{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "128WX7HXAT"
    },
    {
      "field": "uses",
      "target": "C0KFX2CMRR"
    },
    {
      "field": "uses",
      "target": "NW2WW3WSH8"
    },
    {
      "field": "defines",
      "target": "C0GMFP0RF9"
    }
  ],
  "body": "\n## Summary\n\nA zero object of a [[category|A2DE66937D]] is an object that is both [[initial and terminal|EK85VSSRMR]].\nCategories with a zero object, such as groups, modules and pointed sets, have a distinguished zero morphism between any two objects.\nThis is the starting point for kernels, cokernels and exact sequences in homological algebra.\n\n## Explanations\n\n### L1 human\n\nA zero object has exactly one arrow to every object and exactly one arrow from every object.\nThe [[trivial group|0BNPC9850Y]] $\\{e\\}$ is a zero object among groups.\nThere is one homomorphism from it to any group, sending $e$ to the identity.\nThere is one homomorphism to it from any group, sending everything to $e$.\nThe zero vector space plays the same role among vector spaces.\nAmong sets there is no zero object: the empty set is initial but not terminal, and a one-point set is terminal but not initial.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "18A30",
      "18A20"
    ],
    "name": "Zero object",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Zero_object_(algebra)"
      }
    ],
    "specializes": [
      "EK85VSSRMR"
    ],
    "tag": "AQET9YHH6F"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/AQET9YHH6F/",
  "links": [
    {
      "field": "specializes",
      "target": "EK85VSSRMR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AQET9YHH6F.md",
  "msc": [
    "18A30",
    "18A20"
  ],
  "name": "Zero object",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AQE/AQET9YHH6F.md",
  "status": null,
  "tag": "AQET9YHH6F",
  "uri": "m2://AQET9YHH6F",
  "url": "https://tamnd.github.io/math2/wiki/zero-object/",
  "writing": {
    "tag": "AQET9YHH6F",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}