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 C ^ = F c t ( C op , S e t ) {\displaystyle {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )} . The category C ^ {\displaystyle {\widehat {C}}} admits small limits and small colimits. Explicitly, if f : I → C ^ {\displaystyle f:I\to {\widehat {C}}} is a functor from a small category I and U is an object in C, then lim → i ∈ I f ( i ) {\displaystyle \varinjlim _{i\in I}f(i)} is computed pointwise:
( lim → f ( i ) ) ( U ) = lim → f ( i ) ( U ) . {\displaystyle (\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 a full subcategory of C ^ {\displaystyle {\widehat {C}}} . If η : C → D {\displaystyle \eta :C\to D} is a functor, if f : I → C {\displaystyle f:I\to C} is a functor from a small category I and if the colimit lim → f {\displaystyle \varinjlim f} in C ^ {\displaystyle {\widehat {C}}} is representable; i.e., isomorphic to an object in C, then, in D,
η ( lim → f ) ≃ lim → η ∘ f {\displaystyle \eta (\varinjlim f)\simeq \varinjlim \eta \circ f}
(in particular the colimit on the right exists in D). In other words, if the Yoneda embedding of C preserves the colimit of a functor into C, then every functor out of C preserves that colimit. The density theorem states that every presheaf is a colimit of representable presheaves.
Notes
References Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves.
