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Abstract: This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, \(\sigma \) - and \(\kappa \) -frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these. PubDate: 2023-03-15
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Abstract: We give some characterizations of commutative objects in a subtractive category and central morphisms in a regular subtractive category. In particular, we show that commutative objects, i.e., internal unitary magmas, are the same as internal abelian groups in a subtractive category and that analogously, centrality has an alternative description in terms of so-called “subtractors” in a regular subtractive category. PubDate: 2023-03-12
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Abstract: \({\textbf {W}}\) denotes the category, or class of algebras, in the title. A hull operator (ho) in \({\textbf {W}}\) is a function \(\textbf{ho} {\textbf {W}}\overset{h}{\longrightarrow }\ {\textbf {W}}\) which can be called an essential closure operator. The family of these, denoted \(\textbf{ho} {\textbf {W}}\) , is a proper class and a complete lattice in the ordering as functions “pointwise", with the bottom \({{\,\textrm{Id}\,}}_{{\textbf {W}}}\) and top Conrad’s essential completion e. Other much studied hull operators are the divisible hull, maximum essential reflection, projectable hull, and Dedekind completion. This paper is the authors’ latest efforts to understand/create structure in \(\textbf{ho} {\textbf {W}}\) through the nature of the interaction that an h might have with B, the bounded monocoreflection in \({\textbf {W}}\) (e.g., Bh=hB). We define and investigate three functions \(\textbf{ho} {\textbf {W}}\longrightarrow \textbf{ho} {\textbf {W}}\) which stand in the relation $$\begin{aligned} {{\,\textrm{Id}\,}}_{{\textbf {W}}} \le \overline{\alpha }(h) \le \overline{\lambda }(h) \le \overline{c}(h) \le h. \end{aligned}$$ General properties that an h might have, and particular choices of h, show various assignments of < and \(=\) in this chain. PubDate: 2023-02-28
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Abstract: We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal multicategories as an equalizer of categories of diagrams. The second approach extends the techniques developed by Ivan Le Creurer in his study of descent for internal essentially algebraic structures. PubDate: 2023-01-17
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Abstract: We prove that the trace of categorified quantum \(\mathfrak {sl}_3\) introduced by Khovanov and Lauda can also be identified with quantum \(\mathfrak {sl}_3\) , thus providing an alternative way of decategorification. This is the second step of trace decategorification of quantum \(\mathfrak {sl}_n\) groups over the integers, the first being the \(\mathfrak {sl}_2\) case. The main technique used is decoupling of categorified quantum group into its positive and negative part. This technique can be used for more general categorified quantum groups to reduce the problem to the trace decategorification of its positive part. In the case of quantum \(\mathfrak {sl}_3\) , there is an explicit form of the canonical basis of the positive (and isomorphically negative) part of it based on indecomposables found by Stošić, leading to the full result in this case. PubDate: 2023-01-11
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Abstract: In the context of a tower of (strongly Birkhoff) Galois structures in the sense of categorical Galois theory, we show that the concept of a higher covering admits a characterisation which is at the same time absolute (with respect to the base level in the tower), rather than inductively defined relative to extensions of a lower order; and symmetric, rather than depending on a perspective in terms of arrows pointing in a certain chosen direction. This result applies to the Galois theory of quandles, for instance, where it helps us characterising the higher coverings in purely algebraic terms. PubDate: 2023-01-11
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Abstract: Let A be a commutative noetherian local DG-ring with bounded cohomology. For local Cohen–Macaulay DG-modules with constant amplitude, we obtain an explicit formula for the sequential depth, show that Cohen–Macaulayness is stable under localization and give several equivalent definitions of maximal local Cohen–Macaulay DG-modules over local Cohen–Macaulay DG-rings. We also provide some characterizations of Gorenstein DG-rings by projective and injective dimensions of DG-modules. PubDate: 2023-01-03 DOI: 10.1007/s10485-022-09703-y
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Abstract: Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions. PubDate: 2023-01-03 DOI: 10.1007/s10485-022-09708-7
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Abstract: In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M. Sokić that the free product of Ramsey classes is a Ramsey class, and the result of M. Bodirsky that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees. PubDate: 2022-12-29 DOI: 10.1007/s10485-022-09700-1
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Abstract: This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category. Using this product construction we can define a homotopy theory on coarse metric spaces in a natural way. We prove that all finite colimits exist in the coarse category. PubDate: 2022-12-27 DOI: 10.1007/s10485-022-09707-8
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Abstract: In this work we propose a realization of Lurie’s prediction that inner fibrations \(p: X \rightarrow A\) are classified by A-indexed diagrams in a “higher category” whose objects are \(\infty \) -categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and \(\infty \) -categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well. PubDate: 2022-12-27 DOI: 10.1007/s10485-022-09705-w
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Abstract: It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices \(M_n\) and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations. PubDate: 2022-12-27 DOI: 10.1007/s10485-022-09699-5
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Abstract: We introduce intermediate commutators and study their degrees. We define \((q, \{\})\) -capable groups and prove that a group G is \((q, \{\})\) -capable if and only if \(Z^{\wedge }_{(q, \{\})}(G)=1\) . PubDate: 2022-12-26 DOI: 10.1007/s10485-022-09701-0
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Abstract: Given a preradical we obtain operators between classes modules with closure properties. These operators turn out to be prenuclei and under suitable conditions nuclei. In particular, for the lattices of natural and conatural classes we obtain some nice properties and characterize some classes of rings. PubDate: 2022-12-19 DOI: 10.1007/s10485-022-09702-z
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Abstract: The aim of this paper is to introduce the concept of n-Gorenstein tilting comodules and study its main properties. This concept generalizes the notion of n-tilting comodules of finite injective dimensions to the case of finite Gorenstein injective dimensions. As an application of our results, we discuss the problem of existence of complements to partial n-Gorenstein tilting comodules. PubDate: 2022-12-01 DOI: 10.1007/s10485-022-09688-8
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Abstract: In a regular category \(\mathbb {E}\) , the direct image along a regular epimorphism f of a preorder is not a preorder in general. In Set, its best preorder approximation is then its cocartesian image above f. In a regular category, the existence of such a cocartesian image above f of a preorder S is actually equivalent to the existence of the supremum \(R[f]\vee S\) among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They apply to two very dissimilar contexts: any topos \(\mathbb {E}\) with suprema of countable chains of subobjects or any n-permutable regular category. PubDate: 2022-12-01 DOI: 10.1007/s10485-022-09686-w
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Abstract: Bernhard Banaschewski at work in his Study. Painted by Patricia Murphy (reprinted with permission) PubDate: 2022-11-29 DOI: 10.1007/s10485-022-09697-7
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Abstract: In this paper, we provide a notion of \(\infty \) -bicategories fibred in \(\infty \) -bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling \(\infty \) -bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an \(\infty \) -bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \({\text {Set}}^+_{\Delta }\) -enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories. PubDate: 2022-09-28 DOI: 10.1007/s10485-022-09693-x
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Abstract: In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them. PubDate: 2022-09-21 DOI: 10.1007/s10485-022-09692-y
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Abstract: In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies. PubDate: 2022-08-29 DOI: 10.1007/s10485-022-09690-0
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