{
  "id": "2108.04596",
  "version": "v1",
  "published": "2021-08-10T11:14:12.000Z",
  "updated": "2021-08-10T11:14:12.000Z",
  "title": "$n$-Exact categories arising from $(n+2)$-angulated categories",
  "authors": [
    "Carlo Klapproth"
  ],
  "categories": [
    "math.RT",
    "math.CT"
  ],
  "abstract": "Let $\\mathscr{F}$ be an $(n+2)$-angulated Krull-Schmidt category and $\\mathscr{A} \\subset \\mathscr{F}$ an $n$-extension closed, additive and full subcategory with $\\operatorname{Hom}_{\\mathscr{F}}(\\Sigma_n \\mathscr{A}, \\mathscr{A}) = 0$. Then $\\mathscr{A}$ naturally carries the structure of an $n$-exact category in the sense of Jasso, arising from short $(n+2)$-angles in $\\mathscr{F}$ with objects in $\\mathscr{A}$ and there is a binatural and bilinear isomorphism $\\operatorname{YExt}^{n}_{(\\mathscr{A},\\mathscr{E}_{\\mathscr{A}})}(A_{n+1},A_0) \\cong \\operatorname{Hom}_{\\mathscr{F}}(A_{n+1}, \\Sigma_n A_{0})$ for $A_0, A_{n+1} \\in \\mathscr{A}$. For $n = 1$ this has been shown by Dyer and we generalize this result to the case $n > 1$. On the journey to this result, we also develop a technique for harvesting information from the higher octahedral axiom (N4*) as defined by Bergh and Thaule. Additionally, we show that the axiom (F3) for pre-$(n+2)$-angulated categories, introduced by Geiss, Keller and Oppermann and stating that a commutative square can be extended to a morphism of $(n+2)$-angles, implies a stronger version of itself.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-10T11:14:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18G99",
      "18G80"
    ],
    "keywords": [
      "exact category",
      "exact categories arising",
      "angulated categories",
      "higher octahedral axiom",
      "angulated krull-schmidt category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}