Abstract: Abstract Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of “connection” as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set. PubDate: 2021-01-08

Abstract: Abstract Linkage of ideals is a very well-studied topic in algebra. It has lead to the development of module linkage which looks to extend the ideas and results of the former. Although linkage has been used extensively to find many interesting and impactful results, it has only been extended to schemes and modules. This paper builds a framework in which to perform linkage from a categorical perspective. This allows a generalization of many theories of linkage including complete intersection ideal linkage, Gorenstein ideal linkage, linkage of schemes and module linkage. Moreover, this construction brings together many different robust fields of homological algebra including linkage, homological dimensions, and duality. After defining linkage and showing results concerning linkage directly, we explore the connection between linkage, homological dimensions, and duality. Applications of this new framework are sprinkled throughout the paper investigating topics including module linkage, horizontal linkage, module theoretic invariants, and Auslander and Bass classes. PubDate: 2021-01-07

Abstract: Abstract We define the Grothendieck group of an n-exangulated category. For n odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete subcategories of an n-exangulated category with an n-(co)generator in terms of subgroups of the Grothendieck group. This unifies and extends results of Thomason, Bergh–Thaule, Matsui and Zhu–Zhuang for triangulated, \((n+2)\) -angulated, exact and extriangulated categories, respectively. We also introduce the notion of an n-exangulated subcategory and prove that the subcategories in our classification theorem carry this structure. PubDate: 2021-01-06

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

Abstract: Abstract We systematically investigate, for a monoid M, how topos-theoretic properties of \({{\,\mathrm{\mathbf {PSh}}\,}}(M)\) , including the properties of being atomic, strongly compact, local, totally connected or cohesive, correspond to semigroup-theoretic properties of M. PubDate: 2020-12-03

Abstract: 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: 2020-12-02

Abstract: Abstract A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame \({\mathrm{Filt}}_{{\textsf {E}}}(L)\) somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters \({\mathrm{Filt}}_{{\textsf {sE}}}(L)\) . PubDate: 2020-12-01

Abstract: Abstract We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable. PubDate: 2020-12-01

Abstract: 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: 2020-11-19

Abstract: 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: 2020-11-19

Abstract: 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: 2020-11-04

Abstract: 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: 2020-10-26

Abstract: 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: 2020-10-24

Abstract: 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

Abstract: 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

Abstract: Abstract As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of \(\textsf {Set}\) -functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider “powerset-like” functors based on the Hausdorff \({\mathcal {V}}\) -category structure. As a starting point, we show that for a lifting of a \(\textsf {Set}\) -functor to a topological category \(\textsf {X}\) over \(\textsf {Set}\) that commutes with the forgetful functor, the corresponding category of coalgebras over \(\textsf {X}\) is topological over the category of coalgebras over \(\textsf {Set}\) and, therefore, it is “as complete” but cannot be “more complete”. Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these “negative” results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate “powerset-like” functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of “Kripke polynomial” functors are (co)complete. PubDate: 2020-10-01

Abstract: Abstract We provide a direct and elementary proof of the fact that the category of Nachbin’s compact ordered spaces is dually equivalent to an \(\aleph _1\) -ary variety of algebras. Further, we show that \(\aleph _1\) is a sharp bound: compact ordered spaces are not dually equivalent to any \(\mathrm{SP}\) -class of finitary algebras. PubDate: 2020-08-20

Abstract: Abstract Let \(\kappa \) be a regular cardinal. We study Gabriel–Ulmer duality when one restricts the 2-category of locally \(\kappa \) -presentable categories with \(\kappa \) -accessible right adjoints to its locally full sub-2-category of \(\kappa \) -presentable Grothendieck topoi with geometric \(\kappa \) -accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of \(\kappa \) -small cocomplete categories with \(\kappa \) -small colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally \(\kappa \) -presentable Grothendieck topoi with geometric \(\kappa \) -accessible morphisms is a reflective sub-bicategory of the 2-category of weakly \(\kappa \) -ary sites [in the sense of Shulman (Theory Appl Categ 27:97–173, 2012)] with morphisms of sites. PubDate: 2020-08-20

Abstract: Abstract We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid A, which we denote \(\mathcal {F}(A)\) . The objects of \(\mathcal {F}(A)\) are factorizations of elements of A, and the morphisms in \(\mathcal {F}(A)\) encode combinatorial similarities and differences between the factorizations. We pay particular attention to the divisibility pre-order and to the monoid \(A=D{\setminus }\{0\}\) where D is an integral domain. Among other results, we show that \(\mathcal {F}(A)\) is a symmetric and strict monoidal category with weak equivalences and compute the associated category of fractions obtained by inverting the weak equivalences. Also, we use this construction to characterize various factorization properties of integral domains: atomicity, unique factorization, and so on. PubDate: 2020-08-17

Abstract: Abstract In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of \(T_0\) -spaces. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints completely generate the lattice. It is shown that there is a dual adjunction between the category of topological spaces and the category of Raney algebras that restricts to a dual equivalence between \(T_0\) -spaces and Raney algebras. The underlying idea is to take the lattice of upsets of the specialization order with the restriction of the interior operator of a space as the Raney algebra associated to a topological space. Further properties of topological spaces are explored in the dual setting of Raney algebras. Spaces that are \(T_1\) correspond to Raney algebras whose underlying lattices are Boolean, and Alexandroff \(T_0\) -spaces correspond to Raney algebras whose interior operator is the identity. Algebraic description of sober spaces results in algebraic considerations that lead to a generalization of sober that lies strictly between \(T_0\) and sober. PubDate: 2020-08-09