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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn object $0$ of $\\mathcal{C}$ is a *zero object* if it is both an [[initial object and a terminal object|EK85VSSRMR]].\nSo for every object $X$ there is exactly one [[morphism|859F0TFVA3]] $0 \\to X$ and exactly one morphism $X \\to 0$.\n\n## Remarks\n\nA zero object is unique up to unique isomorphism, because an initial object is.\nA category with a zero object is called pointed.\n\nExample: in the category of groups, the [[trivial group|0BNPC9850Y]] $\\{e\\}$ is a zero object.\nThe only homomorphism $\\{e\\} \\to G$ sends $e$ to the identity of $G$, and the only map $G \\to \\{e\\}$ is the constant map, which is a homomorphism.\nThe same check shows that the zero module is a zero object in the category of modules over a ring.\nIt also shows that a one-point set is a zero object in the category of pointed sets.\n\nNon-example: in the category of sets, a one-point set $\\{*\\}$ is terminal but not initial, since there are two functions $\\{*\\} \\to \\{0, 1\\}$.\nThe empty set is initial but not terminal, since there is no function from a non-empty set to it.\nSo the category of sets has no zero object.\n",
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