Abelian quotients of categories of $n$-exangles
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915538828001280 |
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| author | Zhou, Yutong |
| author_facet | Zhou, Yutong |
| contents | The notion of $n$-exangulated categories was introduced by Herschend-Liu-Nakaoka, which is a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated categories in the sense of Geiss-Kelier-Oppermann. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ a full subcategory of $\mathscr{C}$ containing $\mathcal{P}$. In the present paper, It is proved that a certian quotient category of $\mathfrak{s}$-def$(\mathcal{M})$ is abelian. We denoted by $S(\mathscr{C})$ the category of $n$-exangles, whose object are given by distinguished $n$-exangles in $\mathscr{C}$. If $\mathcal{M}=\mathscr{C}$, we obtain that a certain ideal quotient category $S(\mathscr{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules mod-$(\mathscr{C}/[\mathcal{P}])$. Furthermore, we present the quotient category $S(\mathscr{C})/\mathcal{R}_2$ always has an abelian structure when taking $n$ as an even number. The abelian quotient $S(\mathscr{C})/\mathcal{R}_2$ admits some nice properties. We describe the projective objects in $S(\mathscr{C})/\mathcal{R}_2$ and characterize the simple objects in $S(\mathscr{C})/\mathcal{R}_2$ as Auslander-Reiten $n$-exangle sequences in $\mathscr{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abelian quotients of categories of $n$-exangles Zhou, Yutong Representation Theory Category Theory The notion of $n$-exangulated categories was introduced by Herschend-Liu-Nakaoka, which is a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated categories in the sense of Geiss-Kelier-Oppermann. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ a full subcategory of $\mathscr{C}$ containing $\mathcal{P}$. In the present paper, It is proved that a certian quotient category of $\mathfrak{s}$-def$(\mathcal{M})$ is abelian. We denoted by $S(\mathscr{C})$ the category of $n$-exangles, whose object are given by distinguished $n$-exangles in $\mathscr{C}$. If $\mathcal{M}=\mathscr{C}$, we obtain that a certain ideal quotient category $S(\mathscr{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules mod-$(\mathscr{C}/[\mathcal{P}])$. Furthermore, we present the quotient category $S(\mathscr{C})/\mathcal{R}_2$ always has an abelian structure when taking $n$ as an even number. The abelian quotient $S(\mathscr{C})/\mathcal{R}_2$ admits some nice properties. We describe the projective objects in $S(\mathscr{C})/\mathcal{R}_2$ and characterize the simple objects in $S(\mathscr{C})/\mathcal{R}_2$ as Auslander-Reiten $n$-exangle sequences in $\mathscr{C}$. |
| title | Abelian quotients of categories of $n$-exangles |
| topic | Representation Theory Category Theory |
| url | https://arxiv.org/abs/2510.06626 |