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 [math]\displaystyle{ \widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}) }[/math].[1] The category [math]\displaystyle{ \widehat{C} }[/math] admits small limits and small colimits.[2] Explicitly, if [math]\displaystyle{ f: I \to \widehat{C} }[/math] is a functor from a small category I and U is an object in C, then [math]\displaystyle{ \varinjlim_{i \in I} f(i) }[/math] is computed pointwise:
- [math]\displaystyle{ (\varinjlim f(i))(U) = \varinjlim f(i)(U). }[/math]
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 [math]\displaystyle{ \widehat{C} }[/math]. If [math]\displaystyle{ \eta: C \to D }[/math] is a functor, if [math]\displaystyle{ f: I \to C }[/math] is a functor from a small category I and if the colimit [math]\displaystyle{ \varinjlim f }[/math] in [math]\displaystyle{ \widehat{C} }[/math] is representable; i.e., isomorphic to an object in C, then,[3] in D,
- [math]\displaystyle{ \eta(\varinjlim f) \simeq \varinjlim \eta \circ f, }[/math]
(in particular the colimit on the right exists in D.)
The density theorem states that every presheaf is a colimit of representable presheaves.
Notes
- ↑ 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.
- ↑ Kashiwara & Schapira 2006, Corollary 2.4.3.
- ↑ Kashiwara & Schapira 2006, Proposition 2.6.4.
References
- {{Cite book
| last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | title=Categories and sheaves | year=2006
 |  |
