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 $$\widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$$.

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