= Presheaf
{disambiguate=category theory}
{title2=$X:\mathcal C^{\mathrm{op}}\to\mathbf{Set}$}
{wiki}
= Presheaf on a category
{synonym}
= Categorical presheaf
{synonym}
A presheaf on a <category> $\mathcal C$ is a <functor> $X:\mathcal C^{\mathrm{op}}\to\mathbf{Set}$. A <morphism> $f:C\to D$ induces the restriction map $X(f):X(D)\to X(C)$, reversing direction. The indexing <category> need not be small. For the <category> of open subsets of a <topological space>, this specializes to a <presheaf of sets on a topological space>.
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