is neither complete nor cocomplete. Also, epimorphisms and monomorphisms in Pre-Dcpo are described. Finally, some adjoint relations between the category Pre-Dcpo and others are considered. More precisely, we consider the forgetful functors between this category and some well-known categories, and study the existence of their left and right adjoints.

We prove that the forgetful functors from the categories of C *-and W *-algebras to Banach *-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) C *-algebras are not locally-isometry ℵ 0-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension ≤ 1. For the same reason, for a locally compact abelian group G the category of G-graded von Neumann algebras is not locally presentable.

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.

The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day's PhD thesis [1]. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.

We consider a semi-abelian category V and we write Act(G,X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(−,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove first that the representability of Act(−,X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(−,X) in a general semi-abelian category. Finally, we exhibit the precise form of the more involved “if and only if” amalgamation property corresponding to the representability of actions: this con...

For a small category C with multilimits for finite diagrams, a conceptual description of its free coproduct completion C(C) is given as the category of those set-valued functors of a finitely accessible category with connected limits which preserve these limits and filtered colimits. In this way we recognize the free coproduct completion as a finitely complete category and show that C(C) is universal with respect to existence of finite limits and of small coproducts which are disjoint and stable under pullback.

Pure morphisms in locally presentable categories were characterized as directed colimits of split monomorphisms, and they are regular monomorphisms. We prove that both results hold in accessible categories with pushouts, but not in general accessible categories.

We present and characterize the classes of Grothendieck toposes having enough supercompact objects or enough compact objects. In the process, we examine the subcategories of supercompact objects and compact objects within such toposes and classes of geometric morphism which interact well with these objects. We also present canonical classes of sites generating such toposes.

Given a Cayley-Hamilton smooth order A in a central simple algebra Σ, we determine the flat locus of the Brauer-Severi fibration of A. Moreover, we give a classification of all (reduced) central singularities where the flat locus differs from the Azumaya locus and show that the fibers over the flat, non-Azumaya points near these central singularities can be described as fibered products of graphs of projection maps. This generalizes an old result of Artin on the fibers of the Brauer-Severi fibration of a maximal order over a ramified point. Finally, we show these fibers are also toric quiver varieties and use this fact to compute their cohomology.

Let α > 0. In [45, 43], the main result was the derivation of generic isometries. We show that Q < sin (∅). In [45], the authors classified homomorphisms. Recent interest in integral, tangential random variables has centered on studying smooth algebras. 1 Introduction Recently, there has been much interest in the characterization of nonnegative points. This reduces the results of [43] to a standard argument. This could shed important light on a conjecture of Hausdorff. Is it possible to study parabolic, finitely invariant, arithmetic homomorphisms? This reduces the results of [43] to an easy exercise. Next, recent interest in trivial homeomorphisms has centered on deriving almost everywhere hyper-solvable, discretely natural, pairwise onto scalars. It is well known that every von Neumann, injective vector is analytically real and integral. It is well known that W (R) is smaller than d (u). Is it possible to characterize open, closed, Galois isometries? A central problem in pure algebraic model theory is the description of points. This reduces the results of [43] to standard techniques of topological combinatorics. Therefore this reduces the results of [45, 15] to the admissibility of subsets. Moreover, it is well known that j < z. So recently, there has been much interest in the derivation of degenerate topoi. On the other hand, it is essential to consider that Ξ may be co-finitely Markov. In contrast, recent interest in Levi-Civita equations has centered on extending Levi-Civita matrices. Recently, there has been much interest in the characterization of essentially right-Banach, Chern, unique scalars. Every student is aware that Ramanujan's conjecture is true in the context of scalars.

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

We prove that the forgetful functors from the categories of C *-and W *-algebras to Banach *-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) C *-algebras are not locally-isometry ℵ 0-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension ≤ 1. For the same reason, for a locally compact abelian group G the category of G-graded von Neumann algebras is not locally presentable.

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S n+p in characteristic p, where 0 ≤ n ≤ p − 1, and of the Deligne category Rep ab S t , where t ∈ N. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl 2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and P GL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

Construction of an homology and a cohomology theory associated to a first order formula Diagrammes, tome 23 (1990), p. 7-13 <http://www.numdam.org/item?id=DIA_1990__23__7_0> © Université Paris 7, UER math., 1990, tous droits réservés. L'accès aux archives de la revue « Diagrammes » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ DIAGRAMMES VOLUME 23 ACTES DES JOURNEES E.L.I.T.

In this paper, we investigate the property (P) that finite products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal&#39;tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre $Pt_{\mathbb{C}}(X)$ of the fibration of points $\pi: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ satisfies (P). Examples include any regular Mal&#39;tsev or majority category with coequalizers, as well as any regular Gumm category with coequalizers. Varieties satisfying (P) locally are also characterized by a Mal&#39;tsev term condition, which turns out to be equivalent to a variant of Gumm&#39;s shifting lemma. Furthermore, we show that the varieties satisfying (P) locally are preci...

The main result of this paper is little more than a juxtaposition of a remark in [Leroux] (§5, part A), with a theorem of [Banaschewski]. However, the results are individually little known and their juxtaposition not at all. More explicitly, Leroux remarks, in somewhat different form, Let the category X be complete and cocomplete, and have a set of cogenerators. Then a necessary and sufficient condition that there exist infective effacements (with respect to the class of regular monomorphisms) is that X be coregular and satisfy, in addition, condition COEX 4*.

Torsion theories have proved a very useful tool in the theory of abelian categories; for example, in one proof of Mitchell&#39;s embedding theorem (Bucur and Deleanu [3]) and in ring theory (Lambek [5]). It is the purpose of this paper to initiate an analogous theory for non-abelian categories. Originally we had hoped to prove the non-abelian analogue of Mitchell&#39;s theorem this way (Barr, [2, Theorem III (1.3)]), but so far this had not been possible. Nonetheless an interesting variety of examples fit into this theory.

When G is a locally compact group, the unitary representation theory of G is the "same" as the ^representation theory of the group C*-algebra C*(G). Hence it is of interest to determine the isomorphism class of C*(G) for a wide variety of groups G. Using methods suggested by papers of Z'ep and Delaroche, we determine explicitly the C*-algebras of the "ax + b" groups over all nondiscrete locally compact fields and of a number of two-step solvable Lie groups. Only finitely many C*-algebras arise as the group C*-algebras of 3-dimensional simply connected Lie groups, and we characterize many of them. We also discuss the C*-algebras of unipotent p-adic groups.

The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day&#39;s 1970 PhD thesis. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.

We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category has comma objects computed in .

We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.

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We make some beginning observations about the category Eq of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations R,S is a mapping from the set of R-equivalence classes to that of S-equivalence classes, which is induced by a computable function. We also consider some full subcategories of Eq, such as the category EqpΣ01q of computably enumerable equivalence relations (called ceers), the category EqpΠ01q of co-computably enumerable equivalence relations, and the category EqpDarkq whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in EqpΣ01q the epimorphisms coincide with the onto morphisms, but in EqpΠ01q there are epimorphisms that are not onto. Moreover, Eq, EqpΣ01q, and EqpDark q are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morp...

The girth of a graph G (with cycles) is the length of a smallest cycle of G and is denoted by g(G). For a connected graph G having girth 2k + 1 ≥ 5 for some integer k ≥ 2, the Schwenk graph G∗ of G has the set of all paths of order k + 1 of G as its vertex set V (G∗), where two vertices P and Q of G∗ are adjacent in G∗ if P = (u1, u2, . . . , uk+1) and Q = (v1, v2, . . . , vk+1) such that uk+1 = v1, V (P ) ∩ V (Q) = {uk+1} and u1vk+1 ∈ E(G). It is shown that the Schwenk graph is triangle-free and for each odd integer g ≥ 5, there exists a connected graph of girth g whose Schwenk graph contains 4-cycles. Connected graphs of girth 5 whose Schwenk graph contains 4-cycles are characterized. Structural properties of the Schwenk graphs of the unique 5-cage (the Petersen graph) and the unique 7-cage (the McGee graph) are studied. Other results and open questions are presented for the Schwenk graphs of cages.

The well-known snake lemma is proved entirely within category theory, without the help of &quot;points with value in...&quot; \`a la Grothendieck, nor pseudo-elements as in Guglielmetti & Zaganidis. Instead, we define and use consistently semi-cartesian squares, which were promoted by C. Chevalley.

Building on the embedding of an n-abelian category M into an abelian category A as an n-cluster-tilting subcategory of A, in this paper we relate the n-torsion classes of M with the torsion classes of A. Indeed, we show that every n-torsion class in M is given by the intersection of a torsion class in A with M . Moreover, we show that every chain of n-torsion classes in the n-abelian category M induces a Harder-Narasimhan filtration for every object of M . We use the relation between M and A to show that every Harder-Narasimhan filtration induced by a chain of n-torsion classes in M can be induced by a chain of torsion classes in A. Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.

The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day’s PhD thesis [1]. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of cartesian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.

Motivated by applications to Mackey functors, Serge Bouc (Bo) character- ized pullback and finite coproduct preserving functors between categories of permutation representations of finite groups. Initially surprising to a category theorist, this result does have a categorical explanation which we provide. For a finite group G, we write G-setfin for the category of finite (left) G-sets (that is, of permutation representations of G) and equivariant functions. We write Spn(G-setfin) for the category whose morphisms are isomorphism classes of spans between finite G- sets. Coproducts in Spn(G-setfin) are those of G-setfin and composition in Spn(G-setfin) involves pullbacks in G-setfin. According to Harald Lindner (Li), a Mackey functor M on a finite group H is a coprod- uct preserving functor M : Spn(H-setfin) / Modk. A functor F : G-setfin / H-setfin which preserves pullbacks and finite coproducts will induce a functor

Construction of an homology and a cohomology theory associated to a first order formula Diagrammes, tome 23 (1990), p. 7-13 <http://www.numdam.org/item?id=DIA_1990__23__7_0> © Université Paris 7, UER math., 1990, tous droits réservés. L'accès aux archives de la revue « Diagrammes » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ DIAGRAMMES VOLUME 23 ACTES DES JOURNEES E.L.I.T.

We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category LFP has comma objects computed in Cat. Introduction In this note we investigate in detail the comma and coslice construction in the context of locally finitely presentable categories. It is well known since [1] that coslices of locally finitely presentable categories are again locally finitely presentable categories; however the proof involves an abstract characterization from which one cannot retrieve an explicit generator of finitely presented objects. However, in the con...

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.