Applied Categorical Structures最新文献

Given a tensor triangulated category we investigate the geometry of the Balmer spectrum as a locally ringed space by constructing functors assigning to every object in the category a corresponding sheaf of modules over the structure sheaf of the spectrum. Taking the support of these associated sheaves recovers a notion of support based on local categories. We compare this support to the usual support in tt-geometry and show that under reasonable conditions they agree on compact objects. We show that when the tt-category satisfies a scheme-like property, then the sheaves associated to objects are quasi-coherent, and that in the presence of an appropriate t-structure and affine assumption, this sheaf is in fact the sheaf associated to the object’s zeroth cohomology.

In Igusa, Todorov and Weyman (Picture groups of finite type and cohomology in type A_n arXiv:1609.02636), we introduced “picture groups” and computed the cohomology of the picture group of type (A_n). This is the same group what was introduced by Loday (Contemp Math 265: 99–127, 2000) where he called it the “Stasheff group”. In this paper, we give an elementary combinatorial interpretation of the “cluster morphism category” constructed in as reported by Igusa and Todorov, (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041) in the special case of the linearly oriented quiver of type (A_n). We prove that the classifying space of this category is locally CAT(0) and thus a (K(pi ,1)). We prove a more general statement that classifying spaces of certain “cubical categories” are locally CAT(0). The objects of our category are the classical noncrossing partitions introduced by Kreweras (Discrete Math 1: 333–350, 1972) . The morphisms are binary forests. This paper is independent of as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041)and as reported by Igusa, Todorov and Weyman (in: Picture groups of finite type and cohomology in type A_n arXiv:1609.02636)except in the last section where we use as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041) to compare our category with the category with the same name given by Hubery and Krause (J Eur Math Soc 18: 2273–2313, 2016).

Double categories have been extended to (co)lax and virtual double categories. We want to show that the first extension still has a general theory of adjunctions, with examples related to homotopy theory, while the second, wider extension has not. Lax and colax double categories have a finitary weak composition, with associativity comparisons which are not assumed to be invertible. We deal with the colax form (also called oplax), which is related to tensor products of topological ‘algebras’. Double adjunctions can be extended to these structures, in the general ‘colax-lax’ form already studied for (weak) double categories: the left adjoint is colax and the right adjoint is lax. For instance, this is the case of the cylinder-cocylinder adjunction. Now, a normal colax double category is known to be essentially the same as a representable virtual double category. Functors of virtual double categories correspond to lax functors of colax double categories, and can only have adjunctions of the weak-lax form; typically, homotopies will not be represented by a cylinder endofunctor, as we show in a class of examples.

Algebraic structures with replicate operations interrelated by various compatibility conditions have long been studied in mathematics and mathematical physics. They are broadly referred as linearly compatible, matching, and totally compatible structures. This paper gives a unified approach to these structures in the context of operads. Generalizing polarizations for polynomials in invariant theory to operads leads to linearly compatible operads. Partitioning polarizations into foliations gives matching operads which further yields total compatible operads under an invariance condition. For unary/binary quadratic operads, linear compatibility and total compatibility are in Koszul dual, and the matching compatibilities are Koszul self-dual among themselves. For binary quadratic operads, these three compatible operads can be achieved by taking Manin products. For some finitely generated binary quadratic operad, Koszulity is preserved under taking the compatibilities.

Let (mathbf{C}) be a small category. The subtoposes of ([mathbf{C}^textrm{op},mathbf{Set}]) are sometimes all of the form ([mathbf{D}^textrm{op},mathbf{Set}]) where (mathbf{D}) is a full subcategory of (mathbf{C}). This is the case for instance when (mathbf{C}) is Cauchy-complete and finite, an Artinian poset, or the simplex category. We call such a category universally rigid. A universally rigid category whose slices are also universally rigid, such as the aforementioned examples, is called stably universally rigid. We provide two equivalent characterizations of such categories. The first one stipulates the existence of a winning strategy in a two-player game, and the second one combines two “local” properties of (mathbf{C}) involving respectively the poset reflections of its slices and its endomorphism monoids.

Using techniques developed (Batanin and Leger in J Noncommutative Geom 13:1521–1576, 2019), we extend the Turchin/Dwyer–Hess double delooping result to further iterations of the Baez–Dolan plus construction. For (0 le m le n), we introduce a notion of (m, n)-bimodules which extends the notions of bimodules and infinitesimal bimodules over the terminal non-symmetric operad. We show that a double delooping always exists for these bimodules. For the triple iteration of the Baez-Dolan construction starting from the initial 1-coloured operad, we provide a further reduceness condition to have a third delooping.

Let b, (b') be commutative monoids in a Bénabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over (botimes b') is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over b, (b').

Compactness of the homotopy categories (textrm{K}(mathcal {P})) and (textrm{K}(mathcal {I})) of graded projective and graded injective dg modules over a dg ring are investigated in view of pure acyclic dg modules. For sufficiently nice non-positive dg rings, we show that the two subcategories (textrm{K}(mathcal {P})) and (textrm{K}(mathcal {I})) are compactly generated.

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector fields on the groupoid, which are closed under the Lie bracket. We generalize this differentiation procedure to groupoid objects in any category with an abstract tangent structure in the sense of Rosický and a scalar multiplication by a ring object that plays the role of the real numbers. We identify the categorical conditions that the groupoid object must satisfy to admit a natural notion of invariant vector fields. Then we show that invariant vector fields are closed under the Lie bracket defined by Rosický and satisfy the Leibniz rule with respect to ring-valued morphisms on the base of the groupoid. The result is what we define axiomatically as an abstract Lie algebroid, by generalizing the underlying vector bundle to a module object in the slice category over its base. Examples include diffeomorphism groups, bisection groups of Lie groupoids, the diffeological symmetry groupoids of general relativity (Blohmann/Fernandes/Weinstein), symmetry groupoids in Lagrangian Field Theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, groupoid objects in differentiable stacks, and affine groupoid schemes.

Track bicategories, where each hom-category is a groupoid, appear in various mathematical and physical contexts. In this paper, we establish a cohomological classification of track bicategories and track categories using group-valued 3-cocycles on small categories, formulated as lax functors into the one-object 3-category of groups. In the abelian case, this classification aligns with Baues-Wirsching cohomology for small categories with coefficients in natural systems, recovering previously known classification results.