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
{\displaystyle {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}
is a functor from a small category I and U is an object in C, then
is computed pointwise: 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
is a functor from a small category I and if the colimit
is representable; i.e., isomorphic to an object in C, then,[3] in D, (in particular the colimit on the right exists in D.) The density theorem states that every presheaf is a colimit of representable presheaves.
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