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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

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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

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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

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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

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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

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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

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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

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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

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Abstract: Abstract We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring. PubDate: 2023-12-02 DOI: 10.1007/s10485-023-09752-x

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Abstract: Abstract We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality. PubDate: 2023-11-27 DOI: 10.1007/s10485-023-09747-8

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Abstract: Abstract \({3/2}\) -Institutions have been introduced as an extension of institution theory that accommodates implicitly partiality of the signature morphisms together with its syntactic and semantic effects. In this paper we show that ordinary institutions that are equipped with an inclusion system for their categories of signatures generate naturally \({3/2}\) -institutions with explicit partiality for their signature morphisms. This provides a general uniform way to build \({3/2}\) -institutions for the foundations of conceptual blending and software evolution. Moreover our general construction allows for an uniform derivation of some useful technical properties. PubDate: 2023-11-15 DOI: 10.1007/s10485-023-09753-w

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Abstract: Abstract We prove that for any small category \({\mathcal {C}}\) , the category \(\textbf{KHausLoc}_{\hat{{\mathcal {C}}}}\) of compact Hausdorff locales in the presheaf topos \(\hat{{\mathcal {C}}}\) , is equivalent to the category of functors \({\mathcal {C}} \rightarrow \textbf{KHausLoc}\) . PubDate: 2023-10-19 DOI: 10.1007/s10485-023-09751-y

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Abstract: Abstract We introduce continuous analogues of Nakayama algebras. In particular, we introduce the notion of (pre-)Kupisch functions, which play a role as Kupisch series of Nakayama algebras, and view a continuous Nakayama representation as a special type of representation of \({\mathbb {R}}\) or \({\mathbb {S}}^1\) . We investigate equivalences and connectedness of the categories of Nakayama representations. Specifically, we prove that orientation-preserving homeomorphisms on \({\mathbb {R}}\) and on \({\mathbb {S}}^1\) induce equivalences between these categories. Connectedness is characterized by a special type of points called separation points determined by (pre-)Kupisch functions. We also construct an exact embedding from the category of finite-dimensional representations for any finite-dimensional Nakayama algebra, to a category of continuous Nakayama representaitons. PubDate: 2023-10-03 DOI: 10.1007/s10485-023-09748-7

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Abstract: Abstract A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of \(T_0\) complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr’s characterizations of sober and \(T_D\) topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions. PubDate: 2023-09-30 DOI: 10.1007/s10485-023-09749-6

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Abstract: Abstract Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to \(\textbf{Rel}\) . PubDate: 2023-09-28 DOI: 10.1007/s10485-023-09750-z

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Abstract: Abstract We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit colimits. In particular, we settle in the negative the question of whether the A-homotopy equivalences in the category of graphs are part of a model structure. PubDate: 2023-09-27 DOI: 10.1007/s10485-023-09746-9

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Abstract: Abstract The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a \(\mathscr {V}\) -category \(\mathscr {C}\) in terms of parametrized \(\mathscr {J}\) -ary operations and diagrammatic equations for a suitable subcategory of arities \(\mathscr {J}\hookrightarrow \mathscr {C}\) . On this basis we introduce the notions of diagrammatic \(\mathscr {J}\) -presentation and \(\mathscr {J}\) -ary variety, and we show that the category of \(\mathscr {J}\) -ary varieties is dually equivalent to the category of \(\mathscr {J}\) -ary \(\mathscr {V}\) -monads. We establish several examples of diagrammatic \(\mathscr {J}\) -presentations and \(\mathscr {J}\) -ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic \(\mathscr {J}\) -presentations. We show that both \(\mathscr {J}\) -relative monads and \(\mathscr {J}\) -pretheories give rise to diagrammatic \(\mathscr {J}\) -presentations that directly describe their algebras. Using diagrammatic \(\mathscr {J}\) -presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting. PubDate: 2023-09-22 DOI: 10.1007/s10485-023-09735-y

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Abstract: Abstract The category of internal groupoids (in an arbitrary category) is shown to be equivalent to the full subcategory of so called involutive-2-links that are unital and associative. PubDate: 2023-09-18 DOI: 10.1007/s10485-023-09740-1

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Abstract: Abstract The correspondence between the concept of conditional flatness and admissibility in the sense of Galois appears in the context of localization functors in any semi-abelian category admitting a fiberwise localization. It is then natural to wonder what happens in the category of crossed modules where fiberwise localization is not always available. In this article, we establish an equivalence between conditional flatness and admissibility in the sense of Galois (for the class of regular epimorphisms) for regular-epi localization functors. We use this equivalence to prove that nullification functors are admissible for the class of regular epimorphisms, even if the kernels of their localization morphisms are not acyclic. PubDate: 2023-09-08 DOI: 10.1007/s10485-023-09738-9