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: 2021-06-01
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: 2021-06-01
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: 2021-06-01
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: 2021-06-01
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: 2021-06-01
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: 2021-06-01
Abstract: We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus. PubDate: 2021-05-05
Abstract: In this work, given two crossed modules \(\mathcal {M=}\left( \mu :\mathrm {M} \rightarrow \mathrm {A}\right) \) and \({\mathcal {N}}=\left( \eta :\mathrm {N} \rightarrow \mathrm {B}\right) \) of R-algebroids and a crossed module morphism \(f:{\mathcal {M}}\rightarrow {\mathcal {N}}\) , we introduce an f-derivation as an ordered pair \(H=\left( H_{1},H_{0}\right) \) of maps \(H_{1}: \mathrm {Mor}\left( \mathrm {A}\right) \rightarrow \mathrm {Mor}\left( \mathrm {N }\right) \) and \(H_{0}:\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {B} \right) \) which are subject to satisfy certain axioms and show that f and H determine a crossed module morphism \(g:{\mathcal {M}}\rightarrow \mathcal { N}\) . Then calling such a pair \(\left( H,f\right) \) a homotopy from f to g we prove that there exists a groupoid structure of which objects are crossed module morphisms from \({\mathcal {M}}\) to \({\mathcal {N}} \) and morphisms are homotopies between crossed module morphisms. Moreover, given two crossed module morphisms \(f,g:{\mathcal {M}}\rightarrow {\mathcal {N}}\) , we introduce an fg-map as a map \(\varLambda :\mathrm {A}_{0}\rightarrow \mathrm {Mor}\left( \mathrm {N}\right) \) subject to some conditions and then show that \(\varLambda \) determines for each homotopy \(\left( H,f\right) \) from f to g a homotopy \(\left( H^{\prime },f\right) \) from f to g. Furthermore, calling such a pair \(\left( \varLambda ,\left( H,f\right) \right) \) a 2-fold homotopy from \(\left( H,f\right) \) to \(\left( H^{\prime },f\right) \) we prove that the groupoid structure constructed by crossed module morphisms from \({\mathcal {M}}\) to \({\mathcal {N}}\) and homotopies between them is upgraded by 2-fold homotopies to a 2-groupoid structure. Besides, in order to see reduced versions of all general constructions mentioned, we examine homotopies of crossed modules of associative R-algebras, as a pre-stage. PubDate: 2021-04-23
Abstract: Let k be a field. We show that locally presentable, k-linear categories \({\mathcal {C}}\) dualizable in the sense that the identity functor can be recovered as \(\coprod _i x_i\otimes f_i\) for objects \(x_i\in {\mathcal {C}}\) and left adjoints \(f_i\) from \({\mathcal {C}}\) to \(\mathrm {Vect}_k\) are products of copies of \(\mathrm {Vect}_k\) . This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III. PubDate: 2021-04-22
Abstract: Let \(H_1\) and \(H_2\) be Hopf algebras which are not necessarily finite dimensional and \(\alpha ,\beta \in Aut_{Hopf}(H_1),\gamma ,\delta \in Aut_{Hopf}(H_2)\) . In this paper, we introduce a category \(_{H_1}\mathcal {LR}_{H_2}(\alpha ,\beta ,\gamma ,\delta )\) , generalizing Yetter–Drinfeld–Long bimodules and construct a braided T-category \(\mathcal {LR}(H_1,H_2)\) containing all the categories \(_{H_1}\mathcal {LR}_{H_2}(\alpha ,\beta ,\gamma ,\delta )\) as components. We also prove that if \((\alpha ,\beta ,\gamma ,\delta )\) admits a quadruple in involution, then \(_{H_1}\mathcal {LR}_{H_2}(\alpha ,\beta ,\gamma ,\delta )\) is isomorphic to the usual category \(_{H_1}\mathcal {LR}_{H_2}\) of Yetter–Drinfeld–Long bimodules. PubDate: 2021-04-21
Abstract: In a recent article, the authors established an explicit description of kernels in the category of the formal group laws over the ring of Witt vectors over a finite field in terms of Fontaine’s triples. The present research is devoted to an adjacent problem of explicit description of cokernels. The technique developed is applied to a natural monomorphism from \(F_m\) to the Weil restriction of \(F_m\) with respect to certain ring extensions. Besides, we investigate some properties of the category of formal group laws over the ring of Witt vectors such as left and right integrability and left and right semi-abelianity. PubDate: 2021-04-21
Abstract: Let \({\mathcal {C}}\) be an n-angulated category. We prove that its idempotent completion \(\widetilde{{\mathcal {C}}}\) admits a unique n-angulated structure such that the canonical functor \(\iota : {\mathcal {C}}\rightarrow \widetilde{{\mathcal {C}}}\) is n-angulated. Moreover, the functor \(\iota \) induces an equivalence \(Hom _{n-ang }(\widetilde{{\mathcal {C}}},{\mathcal {D}})\cong Hom _{n-ang }({\mathcal {C}},{\mathcal {D}})\) for any idempotent complete n-angulated category \({\mathcal {D}}\) , where \(Hom _{n-ang }\) denotes the category of n-angulated functors. PubDate: 2021-04-13
Abstract: We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author’s classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abelian category. PubDate: 2021-04-04
Abstract: Let \(V_*\otimes V\rightarrow {\mathbb {C}}\) be a non-degenerate pairing of countable-dimensional complex vector spaces V and \(V_*\) . The Mackey Lie algebra \({\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)\) corresponding to this pairing consists of all endomorphisms \(\varphi \) of V for which the space \(V_*\) is stable under the dual endomorphism \(\varphi ^*: V^*\rightarrow V^*\) . We study the tensor Grothendieck category \({\mathbb {T}}\) generated by the \({\mathfrak {g}}\) -modules V, \(V_*\) and their algebraic duals \(V^*\) and \(V^*_*\) . The category \({{\mathbb {T}}}\) is an analogue of categories considered in prior literature, the main difference being that the trivial module \({\mathbb {C}}\) is no longer injective in \({\mathbb {T}}\) . We describe the injective hull I of \({\mathbb {C}}\) in \({\mathbb {T}}\) , and show that the category \({\mathbb {T}}\) is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category \(_I{\mathbb {T}}\) of objects in \({\mathbb {T}}\) which are free as I-modules. Our main result is that the category \({}_I{\mathbb {T}}\) is also Koszul, and moreover that \({}_I{\mathbb {T}}\) is universal among abelian \({\mathbb {C}}\) -linear tensor categories generated by two objects X, Y with fixed subobjects \(X'\hookrightarrow X\) , \(Y'\hookrightarrow Y\) and a pairing \(X\otimes Y\rightarrow {\mathbf{1 }}\) where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories \({\mathbb {T}}\) and PubDate: 2021-04-03
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 DOI: 10.1007/s10485-020-09613-x
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 DOI: 10.1007/s10485-020-09618-6
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 DOI: 10.1007/s10485-020-09616-8
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 DOI: 10.1007/s10485-020-09614-w
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 DOI: 10.1007/s10485-020-09619-5
Abstract: In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the relative group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over vec. We prove that the latter two groups are isomorphic. PubDate: 2021-03-18
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