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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Abstract: Abstract We give a new categorical approach to the Halmos–von Neumann theorem for actions of general topological groups. As a first step, we establish that the categories of topological and measure-preserving irreducible systems with discrete spectrum are equivalent. This allows to prove the Halmos–von Neumann theorem in the framework of topological dynamics. We then use the Pontryagin and Tannaka–Krein duality theories to obtain classification results for topological and then measure-preserving systems with discrete spectrum. As a byproduct, we obtain a complete isomorphism invariant for compactifications of a fixed topological group. PubDate: 2023-09-08 DOI: 10.1007/s10485-023-09743-y
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Abstract: Abstract It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties of the maximalization functor for coactions. We also provide a dual proof for the universal property of normalization of coactions. PubDate: 2023-09-07 DOI: 10.1007/s10485-023-09741-0
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Abstract: Abstract There are several prominent duality results in pointfree topology. Hofmann–Lawson duality establishes that the category of continuous frames is dually equivalent to the category of locally compact sober spaces. This restricts to a dual equivalence between the categories of stably continuous frames and stably locally compact spaces, which further restricts to Isbell duality between the categories of compact regular frames and compact Hausdorff spaces. We show how to derive these dualities from Priestley duality for distributive lattices, thus shedding new light on these classic results. PubDate: 2023-09-01 DOI: 10.1007/s10485-023-09739-8
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Abstract: Abstract In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Hurewicz (resp. Serre) stack. PubDate: 2023-08-28 DOI: 10.1007/s10485-023-09744-x
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Abstract: Abstract We survey the theory of Hopf monads on monoidal categories, and present new examples and applications. As applications, we utilise this machinery to present a new theory of cross products, as well as analogues of the Fundamental Theorem of Hopf algebras and Radford’s biproduct Theorem for Hopf algebroids. Additionally, we describe new examples of Hopf monads which arise from Galois and Ore extensions of bialgebras. We also classify Lawvere theories whose corresponding monads on the category of sets and functions become Hopf, as well as Hopf monads on the poset of natural numbers. PubDate: 2023-08-27 DOI: 10.1007/s10485-023-09732-1
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Abstract: Abstract We introduce maximal ordered groupoids and study some of their properties. Also, we use the Ehresmann–Schein–Nambooripad Theorem, which establishes a one-to-one correspondence between inverse semigroups and a class of ordered groupoids, to prove a Galois correspondence for the case of inverse semigroups acting orthogonally on commutative rings. PubDate: 2023-08-21 DOI: 10.1007/s10485-023-09742-z
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Abstract: Abstract We construct the main object of the Partite Lemma as the colimit over a certain diagram. This gives a purely category theoretic take on the Partite Lemma and establishes the canonicity of the object. Additionally, the categorical point of view allows us to unify the direct Partite Lemma in Nešetřil and Rödl (J Comb Theory Ser A 22(3):289–312, 1977; J Comb Theory Ser A 34(2):183–201, 1983; Discrete Math 75(1–3):327–334, 1989) with the dual Paritite Lemma in Solecki (J Comb Theory Ser A 117(6):704–714, 2010). PubDate: 2023-08-04 DOI: 10.1007/s10485-023-09733-0
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