Applied Categorical Structures最新文献

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.

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.

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.

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.

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.

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, M. Pinsker and T. Tsankov 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.

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.

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.

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.