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Limit and colimit of presheaves

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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 .[1]

The category admits small limits and small colimits.[2] Explicitly, if 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 . If is a functor, if is a functor from a small category I and if the colimit in 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.

Notes

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  1. ^ Notes on the foundation: the notation Set implicitly assumes that there is the notion of a small set; i.e., one has made a choice of a Grothendieck universe.
  2. ^ Kashiwara & Schapira 2006, Corollary 2.4.3.
  3. ^ Kashiwara & Schapira 2006, Proposition 2.6.4.

References

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  • Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves.