Homological Category Research Articles

The Lambek Invariants of Commutative Squares in a Homological Category

The Lambek Invariants of Commutative Squares in a Homological Category

CONDITIONAL FLATNESS, FIBERWISE LOCALIZATIONS, AND ADMISSIBLE REFLECTIONS

Abstract We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.

Non-abelian and ε-curved homological algebra with arrow categories

Grandis's non-abelian homological algebra generalizes standard homological algebra in abelian categories to homological categories, which are a broader class of categories including for example the category of lattices and Galois connections. Here, we prove that if C is any category with an ideal of null morphisms with respect to which (co)kernels exist, then the arrow category of C is a homological category. This broadens the applicability of Grandis's framework substantially. In particular, one can form the homology of chain complexes in C by taking the homology objects to be morphisms of C, which one may think of as maps from an object of cycles to an object of chains modulo boundaries.One situation to which Grandis's original framework does not apply directly is ε-curved homological algebra. This refers to chain complexes of normed spaces whose differential squares to zero only approximately, in the sense that ‖d2‖≤ε for some ε>0. This is relevant for example in the theory of approximate representations of groups, where Kazhdan has successfully employed ε-curved homological techniques in an ad-hoc manner. We develop some basics of ε-curved homological algebra and note that our result on arrow categories facilitates the application of Grandis's theory.

Algebraic logoi

We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context of a homological category. We prove that, if the category is moreover well-powered with (small) joins, then the existence of split extension cores is equivalent to the condition that the change-of-base functors in the fibration of points are geometric. We call a finitely complete category that satisfies this condition an algebraic logos. We give examples of such categories, compare them with algebraically coherent ones, and study equivalent conditions as well as stability under common categorical operations.

English

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on Bmod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category Bmod^s is simply Bmod in the topos of sets endowed with an involution and as such it shares with Bmod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category Bmod^s is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between Bmod^s and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of Bmod^s the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.

Fundamental group functors in descent-exact homological categories

We study the notion of a fundamental group in the framework of descent-exact homological categories. This setting is sufficiently large to include several categories of “algebraic” nature such as almost abelian categories, semi-abelian categories, and categories of topological semi-abelian algebras. For many adjunctions in this context, we describe the fundamental groups by generalised Brown–Ellis–Hopf formulae for the integral homology of groups.

Composites of Central Extensions Form a Relative Semi-Abelian Category

We consider trivial and central extensions, in the sense of G. Janelidze and G. M. Kelly, which are defined with respect to an adjunction between a Barr-exact category C and a Birkhoff subcategory X of C. Assuming in addition that C is a pointed Mal’tsev category with cokernels, and that X is protomodular, we prove that: (a) the class of all trivial extensions and the class of all finite composites of central extensions form relative homological category structures on C; (b) if C has finite coproducts, then the class of all finite composites of central extensions forms a relative semi-abelian category structure on C.

The ternary commutator obstruction for internal crossed modules

In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows us to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones.Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.

Relative Semi-abelian Categories

We define a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of regular epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category. Some results on the equivalence of the so-called old-style and new-style axioms for the semi-abelian categories are extended to the relative case.

Homological category weights and estimates for $\mathrm{cat}^1(X,\xi)$

In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological category weight of E. Fadell and S. Husseini but these two notions are distinct already for Poincaré complexes. An important advantage of the homological category weight is its homotopy invariance. We use the notion of homological category weight to study various generalizations of the Lusternik - Schnirelmann category which appeared in the theory of closed one-forms and have applications in dynamics. Our primary goal is to compare two such invariants \mathrm{cat}(X,\xi) and \mathrm{cat}^1(X,\xi) which are defined similarly with reversion of the order of quantifiers. We compute these invariants explicitly for products of surfaces and show that they may differ by an arbitrarily large quantity. The proof of one of our main results uses an algebraic characterization of homology classes z\in H_i(\tilde X;\mathbb{Z}) (where \tilde X\to X is a free abelian covering) which are movable to infinity of \tilde X with respect to a prescribed cohomology class \xi\in H^1(X;\mathbb{R}) . This result is established in Part II which can be read independently of the rest of the paper.

Torsion theories in homological categories

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.

Baer sums in homological categories

We give a unified treatment of the Baer sums in the context of efficiently homological categories which, on the one hand, contains any category of groups with multiple operators and more generally any semi-abelian variety and, on the other hand, the category of Hausdorff groups and more generally any category of semi-abelian Hausdorff algebras. This gives rise to a generalized “Euclide's Postulate” and a five terms exact sequence.

Torsion theories and Galois coverings of topological groups

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.

Torsion theories and radicals in normal categories

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 complement 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.

On the direct image of intersections in exact homological categories

Given a regular epimorphism f : X ↠ Y in an exact homological category C , and a pair ( U , V ) of kernel subobjects of X, we show that the quotient ( f ( U ) ∩ f ( V ) ) / f ( U ∩ V ) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair ( R , S ) of equivalence relations on X, the difference mapping δ : Y / f ( R ∩ S ) ↠ Y / ( f ( R ) ∩ f ( S ) ) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y = X / T , this result says that the difference mapping determined by the inclusion T ∪ ( R ∩ S ) ⩽ ( T ∪ R ) ∩ ( T ∪ S ) has an abelian kernel relation, which casts a new light on the congruence distributive property.

Weak subobjects and the epi-monic completion of a category

The notion of weak subobject, or variation, was introduced by Grandis (Cahiers Topologie Géom. Différentielle Catégoriques 38 (1997) 301–326) as an extension of the notion of subobject, adapted to homotopy categories or triangulated categories, and well linked with their weak limits. We study here some formal properties of this notion. Variations in the category X can be identified with (distinguished) subobjects in the Freyd completion Fr X , the free category with epi-monic factorisation system over X , which extends the Freyd embedding of the stable homotopy category of spaces in an abelian category (Freyd, in: Proceedings of Conference on Categ. Algebra, La Jolla, 1965, Springer, Berlin, 1966, pp. 121–176). If X has products and weak equalisers, as Ho Top and various other homotopy categories, Fr X is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then Fr X is a homological category (Grandis, Cahiers Topologie Géom. Différentielle Catégoriques 33 (1992) 135–175); finally, if X is triangulated, Fr X is abelian and the embedding X → Fr X is the universal homological functor on X , as in Freyd's original case. These facts have consequences on the ordered sets of variations.

On the Homological Category of 3-Manifolds

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Modules over rings with a power basis

A homological category is constructed and also a functor into it from the category of finitely generated modules over a ring with a power basis such that from an object that corresponds to a module the latter is uniquely determined to within free direct summands, and from a morphism corresponding to a module homomorphism this homomorphism can be uniquely reconstructed to within norms.