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Abstract: The concepts of Booleanization and of smallest dense quotient are explored in the context of d-frames. These two notions coincide for frames, but bitopologically they turn out to be different. The approach followed here is different from that in (Moshier in On Isbell’s density theorem for bitopological pointfree spaces I 273:106962, 2020), in which the smallest dense extremal epimorphism of a d-frame is described. Here, we consider the lattice of all quotients of a d-frame, extremal and nonextremal. Here, it is shown that for corrigible d-frames there is a smallest dense quotient, and it is shown that this coincides with the Booleanization of a d-frame: this is a construction which, as in the frame theoretical case, gives us the Boolean reflection of a d-frame in a suitable subcategory of the category of d-frames. PubDate: 2022-06-01

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Abstract: We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of two new classes of algebras: string almost gentle (SAG) algebras and SUMP algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by Bekkert and Merklen (Algebras Rep Theory 6:285–302, 2003). With this description, we give a necessary and sufficient condition for a given string complex to have infinite minimal projective resolution and we extend this condition for the case of string algebras. Using this characterization we establish a sufficient condition for a string almost gentle algebra (or a string algebra) to have infinite global dimension. PubDate: 2022-06-01

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Abstract: We introduce, comment and develop the Scott adjunction, mostly from the point of view of a category theorist. Besides its technical and conceptual aspects, in a nutshell we provide a categorification of the Scott topology over a posets with directed suprema. From a technical point of view we establish an adjunction between accessible categories with directed colimits and Grothendieck topoi. We show that the bicategory of topoi is enriched over the 2-category of accessible categories with directed colimits and it has tensors with respect to this enrichment. The Scott adjunction (re-)emerges naturally from this observation. PubDate: 2022-06-01

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Abstract: We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated. PubDate: 2022-06-01

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Abstract: A given monoid usually admits many presentations by generators and relations and the notion of Tietze equivalence characterizes when two presentations describe the same monoid: it is the case when one can transform one presentation into the other using the two families of so-called Tietze transformations. The goal of this article is to provide an abstract and geometrical understanding of this well-known fact, by constructing a model structure on the category of presentations, in which two presentations are weakly equivalent when they present the same monoid. We show that Tietze transformations form a pseudo-generating family of trivial cofibrations and give a proof of the completeness of these transformations by an abstract argument in this setting. PubDate: 2022-06-01

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Abstract: Silting theorem gives a generalization of the classical tilting theorem of Brenner and Butler for a 2-term silting complex. In this paper, we give a relative version of a silting theorem for any abelian category which is a finite R-variety over some commutative Artinian ring R. To this end, the notion of relative silting complexes is introduced and it is shown that they play a similar role as silting complexes. It is shown that if \({\mathcal {X}} \) is a subcategory of \({\mathcal {A}} \) which is a dualizing R-variety and \({\mathbf {X}}\in {\mathbb {K}} ^{{\mathrm{b}}}({\mathcal {X}} )\) is a 2-term \({\mathcal {X}} \) -relative silting complex, then there are two torsion pairs, in \({\mathcal {A}} \) and in \({\mathrm{{mod{-}}}}{\mathrm{End}}({\mathbf {X}})^{\mathrm{op}}\) together with a pair of crosswise equivalences between torsion and torsion-free classes. PubDate: 2022-06-01

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Abstract: We describe Dold–Kan correspondence for an idempotent complete additive category \({{\mathscr {A}}}\) . Our approach is based on a family of idempotents in \({\mathbb {Z}}\Delta \) . We represent the obtained normalised complex equivalence of the category of simplicial objects in \({{\mathscr {A}}}\) and the category of non-negatively graded chain complexes in \({{\mathscr {A}}}\) , \(N:s{{\mathscr {A}}}\rightarrow \text {Ch}_{\geqslant 0}({{\mathscr {A}}})\) , as a coend. Explicit formulae for the right adjoint equivalence \(K:\text {Ch}_{\geqslant 0}({{\mathscr {A}}})\rightarrow s{{\mathscr {A}}}\) are obtained. It is shown that the functors N, K preserve the homotopy relation. Similar results are obtained for cosimplicial objects. PubDate: 2022-06-01

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Abstract: \(\mathfrak {KNJ}\) is the category of compact normal joinfit frames and frame homomorphisms and \(\mathfrak {KReg}\) is the coreflective subcategory of compact regular frames. This work investigates \(\mathfrak {KNJ}\) through its interaction with \(\mathfrak {KReg}\) via the coreflection \(\rho \) . A \(\mathfrak {KNJ}\) morphism \(\phi : F \longrightarrow M\) is \(\mathcal {P}\) -essential if \(\phi \) is skeletal and the map between the frames of polars, \(\mathcal {P}(\phi ): \mathcal {P}F \longrightarrow \mathcal {P}M\) defined by \(\mathcal {P}(\phi )(p)=\phi (p)^{\perp \perp }\) , is a boolean isomorphism. The \(\mathcal {P}\) -essential morphisms in \(\mathfrak {KNJ}\) are closely related to the essential embeddings in \(\mathfrak {KReg}\) . We provide a characterization of the \(\mathcal {P}\) -essential morphisms in \(\mathfrak {KNJ}\) and a connection to the essential embeddings in \(\mathfrak {KReg}\) . Further results about the preservation of joinfitness, the factorization of morphisms, and monomorphisms in \(\mathfrak {KNJ}\) are provided. Moreover, in the category of \(\mathfrak {KNJ}\) objects and skeletal frame homomorphisms, \(\mathfrak {KNJS}\) , we construct for \(F \in \mathfrak {KNJ}\) and \(\phi :\rho F \longrightarrow H\) (an arbitrary \(\mathfrak {KReg}\) essential embedding of \(\rho F\) ) the \(\mathfrak {KNJS}\) pushout of \(\rho _F: \rho F \longrightarrow F\) and \(\phi : \rho F \longrightarrow H\) . Lastly, we investigate the epimorphisms and epicomplete objects in \(\mathfrak {KNJS}\) . PubDate: 2022-05-05

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Abstract: Abstract In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category \(\mathcal {V}\) , generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as \(\mathcal {V}\) the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories. PubDate: 2022-05-02

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Abstract: Abstract Let \(\Lambda \) be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in \({{\mathbf {K}}^{b}(\mathrm {proj}\,\Lambda )}\) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes. PubDate: 2022-04-26

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Abstract: A Correction to this paper has been published: 10.1007/s10485-021-09664-8 PubDate: 2022-04-22

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Abstract: Abstract We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. Then, we contextualize the new notion by comparing it to another known generalization of enrichment: that of enrichment for indexed categories. It turns out that the two notions are closely related. PubDate: 2022-04-19

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Abstract: Abstract Let \({p : \mathcal {E}\rightarrow \mathcal S}\) be a hyperconnected geometric morphism. For each X in the ‘gros’ topos \(\mathcal {E}\) , there is a hyperconnected geometric morphism \({p_X : \mathcal {E}/X \rightarrow \mathcal S(X)}\) from the slice over X to the ‘petit’ topos of maps (over X) with discrete fibers. We show that if p is essential then \(p_X\) is essential for every X. The proof involves the idea of collapsing a connected subspace to a ‘basepoint’, as in Algebraic Topology, but formulated in topos-theoretic terms. In case p is local, we characterize when \({p_X}\) is local for every X. This is a very restrictive property, typical of toposes of spaces of dimension \({\le 1}\) . PubDate: 2022-04-08

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Abstract: Abstract We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions. PubDate: 2022-04-08

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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: 2022-04-01 DOI: 10.1007/s10485-021-09654-w

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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: 2022-04-01 DOI: 10.1007/s10485-021-09659-5

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Abstract: Abstract Let \({\mathcal {P}}\) be a property of subobjects relevant in a category \({\mathcal {C}}\) . An object \(X\in {\mathcal {C}}\) is \({\mathcal {P}}\) -separated if the diagonal in \(X\times X\) has \({\mathcal {P}}\) ; thus e.g. closedness in the category of topological spaces (resp. locales) induces the Hausdorff (resp. strong Hausdorff) axiom. In this paper we study the locales (frames) in which the diagonal is fitted (i.e., an intersection of open sublocales—we speak about \({\mathcal {F}}\) -separated locales). Recall that a locale is fit if each of its sublocales is fitted. Since this property is inherited by products and sublocales, fitness implies ( \({\mathcal {F}}\) sep) which is shown to be strictly weaker (one of the results of this paper). We show that ( \({\mathcal {F}}\) sep) is in a parallel with the strong Hausdorff axiom (sH): (1) it is characterized by a Dowker-Strauss type property of the combinatorial structure of the systems of frame homomorphisms \(L\rightarrow M\) (and therefore, in particular, it implies \((T_U)\) for analogous reasons like (sH) does), and (2) in a certain duality with (sH) it is characterized in L by all almost homomorphisms (frame homomorphisms with slightly relaxed join-requirement) \(L\rightarrow M\) being frame homomorphisms (while one has such a characteristic of (sH) with weak homomorphisms, where meet-requirement is relaxed). PubDate: 2022-04-01 DOI: 10.1007/s10485-021-09655-9

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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: 2022-04-01 DOI: 10.1007/s10485-021-09658-6

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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: 2022-04-01 DOI: 10.1007/s10485-021-09660-y

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Abstract: Abstract In this investigation we give a module-theoretic counterpart of the well known De Morgan’s laws for rings and topological spaces. We observe that the module-theoretic De Morgan’s laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan’s laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law. PubDate: 2022-04-01 DOI: 10.1007/s10485-021-09656-8