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Abstract: Abstract \(\mathfrak {KNJ}\) is the category of compact normal joinfit frames and frame homomorphisms. \(\mathcal {P}F\) is the complete boolean algebra of polars of the frame F. A function \(\mathfrak {X}\) that assigns to each \(F \in \mathfrak {KNJ}\) a subalgebra \(\mathfrak {X}(F)\) of \(\mathcal {P}F\) that contains the complemented elements of F is a polar function. A polar function \(\mathfrak {X}\) is invariant (resp., functorial) if whenever \(\phi : F \longrightarrow H \in \mathfrak {KNJ}\) is \(\mathcal {P}\) -essential (resp., skeletal) and \(p \in \mathfrak {X}(F)\) , then \(\phi (p)^{\perp \perp } \in \mathfrak {X}(H)\) . \(\phi : F \longrightarrow H \in \mathfrak {KNJ}\) is \(\mathfrak {X}\) -splitting if \(\phi \) is \(\mathcal {P}\) -essential and whenever \(p \in \mathfrak {X}(F)\) , then \(\phi (p)^{\perp \perp }\) is complemented in H. \(F \in \mathfrak {KNJ}\) is \(\mathfrak {X}\) -projectable means that every \(p \in \mathfrak {X}(F)\) is complemented. For a polar function \(\mathfrak {X}\) and \(F \in \mathfrak {KNJ}\) , we construct the least \(\mathfrak {X}\) -splitting frame of F. Moreover, we prove that if \(\mathfrak {X}\) is a functorial polar function, then the class of \(\mathfrak {X}\) -projectable frames is a \(\mathcal {P}\) -essential monoreflective subcategory of \(\mathfrak {KNJS}\) , the category of \(\mathfrak {KNJ}\) -objects and skeletal maps (the case \(\mathfrak {X}= \mathcal {P}\) is the result from Martínez and Zenk, which states that the class of strongly projectable PubDate: 2024-08-23

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Abstract: Abstract Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, \(\mathscr {P}\) -affine schemes can be defined as objects of the opposite of the category of algebras over an operad \(\mathscr {P}\) . An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad \(\mathscr {P}^{(A)}\) over a \(\mathscr {P}\) -affine scheme A is precisely the slice tangent category over A of \(\mathscr {P}\) -affine schemes. We are going to employ this result to show that differential bundles over a \(\mathscr {P}\) -affine scheme A are precisely A-modules in the operadic sense. PubDate: 2024-08-19

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Abstract: Abstract In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories. PubDate: 2024-08-13

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Abstract: Abstract We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient \(A \rightarrow B\) satisfying some finiteness conditions, the Frobenius tensor category \({\mathcal {D}}\) of graded B-comodules with its stable model structure induces a monoidal model structure on \({\mathcal {C}}\) . We consider the corresponding homotopy quotient \(\gamma : {\mathcal {C}} \rightarrow Ho {\mathcal {C}}\) and the induced quotient \({\mathcal {T}} \rightarrow Ho {\mathcal {T}}\) for the tensor category \({\mathcal {T}}\) of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in \(Ho {\mathcal {T}}\) . We apply these results in the Rep(GL(m n))-case and study its homotopy category \(Ho {\mathcal {T}}\) associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of \(Ho{\mathcal {T}}\) by the negligible morphisms is again the representation category of a supergroup scheme. PubDate: 2024-08-12

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Abstract: Abstract We construct a category \({\textrm{HomCob}}\) whose objects are homotopically 1-finitely generated topological spaces, and whose morphisms are cofibrant cospans. Given a manifold submanifold pair (M, A), we prove that there exists functors into \({\textrm{HomCob}}\) from the full subgroupoid of the mapping class groupoid \(\textrm{MCG}_{M}^{A}\) , and from the full subgroupoid of the motion groupoid \(\textrm{Mot}_{M}^{A}\) , whose objects are homotopically 1-finitely generated. We also construct a family of functors \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}\) , one for each finite group G. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf–Witten. Given a space X, we prove that \({\textsf{Z}}_G(X)\) can be expressed as the \({\mathbb {C}}\) -vector space with basis natural transformation classes of maps from \(\pi (X,X_0)\) to G for some finite representative set of points \(X_0\subset X\) , demonstrating that \({\textsf{Z}}_G\) is explicitly calculable. PubDate: 2024-07-31

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Abstract: Abstract This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist. PubDate: 2024-07-30

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Abstract: Abstract A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications. PubDate: 2024-07-26

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Abstract: Abstract Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the \(\ell ^1\) -semi-norm. However, the \(\ell ^1\) -semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms. PubDate: 2024-07-19

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Abstract: Abstract We develop a hierarchy of semilattice bases (S-bases) for frames. For a given (unbounded) meet-semilattice A, we analyze the interval in the coframe of sublocales of the frame of downsets of A formed by all frames with the S-base A. We study various degrees of completeness of A, which generalize the concepts of extremally disconnected and basically disconnected frames. We introduce the concepts of D-bases and L-bases, as well as their bounded counterparts, and show how our results specialize and sharpen in these cases. Classic examples that are covered by our approach include zero-dimensional, completely regular, and coherent frames, allowing us to provide a new perspective on these well-studied classes of frames, as well as their spatial counterparts. PubDate: 2024-07-10

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Abstract: Abstract The Cohen–Macaulay Auslander algebra of an algebra A is defined as the endomorphism algebra of the direct sum of all indecomposable Gorenstein projective A-modules. The Cohen–Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Moreover, it is shown that a class of special string algebras, which are called to be string algebras satisfying the G-condition, are representation-finite if and only if their Cohen–Macaulay Auslander algebras are representation-finite. As applications, it is proved that the derived representation type of gentle algebras coincide with their Cohen–Macaulay Auslander algebras. PubDate: 2024-07-09

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Abstract: Abstract We treat the problem of lifting bicategories into double categories through categories of vertical morphisms. We consider structures on decorated 2-categories allowing us to formally implement arguments of sliding certain squares along vertical subdivisions in double categories. We call these structures \(\pi _2\) -indexings. We present a construction associating, to every \(\pi _2\) -indexing on a decorated 2-category, a length 1 double internalization. PubDate: 2024-06-22

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Abstract: Abstract In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory. PubDate: 2024-06-07

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Abstract: Abstract (Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part. PubDate: 2024-06-04

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Abstract: Abstract We prove that the 2-category of action Lie groupoids localised in the following three different ways yield equivalent bicategories: localising at equivariant weak equivalences à la Pronk, localising using surjective submersive equivariant weak equivalences and anafunctors à la Roberts, and localising at all weak equivalences. These constructions generalise the known case of representable orbifold groupoids. We also show that any weak equivalence between action Lie groupoids is isomorphic to the composition of two particularly nice forms of equivariant weak equivalences. PubDate: 2024-05-24

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Abstract: Abstract It is well-known that DG-enhancements of the unbounded derived category \({\text {D}}_{qc}(X)\) of quasi-coherent sheaves on a scheme X are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf \(\mathcal {F}\) on a finite-dimensional Noetherian separated scheme (even if \(\mathcal {F}\) does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of \(\mathcal {F}\) . PubDate: 2024-05-08

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Abstract: Abstract We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of codescent objects. After prerequisites on pseudomonads and their pseudo-algebras, we give a 2-dimensional Linton theorem reducing bicocompleteness of 2-categories of pseudo-algebras to existence of bicoequalizers of codescent objects. Finally we prove this condition to be fulfilled in the case of a bifinitary pseudomonad, ensuring bicocompleteness. PubDate: 2024-04-09

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Abstract: Abstract We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules. PubDate: 2024-04-05

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Abstract: Abstract In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let k be a field of characteristic zero and \({\mathscr {S}}_{k}\) be a strongly locally noetherian k-linear Grothendieck category. For a commutative noetherian k-algebra R, let \({\mathscr {S}}_R\) denote the category of R-objects in \({\mathscr {S}}_k\) obtained through a non-commutative base change by R of the abelian category \({\mathscr {S}}_{k}\) . First, we establish Grothendieck’s Vanishing Theorem for any object \({\mathscr {M}}\) in \({\mathscr {S}}_{R}\) . Further, if R is local and \({\mathscr {S}}_{k}\) is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object \({\mathscr {M}}\) in \({\mathscr {S}}_R\) . PubDate: 2024-04-04

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Abstract: Abstract Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system—which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit. PubDate: 2024-03-08

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Abstract: Abstract We will show that the Morrison–Walker blob complex appearing in Topological Quantum Field Theory is an operadic bar resolution of a certain operad composed of fields and local relations. As a by-product we develop the theory of unary operadic categories and study some novel and interesting phenomena arising in this context. PubDate: 2024-02-08