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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 DOI: 10.1007/s10485-024-09769-w

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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 DOI: 10.1007/s10485-024-09765-0

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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 DOI: 10.1007/s10485-024-09764-1

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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 DOI: 10.1007/s10485-024-09767-y

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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 DOI: 10.1007/s10485-024-09763-2

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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 DOI: 10.1007/s10485-023-09759-4

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Abstract: Abstract Suppose \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of \(\mathcal {C}\) are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of \(\mathcal {C}\) into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from \((\mathcal {C},\mathbb {E},\mathfrak {s})\) to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if \((\mathcal {C},\mathbb {E},\mathfrak {s})\) is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and \((n+2)\) -angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories. PubDate: 2024-02-08 DOI: 10.1007/s10485-023-09758-5

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Abstract: Abstract Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category. Along the way, we also give an interpretation of these structures in the language of extriangulated categories. PubDate: 2024-02-06 DOI: 10.1007/s10485-023-09755-8

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Abstract: Abstract We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures for dynamic machine learning models (including recurrent neural networks, graph neural networks, and diffusion models) can be embedded. Considered in category theory, we also show that these algebraic models are a natural language for describing the compositionality of dynamic models. Furthermore, we propose that these models provide a template for the generalisation of the above dynamic models to learning problems on structured or non-numerical data, including ‘hybrid symbolic-numeric’ models. PubDate: 2024-01-18 DOI: 10.1007/s10485-023-09762-9

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Abstract: Abstract In this paper, we introduce the concepts of generalized continuous posets and present topological dualities for them. Moreover, we show that the category of generalized continuous posets and continuous morphisms is dually equivalent to the category of F-spaces and F-morphisms. In particular, some special cases are obtained, such as the topological representations for posets, domains, continuous lattices and join-semilattices. PubDate: 2024-01-16 DOI: 10.1007/s10485-023-09761-w

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Abstract: Abstract Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking a uniform approach, this paper presents an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly stronger restraints among the replicated copies. The linear compatibility is in Koszul duality to the total compatibility, while the matching compatibility is self dual. Further, each compatibility condition can be expressed in terms of either one or both of the two Manin square products. Finally it is shown that the operads defined by these compatibility conditions from the associative algebra and differential algebra are Koszul utilizing rewriting systems. PubDate: 2024-01-10 DOI: 10.1007/s10485-023-09760-x

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Abstract: Abstract The homology of a Garside monoid, thus of a Garside group, can be computed efficiently through the use of the order complex defined by Dehornoy and Lafont. We construct a categorical generalization of this complex and we give some computational techniques which are useful for reducing computing time. We then use this construction to complete results of Salvetti, Callegaro and Marin regarding the homology of exceptional complex braid groups. We most notably study the case of the Borchardt braid group \(B(G_{31})\) through its associated Garside category. PubDate: 2023-12-12 DOI: 10.1007/s10485-023-09757-6

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Abstract: Abstract Suppose that we have a bicomplete closed symmetric monoidal quasi-abelian category \(\mathcal {E}\) with enough flat projectives, such as the category of complete bornological spaces \({{\textbf {CBorn}}}_k\) or the category of inductive limits of Banach spaces \({{\textbf {IndBan}}}_k\) . Working with monoids in \(\mathcal {E}\) , we can generalise and extend the Koszul duality theory of Beilinson, Ginzburg, Soergel. We use an element-free approach to define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders’ embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of certain subcategories of the derived categories of graded modules over Koszul monoids and their duals. PubDate: 2023-12-06 DOI: 10.1007/s10485-023-09756-7

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Abstract: Abstract We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any pretopology on the original category induces a canonical pretopology of generalised coverings on the free cocompletion. We show that with respect to these pretopologies the homotopy right Kan extension along the Yoneda embedding preserves homotopy sheaves valued in (sufficiently nice) simplicial model categories. Moreover, we show that this induces an equivalence between sheaves of spaces on the original category and colimit-preserving sheaves of spaces on its free cocompletion. We present three applications in geometry and topology: first, we prove that diffeological vector bundles descend along subductions of diffeological spaces. Second, we deduce that various flavours of bundle gerbes with connection satisfy \((\infty ,2)\) -categorical descent. Finally, we investigate smooth diffeomorphism actions in smooth bordism-type field theories on a manifold. We show how these smooth actions allow us to extract the values of a field theory on any object coherently from its values on generating objects of the bordism category. PubDate: 2023-12-04 DOI: 10.1007/s10485-023-09754-9