Bordg, Anthony
Published in
Applied Categorical Structures

In this article, the author endows the functor category [B(Z2),Gpd]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\mathbf {B}(\mathbb {Z}_2),\mathbf {Gpd}]$$\end{docu...

Jakl, Tomáš Jung, Achim Pultr, Aleš
Published in
Applied Categorical Structures

It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and b...

Kvamme, Sondre Marczinzik, René
Published in
Applied Categorical Structures

We review the theory of Co-Gorenstein algebras, which was introduced in Beligiannis (Commun Algebra 28(10):4547–4596, 2000). We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.

Lin, Daniel
Published in
Applied Categorical Structures

Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is ...

Tanaka, Kohei
Published in
Applied Categorical Structures

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg...

Lu, Xuexing Ye, Yu Hu, Sen
Published in
Applied Categorical Structures

Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to form...

Husainov, Ahmet A.
Published in
Applied Categorical Structures

The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of abelian groups are isomorphic to the homology groups of the normalized complex of the cubical abe...

Ziemiański, Krzysztof
Published in
Applied Categorical Structures

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry inform...

Ogawa, Yasuaki
Published in
Applied Categorical Structures

Given the pair of a dualizing k-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander’s formulas: the first one realizes a module category as a Serre quotient of a suitable functor category. The seco...

Shaumbwa, Vaino Tuhafeni
Published in
Applied Categorical Structures

We show that in an ideal-determined unital category the Higgins commutator can be characterized as the largest binary operation C on subobjects (defined on all subobjects of each object) satisfying the following conditions: (a) C is order-preserving; (b) C(H, K) is always less or equal to the meet of normal closures of H and K; (c) C(f(H),f(K))=f(C...