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 $\hat{C}=𝐅𝐜𝐭(C^{\text{op}},𝐒𝐞𝐭)$.

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

$(\underrightarrow{\lim }f(i))(U)=\underrightarrow{\lim }f(i)(U).$

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 $\hat{C}$. 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 $\underrightarrow{\lim }f$ in $\hat{C}$ is representable; i.e., isomorphic to an object in C, then, in D,

$\eta (\underrightarrow{\lim }f)\simeq \underrightarrow{\lim }\eta \circ f$

(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.

References

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