Methods of constructive category theory

D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Here, employing a notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni’s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using n-X-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in Rep(Q,C) (the category of representations of a quiver Q in an abelian category C) from tilting classes in C, where Q is finite-cone-shape.

Let [Formula: see text] be small category and [Formula: see text] an arbitrary category. Consider the category [Formula: see text] whose objects are functors from [Formula: see text] to [Formula: see text] and whose morphisms are natural transformations. Let [Formula: see text] be another category, and again, consider the category [Formula: see text]. Now, given a functor [Formula: see text] we construct the induced functor [Formula: see text]. Assuming [Formula: see text] and [Formula: see text] to be abelian categories, it follows that the categories [Formula: see text] and [Formula: see text] are also abelian. We have two main goals: first, to find a relationship between the derived category [Formula: see text] and the category [Formula: see text]; second relate the functors [Formula: see text] and [Formula: see text]. We apply the general results obtained to the special case of quiver sheaves.

Let X. be a simplicial set. A cohomological system of coefficients F. on X. with values in an abelian category A is given by an object F σ ∈ A for every σ ∈ X. and by a compatible set of morphisms Fα*σ → Fσ for every map of ordered sets α : m → n, m, n = 0, … and σ ∈ Xn. We denote by SH(X.) the category of cohomological systems F. of coefficients on X, such that the maps Fα * σ → Fσ, are isomorphisms if α is surjective. This category is equivalent to a certain subcategory of the category of sheaves SH(X) on the geometric realization X = ∣X.∣ of the simplicial set X.The present paper gives a construction of a duality theory for SH(X.) which is analogue to the topological Verdier duality for SH(X). It consists of a construction of a right adjoint to the cohomology with compact support functor and a resulting duality theory given by a dualizing complex. These constructions are much more explicit than in the topological case. We compare the simplicial and topological constructions via the embedding SH(X.) → SH(X). In particular, we get an explicit description of the dualizing sheaf complex for topological spaces X = ∣X∣.

An invariant I of quasiprojective K-varieties X with values in a commutative ring R is if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge polynomials. We first define a unique extension I' of I to finite type Artin K-stacks F, which is motivic and satisfies I'([X/G])=I(X)/I(G) when X is a K-variety, G a K-group acting on X, and [X/G] is the quotient stack. This only works if I(G) is invertible in R for all special K-groups G, which excludes I=chi as chi(K*)=0. But we can extend the construction to get round this. Then we develop the theory of on Artin stacks. These are a universal generalization of constructible functions on Artin stacks, as studied in the author's paper math.AG/0403305. There are several versions of the construction: the basic one SF(F), and variants SF(F,I,R),... twisted by motivic invariants. We associate a Q-vector space SF(F) or an R-module SF(F,I,R) to each Artin stack F, with functorial operations of multiplication, pullbacks phi^* and pushforwards phi_* under 1-morphisms phi : F --> G, and so on. They will be important tools in the author's series on Configurations in abelian categories, math.AG/0312190, math.AG/0503029, math.AG/0410267 and math.AG/0410268.

This is the fourth (and last) prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to present several applications of this theory in commutative and noncommutative algebra. The emphasis is on constructions and examples, rather than on axiomatics. Here are the topics covered in the book: - A review of standard facts on abelian categories. - Differential graded algebra (DG rings, DG modules, DG categories and DG functors). - Triangulated categories and triangulated functors between them. How they arise from the DG background. The homotopy category K(A,M) of DG A-modules in M. - Localization of categories. The derived category D(A,M), which is the localization of K(A,M) with respect to the quasi-isomorphisms. - Left and right derived functors of a triangulated functor. - K-injective, K-projective and K-flat DG modules. Their roles, and their existence in several important algebraic situations. - Dualizing and residue complexes over commutative noetherian rings, including Van den Bergh rigidity. - Perfect DG modules and tilting DG bimodules over NC (noncommutative) DG rings. - NC connected graded rings, including Artin-Schelter regular rings. Derived torsion for NC connected graded rings, its relation to the chi condition of Artin-Zhang, and the NC MGM Equivalence. Balanced dualizing complexes, their uniqueness, existence and trace functoriality. - NC rigid dualizing complexes, following Van den Bergh. The uniqueness and existence of these complexes, a few examples, and their relation to Calabi-Yau rings. Readers of this preview version are urged to write to the author with any comments regarding errors, suggestions or questions.

In this chapter, we study complexes (and double complexes) in abelian categories and give tools to compute their cohomology. In particular, we prove the classical “Snake lemma” and we construct the long exact sequence associated with a short exact sequence of complexes.As an application, we discuss Koszul complexes associated to functors defined on a category of finite subsets of a set S, with values in an abelian category C. The main result asserts that such a complex may be obtained as the mapping cone of a morphism acting on a simpler Koszul complex. We apply these results to the study of distributive families of subobjects of an object X in C.We postpone the introduction of derived categories to the next chapter.Note that we avoid the use of spectral sequences, using instead systematically the “truncation functors”.KeywordsExact SequenceCommutative DiagramSpectral SequenceShort Exact SequenceRegular SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

LET ME begin with three assertions. First, causal analysis is an important and viable method for theory construction in sociology. Second, Merton variety of functional analysis in sociology is a unique method of theory construction and is major alternative to causal analysis. Third, inherent methodological difficulties in Mertonian functional ananalysis in sociology argument for its abandonment. first of above assertions constitutes my reason for this attempt to spell out a notion of causal analysis as a method for theory construction in sociology. appreciation of second and third assertions requires some discussion of work of Robert K. Merton and Kingsley Davis. Merton's classic methodological paradigm for functional analysis was put forth in Social Theory and Social Structure in 1949. A number of assertions prior to statement of paradigm imply that Merton saw functional analysis as a unique method of theory construction. He claimed that much had been accomplished with its use and much more could be expected. paradigm was offered as an explicit codification of procedures used by several sociologists and anthropologists. In this light, then, it is interesting to find that ten years later president of American Sociological Association was claiming that uniqueness of functional analysis in sociology was a myth. In paper titled The Myth of Functional Analysis as a Special Method in Sociology and Anthropology, Kingsley Davis (1959) argued that the definitions most commonly agreed upon functionalism synonymous with sociological analysis and non-functionalism synonymous with either reductionist theories or pure description. Davis went on to claim that ambiguities of special terminology of functionalism make myth that it is a special method a liability, and that myth should be abandoned. Davis' argument is essentially that of parsimony. If it can be shown that there are no differences between functional analysis and any other kind of analysis in sociology then it is clearly parsimonious to abandon

Muger proved in 2003 that the center of a spherical fusion category C of nonzero dimension over an algebraically closed field is a modular fusion category whose dimension is the square of that of C. We generalize this theorem to a pivotal fusion category C over an arbitrary commutative ring k, without any condition on the dimension of the category. (In this generalized setting, modularity is understood as 2-modularity in the sense of Lyubashenko.) Our proof is based on an explicit description of the Hopf algebra structure of the coend of the center of C. Moreover we show that the dimension of C is invertible in k if and only if any object of the center of C is a retract of a “free” half-braiding. As a consequence, if k is a field, then the center of C is semisimple (as an abelian category) if and only if the dimension of C is nonzero. If in addition k is algebraically closed, then this condition implies that the center is a fusion category, so that we recover Muger’s result.

As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes necessary to deal with epimorphisms and monomorphisms, with various canonical factorizations of arbitrary morphisms, and with the relations of these things to such limit operations as equalizers and pull-backs.

Introduction: We have previously reported on the findings of a critical realist concurrent triangulated mixed method multilevel study that sought to identify and explain complex perinatal contextual social and psychosocial mechanisms that may influence the developmental origins of health and disease. That study used both Emergent and Construction Phases of a realist Explanatory Theory Building Method (1). The purpose here is to present The Thesis, Theoretical Framework, Propositions and Models explaining neighbourhood context, stress, depression and the developmental origins of health and disease. Theory/Methods: The analysis draws on an extensive extant literature; intensive (qualitative), extensive (quantitative) and multilevel studies used for phenomena detection, description, and emergent phase theory development; and the abductive and retroductive analysis undertaken for the theory construction phase. The theory construction methods used included: “1) defining stratified levels; 2) analytic resolution; 3) abductive reasoning; 4) comparative analysis (triangulation); 5) retroduction; 6) postulate and proposition development; 7) comparison and assessment of theories; and 8) conceptual frameworks and model development”. Results: Global, economic, social and cultural mechanisms were identified that explain maternal stress and depression within family and neighbourhood contexts. There is a complex intertwining of historical, spatial, cultural, material and relational elements that contribute to the experiences of loss and nurturing. Emerging are the centrality of social isolation and “expectation lost” as possible triggers of stress and depression not only for mothers but possibly also others who have their dreams shattered during life’s transitions. Discussion: The theoretical framework takes a critical realist perspective of perinatal social context, stress, depression and the developmental origins of health and disease and builds on the emerging literature on stress process, social isolation, social exclusion, social capital, segregation, acculturation, Globalisation, neighbourhood effects on health, perinatal adversity, and the developmental origins of health and disease. We draw on the philosopher Bhaskar’s (2) articulation of critical realism with its ontological stratification of reality. Such a stratified ontological perspective adds theoretical depth to the layered ecological models advanced by earlier social epidemiologists. Conclusions: The Thesis: In the neighbourhood spatial context, in keeping with critical realist ontology, global-economic, social and cultural level generative powers trigger and condition maternal psychological and biological level stress mechanisms resulting in the phenomenon of maternal depression and alteration of the infants’ developmental trajectory. Lessons learned: The meta-theory of critical realism is used here to generate and construct social epidemiological theory using stratified ontology and both abductive and retroductive analysis. The findings will be applied to the development of a middle range theory and subsequent programme theory for local perinatal child and family interventions. Limitations: The stratified levels of analysis in this study were predominantly social and psychological. The macro and meso levels were not fully analysed. Suggestions for future research: We will use the theories developed here for future confirmatory studies and realist programme theory development.

In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3]. In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC• (A) and HP•(A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ ∈ Λ(n,n) is cyclic of order n + 1 is relaxed to the condition that it be invertible. We call this category the paracyclic category. Given a paracyclic module P , we can define a chain complex C(P ), with differentials b and B, which respectively lower and raise degree. The condition that the module P is paracyclic translates to the condition on C(P ) that 1− (bB +Bb) is invertible. Our main result is to show that there is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. It is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of Feigin and Tsygan. If M is a module over a commutative ring k, we will denote by M (k) the iterated tensor product, defined by M (0) = k and M (k+1) =M (k) ⊗M . If M and N are graded modules, we will denote by M ⊗N their graded tensor product. We would like to thank J. Block, J.-L. Brylinski and C. Kassel for a number of interesting discussions on the results presented here.

An optimized algorithm for optimal power flow based on deep learning

Moss’ theorem, which relates Massey products in the $$E_r$$ E r -page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable simplicial model category, we prove a general version of Moss’ theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. This involves the study of Massey products and Toda brackets in a non-strictly associative context. The theorem has broad applications, e.g., to the computation of the motivic slice spectral sequence and other colocalization towers.

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.