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  "body": "\n## Definition\n\nA *groupoid* is a [[category|A2DE66937D]] $\\mathcal{G}$ in which every morphism is an isomorphism.\nThat is, for every morphism $f \\colon x \\to y$ of $\\mathcal{G}$ there is a morphism $g \\colon y \\to x$ with $g \\circ f = \\id_x$ and $f \\circ g = \\id_y$.\n\n## Remarks\n\nA groupoid is *small* if its objects form a set.\nFor each object $x$, the morphisms $x \\to x$ form a [[group|G5S1M9]], the vertex group at $x$.\n\nExample: let $\\sim$ be an [[equivalence relation|0G8VEZ3WTK]] on a set $X$.\nTake $X$ as the set of objects, with exactly one morphism $x \\to y$ when $x \\sim y$ and none otherwise.\nComposition exists by transitivity, identities by reflexivity, and the inverse of $x \\to y$ is the morphism $y \\to x$, which exists by symmetry.\n\nNon-example: the category with objects $0, 1$ and one non-identity arrow $0 \\to 1$, the poset $0 \\le 1$.\nThere is no morphism $1 \\to 0$, so the arrow $0 \\to 1$ has no inverse.\n\nA group $G$ gives a groupoid with one object whose morphisms are the elements of $G$, and every one-object groupoid arises in this way.\n",
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