From Right (n + 2)-angulated Categories to n-exangulated Categories

Abstract

In this paper, we introduce the notion of a right semi-equivalence for right (n + 2)-angulated categories. Let

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

be an n-exangulated category and

\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{X}$$\end{document}

be a strongly covariantly finite subcategory of

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

. We prove that the right (n + 2)-angulated category

\mathcal{C}/\mathcal{X}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

has an n-suspension functor that is a right semi-equivalence under a natural assumption. As an application, we show that a right (n + 2)-angulated category has an n-exangulated structure if and only if the n-suspension functor is a right semi-equivalence. Furthermore, we also prove that an n-exangulated category

\mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr{C}$$\end{document}

has the structure of a right (n + 2)-angulated category with a right semi-equivalence if and only if for any object

X\in \mathcal{C}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X \in \mathscr{C}$$\end{document}

, the morphism X → 0 is a trivial inflation.

Keywords

n-exangulated categories / extriangulated categories / right (n + 2)-angulated categories / right triangulated categories / right semi-equivalences

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