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Abstract: 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: 2021-12-01

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Abstract: 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: 2021-12-01

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Abstract: Abstract We provide a new method to compute the Balmer spectra of the bounded derived category and the singularity category of the category algebra of a finite EI category by a decomposition trick due to Stevenson. In particular, we reobtain the result on the singularity category given by Wang under a weaker condition. PubDate: 2021-12-01

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Abstract: Abstract The categorical framework for our axioms of quantale-enriched topologies is the theory of modules in the monoidal category \(\textsf {Sup}\) and its free right modules generated by power sets. To express the intersection axiom we introduce the structure of a quasi-magma on a quantale. By selecting appropriate quantales and their corresponding quasi-magmas, we show that some well-established mathematical structures become quantale-enriched topologies. These include, among others, the closed left ideal lattices of non-commutative \(C^*\) algebras, lower regular function frames of approach spaces as well as quantale-valued topological spaces. PubDate: 2021-12-01

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Abstract: 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: 2021-12-01

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Abstract: Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area. PubDate: 2021-12-01

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Abstract: 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: 2021-12-01

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Abstract: Abstract We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, and we examine some examples. PubDate: 2021-10-16

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Abstract: Abstract This paper is a coalgebra version of Positselski (Rendiconti Seminario Matematico Univ. Padova 143: 153–225, 2020) and a sequel to Positselski (Algebras and Represent Theory 21(4):737–767, 2018). We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras \({\mathcal {C}}\) and \({\mathcal {D}}\) . For any such complex \({\mathcal {L}}^{\scriptstyle \bullet }\) , we construct a triangulated category endowed with a pair of (possibly degenerate) t-structures of the derived type, whose hearts are the abelian categories of left \({\mathcal {C}}\) -comodules and left \({\mathcal {D}}\) -contramodules. A weak version of pseudo-derived categories arising out of (co)resolving subcategories in abelian/exact categories with enough homotopy adjusted complexes is also considered. Quasi-finiteness conditions for coalgebras, comodules, and contramodules are discussed as a preliminary material. PubDate: 2021-10-15

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Abstract: 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: 2021-10-01 DOI: 10.1007/s10485-021-09635-z

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Abstract: 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: 2021-10-01 DOI: 10.1007/s10485-021-09638-w

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Abstract: 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 \({}_I{\mathbb {T}}\) . PubDate: 2021-10-01 DOI: 10.1007/s10485-021-09640-2

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Abstract: This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical systems may, in turn, be viewed as determining systems of differential equations. The unique map from the curve object to a dynamical system is a solution of the system, and a dynamical system is said to be complete when for all initial conditions there is a solution. A subtle issue concerns the question of when a dynamical system is complete, and the paper provides abstract conditions for this. This abstract formulation also allows new perspectives on topics such as commutative vector fields and flows. In addition, the stronger notion of a differential curve object, which is the centrepiece of the last section of the paper, has exponential maps and forms a differential exponential rig. This rig then, somewhat surprisingly, has an action on every differential object and bundle in the setting. In this manner, in a very strong sense, such a curve object plays the role of the real numbers in standard differential geometry. PubDate: 2021-10-01 DOI: 10.1007/s10485-021-09629-x

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Abstract: Abstract We prove that any right Quillen functor between arbitrary model categories admits non trivial functorial factorizations that are similar to those of a model structure. Given a monad, operad or a PROP(erad) \(\mathcal {O}\) , if we apply one of the factorizations to the forgetful functor \(\textsf {U}: \mathcal {O}{\text {-Alg}}(\textsf {M}) \longrightarrow \textsf {M}\) , we extend the theory of Quillen–Segal \(\mathcal {O}\) -algebras initiated in Bacard (Higher Struct 4(1):57–114, 2020), without the hypothesis of \(\textsf {M}\) being a combinatorial model category. PubDate: 2021-10-01 DOI: 10.1007/s10485-021-09636-y

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Abstract: Abstract This article is part of a series with the aim of classifying all non-hyperoctahedral categories of two-colored partitions. Those constitute by some Tannaka-Krein type result the representation categories of a specific class of quantum groups. In Part I we introduced a class of parameters which gave rise to many new non-hyperoctahedral categories of partitions. In the present article we show that this class actually contains all possible parameter values of all non-hyperoctahedral categories of partitions. This is an important step towards the classification of all non-hyperoctahedral categories. PubDate: 2021-10-01 DOI: 10.1007/s10485-021-09641-1

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Abstract: 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: 2021-10-01 DOI: 10.1007/s10485-021-09637-x

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Abstract: Abstract We specialise a recently introduced notion of generalised dinaturality for functors \(T : (\mathcal {C}^\mathsf {op})^p \times \mathcal {C}^q \rightarrow \mathcal {D}\) to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein (p, q)-ends (resp., (p, q)-coends). While higher arity co/ends are particular instances of ‘totally symmetrised’ (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory. The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends. The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends. PubDate: 2021-08-30 DOI: 10.1007/s10485-021-09653-x

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Abstract: Abstract Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category Tych of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, KHaus and Tych. PubDate: 2021-08-06 DOI: 10.1007/s10485-021-09658-6

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Abstract: Abstract It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions constructed from suitable associated two-variable adjunctions. Representation theorems are established for fixed points of these adjunctions. PubDate: 2021-08-02 DOI: 10.1007/s10485-021-09654-w

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Abstract: 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: 2021-08-01 DOI: 10.1007/s10485-020-09626-6