Applied Categorical Structures最新文献

In this paper we present a unified proof of the fact that the category of modules over a ring and the category of near-vector spaces in the sense of J. André, over an appropriate scalar system (a ‘scalar group’), are both abelian categories. The unification is possible by viewing each of these categories as subcategories of the (abelian) category of modules over a multiplicative monoid M. Although in the case of near-vector spaces all elements of M except one (the ‘zero’ element) are invertible, we show that this requirement is not necessary for the corresponding category to be abelian in analogy to the well-known fact that modules over a ring form an abelian category even if the ring is not a field (i.e., modules over it are not vector spaces).

Given a functor (F: mathcal {C}rightarrow mathcal {D}) and a model-theoretic independence relation on (mathcal {D}), we can lift that independence relation along F to (mathcal {C}) by declaring a commuting square in (mathcal {C}) to be independent if its image under F is independent. For each property of interest that an independence relation can have we give assumptions on the functor that guarantee the property to be lifted.

We prove some facts about locales L equipped with the Scott topology ({Omega }(L)), in particular studying a canonical frame homomorphism (phi :{Omega }(L)rightarrow L) which is motivated by an application to cognitive science. Such a topological locale L is called a Scott locale if the inclusion of primes (p:{Sigma }(L)rightarrow L) is continuous. We prove that the spectrum ({Sigma }(L)) of a Scott locale L is necessarily (T_1), and that preregular locales (a generalization of regular locales) are Scott locales. If L is the topology of a topological space X we find a (necessarily unique) continuous map (f:Xrightarrow L) such that (f^{-1}=phi ) and compare it with the points-to-primes map (p:Xrightarrow L), showing that (f=p) if and only if X is preregular, and that a sober space X is Hausdorff if and only if X is (T_1) and (f(X)subseteq {Sigma }(L).)

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').