Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and conco...

We study localizing and colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between localizing subcategories and equivalence classes of idempotent elements in the dual algebra C∗. In this framework, we give a useful description of the localization functor by means of the Morita–Takeuchi context defined by the quasi-finite injective cogenerat...

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions betweeen simple objects. The simple objects in the category are parametized by the affine weight lattice. W...

A categorical approach to probability allows to put basic notions of probability into a broader mathematical perspective, to evaluate their roles, and mutual relationships. Classical probability theory and fuzzy probability theory lead to two particular categories and their relationship (in categorical terms) enable us to understand and explicitly formulate the difference between them. Using ou...

In this paper we revisit the problem of determining when heart a t-structure is Grothendieck category, with special attention to case Happel-Reiten-Smalo (HSR) in derived category associated torsion pair latter. We HRS tilting process deriving from it lot information on t-structures which have projective generator or an injective cogenerator, and obtain several bijections between classes pairs ...

In topological data analysis, two-parameter persistence can be studied using the representation theory of 2d commutative grid, tensor product two Dynkin quivers type A. a previous work, we defined interval approximations restrictions to essential vertices intervals together with Mobius inversion. this consider homological resolutions, and show that resolution global dimension is finite for pose...

Let Λ be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of Λ is three in case Λ is representation infinite. Recently there has been a lot of attention to compute the representation dimension of a finite dimensional algebra Λ, a notion introduced by Auslander in [A] in an attempt to measure the complexity of the represent...