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.

References

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