Limit And Colimit Of Presheaves

In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category ''C'' is a limit or colimit in the functor category $\widehat = \mathbf(C^, \mathbf)$.

The category $\widehat$ admits small limits and small colimits. Explicitly, if $f: I \to \widehat$ is a functor from a small category ''I'' and ''U'' is an object in ''C'', then $\varinjlim_ f(i)$ is computed pointwise:

:$(\varinjlim f(i))(U) = \varinjlim f(i)(U).$

The same is true for small limits. Concretely this means that, for example, a fiber product exists and is computed pointwise.

When ''C'' is small, by the Yoneda lemma, one can view ''C'' as the full subcategory of $\widehat$. If $\eta: C \to D$ is a functor, if $f: I \to C$ is a functor from a small category ''I'' and if the colimit $\varinjlim f$ in $\widehat$ is representable; i.e., isomorphic to an object in ''C'', then, in ''D'',

: $\eta(\varinjlim f) \simeq \varinjlim \eta \circ f,$

(in particular the colimit on the right exists in ''D''.)

The density theorem states that every presheaf is a colimit of representable presheaves.

Notes

References

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Category theory
Sheaf theory

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