Abstract: This paper aims at studying the homotopy category of cotorsion flat left modules \({{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}\) over a ring R. We prove that if R is right coherent, then the homotopy category \({\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\) of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair \(({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))\) in the homotopy category \({{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}\) of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence \({\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}\) of triangulated categories where \({{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}\) is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of \({{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}\) over right coherent R. Also we get the unbounded derived category \({\mathbb {D}} (R)\) of R as a Verdier quotient of \({\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\) . PubDate: 2021-04-01
Abstract: We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite étale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter. PubDate: 2021-04-01
Abstract: The fact that equalizers in the context of strongly Hausdorff locales (similarly like those in classical spaces) are closed is a special case of a standard categorical fact connecting diagonals with general equalizers. In this paper we analyze this and related phenomena in the category of locales. Here the mechanism of pullbacks connecting equalizers is based on natural preimages that preserve a number of properties (closedness, openness, fittedness, complementedness, etc.). Also, we have a new simple and transparent formula for equalizers in this category providing very easy proofs for some facts (including the general behavior of diagonals). In particular we discuss some aspects of the closed case (strong Hausdorff property), and the open and clopen one. PubDate: 2021-04-01
Abstract: Working in an arbitrary category endowed with a fixed \(({\mathcal {E}}, {\mathcal {M}})\) -factorization system such that \({\mathcal {M}}\) is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i. Some basic properties of these morphisms are discussed. In particular, it is shown that i-codenseness is preserved under both images and dual images under morphisms in \({\mathcal {M}}\) and \({\mathcal {E}}\) , respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i. Notably, we obtain a characterization of quasi i-open morphisms in terms of i-codense subobjects. Furthermore, we prove that these morphisms are a generalization of the i-open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided. PubDate: 2021-04-01
Abstract: The Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl’s framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl’s concrete positive TFT based on fold maps into the plane. PubDate: 2021-04-01
Abstract: We focus on the transfer of some known orthogonal factorization systems from \(\mathsf {Cat}\) to the 2-category \({\mathsf {Fib}}(B)\) of fibrations over a fixed base category B: the internal version of the comprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in \({\mathsf {Fib}}(B)\) , the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in \(\mathsf {Cat}\) , providing suitable conditions on a 2-category \({\mathcal {C}}\) , that allow the transfer of the construction of coinverters and coidentifiers from \({\mathcal {C}}\) to \({\mathsf {Fib}}_{{\mathcal {C}}}(B)\) . PubDate: 2021-04-01
Abstract: In the last few years, López-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers. PubDate: 2021-03-11
Abstract: We show that free objects on sets do not exist in the category \({\varvec{ba}}\varvec{\ell }\) of bounded archimedean \(\ell \) -algebras. On the other hand, we introduce the category of weighted sets and prove that free objects on weighted sets do exist in \({\varvec{ba}}\varvec{\ell }\) . We conclude by discussing several consequences of this result. PubDate: 2021-03-09
Abstract: In this article, we study model structures on the category of finite graphs with \(\times \) -homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Strøm-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with \(\times \) -homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions. PubDate: 2021-03-04
Abstract: A map that is well known in C(X), dating back to the work of L. Gillman, M. Henriksen and M. Jerison in 1954, is here used to construct a morphism of quantales, whose right adjoint we show to be the map that sends a subset A of \(\beta X\) to the ideal $$\begin{aligned} {\varvec{O}}^A =\{f\in C(X)\mid A\subseteq {{\,\mathrm{int}\,}}_{\beta X}{{\,\mathrm{cl}\,}}_{\beta X}Z(f)\} \end{aligned}$$ of C(X) if and only if X is a P-space. We show that when viewed this way, this map characterizes R.G. Woods’ WN-maps in terms of commutativity of a certain diagram in the category of quantales. All this is achieved most economically by working with locales instead of topological spaces. The Lindelöf reflection allows us to present “countable” analogues of the results just mentioned and we highlight the similarities and disparities. PubDate: 2021-03-03
Abstract: Let \(Q \rightarrow R\) be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as a Q-bimodule admits a finite resolution by modules which are projective on both sides. We define an adjoint pair of functors between the homotopy category of totally acyclic R-complexes and that of Q-complexes. This adjoint pair is analogous to the classical adjoint pair of functors between the module categories of R and Q. As a consequence, we obtain a precise notion of approximations of totally acyclic R-complexes by totally acyclic Q-complexes. PubDate: 2021-02-20
Abstract: The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras (bialgebras, Hopf algebras) in categories of topological vector spaces over a nonarchimedean field K. In particular, our results imply reconstruction and recognition theorems for categories of locally analytic representations of compact p-adic groups, which was the major motivation for this work. PubDate: 2021-02-20
Abstract: Thomas Streicher asked on the category theory mailing list whether every essential, hyperconnected, local geometric morphism is automatically locally connected. We show that this is not the case, by providing a counterexample. PubDate: 2021-02-15
Abstract: Let \(\mathscr {C}\) be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories \(\mathscr {T}\) and \(\mathscr {U}\) . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to \(\mathscr {T}\) provides an isomorphism between the split Grothendieck groups of \(\mathscr {U}\) and \(\mathscr {T}\) . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of \((d+2)\) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the \((d+2)\) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category. PubDate: 2021-02-06 DOI: 10.1007/s10485-020-09625-7
Abstract: We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger–Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer–Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block–Smith’s higher Riemann–Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces. PubDate: 2021-02-06 DOI: 10.1007/s10485-021-09631-3
Abstract: Taking advantage of the quantale-theoretic description of étale groupoids we study principal bundles, Hilsum–Skandalis maps, and Morita equivalence by means of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras. PubDate: 2021-02-01 DOI: 10.1007/s10485-021-09628-y
Abstract: The Kechris–Pestov–Todorčević correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a “direct” result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us. PubDate: 2021-02-01 DOI: 10.1007/s10485-020-09611-z
Abstract: A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentially algebraic theory. PubDate: 2021-01-02 DOI: 10.1007/s10485-020-09617-7
Abstract: We discuss Peter Freyd’s universal way of equipping an additive category \(\mathbf {P}\) with cokernels from a constructive point of view. The so-called Freyd category \(\mathcal {A}(\mathbf {P})\) is abelian if and only if \(\mathbf {P}\) has weak kernels. Moreover, \(\mathcal {A}(\mathbf {P})\) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems \(X \cdot \alpha = \beta \) for morphisms \(\alpha \) and \(\beta \) in \(\mathbf {P}\) . We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for \(\mathbf {P}\) that helps solving linear systems in \(\mathbf {P}\) and even in the iterated Freyd category construction \(\mathcal {A}( \mathcal {A}(\mathbf {P})^{\mathrm {op}} )\) , which can be identified with the category of finitely presented covariant functors on \(\mathcal {A}(\mathbf {P})\) . The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra. PubDate: 2020-10-13 DOI: 10.1007/s10485-020-09612-y
Abstract: Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality. PubDate: 2020-10-10 DOI: 10.1007/s10485-020-09609-7
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