Applied Categorical Structures最新文献

By de Vries duality, the category (textsf {KHaus}) of compact Hausdorff spaces is dually equivalent to the category (textsf {DeV}) of de Vries algebras. There is a similar duality for (textsf {KHaus}), where de Vries algebras are replaced by proximity Baer-Specker algebras. The functor associating with each compact Hausdorff space a proximity Baer-Specker algebra is described by generalizing the notion of a boolean power of a totally ordered domain to that of a de Vries power. It follows that (textsf {DeV}) is equivalent to the category (text {textsf{PBSp}}) of proximity Baer-Specker algebras. The equivalence is obtained by passing through (textsf {KHaus}), and hence is not choice-free. In this paper we give a direct algebraic proof of this equivalence, which is choice-independent. To do so, we give an alternate choice-free description of de Vries powers of a totally ordered domain.

We study how tensor categories can be presented in terms of rigid monoidal categories and Grothendieck topologies and show that such presentations lead to strong universal properties. As the main tool in this study, we define a notion on preadditive categories which plays a role similar to (a generalisation of) the notion of a Grothendieck pretopology on an unenriched category. Each such additive pretopology defines an additive Grothendieck topology and suffices to define the sheaf category. This new notion also allows us to study the noetherian and subcanonical nature of additive topologies, to describe easily the join of a family of additive topologies and to identify useful universal properties of the sheaf category.

We consider profunctors between posets and introduce their graph and ascent. The profunctors (text {Pro}(P,Q)) form themselves a poset, and we consider a partition (mathcal {I}sqcup mathcal {F}) of this into a down-set (mathcal {I}) and up-set (mathcal {F}), called a cut. To elements of (mathcal {F}) we associate their graphs, and to elements of (mathcal {I}) we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of (Q times P). Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study (text {Pro}({mathbb N}, {mathbb N})). Such profunctors identify as order preserving maps (f: {mathbb N}rightarrow {mathbb N}cup {infty }). For our applications when P and Q are infinite, we also introduce a topology on (text {Pro}(P,Q)), in particular on profunctors (text {Pro}({mathbb N},{mathbb N})).

It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category (textsf{Group}^mathcal {J}) of presheaves of groups can be characterized in terms of conjugation-theoretic inner automorphisms of the component groups, together with a natural automorphism of the identity functor on the index category (mathcal {J}). In fact, we deduce such a characterization from a much more general result characterizing the categorical inner automorphisms in any category (mathbb {T}textsf{mod}^mathcal {J}) of presheaves of (mathbb {T})-models for a suitable first-order theory (mathbb {T}).

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ((=)term graphs) as a combinatorial data structure. We formulate the appropriate 2-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category (mathcal {K}). In order to do that, we introduce the 2-category of relative monads in a 2-category (mathcal {K}) with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in (mathcal {K}) defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.

The purpose of this paper is to set up derived categories of sheaves of (E_infty )-rings and modules over non-derived sites, in particular over topological spaces. This theory opens up certain new capabilities in spectral algebra. For example, as outlined in the last section of the present paper, using these concepts, one can conjecture a spectral algebra-based generalization of the geometric Langlands program to manifolds of dimension (>2). As explained in a previous paper (Chen et al. in Theory Appl Categ 32:1363-1396, 2017) the only theory of sheaves of spectra on non-derived sites known to date which has well-behave pushforwards is based on Kan spectra, which, however, are reputed not to possess a smash product rigid enough for discussing (E_infty )-objects. The bulk of this paper is devoted to remedying this situation, i.e. defining a more rigid smash product of Kan spectra, and using it to construct the desired derived categories.

Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this result to the situation of a Krull-Schmidt category on which a finite group acts as automorphisms. This then provides the orbit category introduced by Cibils and Marcos, and studied intensively by Keller in the context of cluster algebras, and by Asashiba in the context of Galois covering functors. We formulate and prove Clifford’s theorem for Krull-Schmidt orbit categories with respect to a finite group (Gamma ) of automorphisms, clarifying this way how the image of an indecomposable object in the original category decomposes in the orbit category. The pair of adjoint functors appears as the Kleisli category of the naturally appearing monad given by (Gamma ).

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type (textrm{FP}_n) and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type (textrm{FP}_n) categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type (textrm{FP}_n), called (textrm{FP}_n)-injective objects, which will be the right half of a complete cotorsion pair. As a generalization of the category of modules over an n-coherent ring, we present the concept of n-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for (n = 0, 1). Such categories will provide a setting in which the (textrm{FP}_n)-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by (textrm{FP}_n)-injective objects.