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 ${\displaystyle {\widehat {C}}=\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}$.^[1]

The category ${\displaystyle {\widehat {C}}}$ admits small limits and small colimits.^[2] Explicitly, if ${\displaystyle f:I\to {\widehat {C}}}$ is a functor from a small category I and U is an object in C, then ${\displaystyle \varinjlim _{i\in I}f(i)}$ is computed pointwise:

${\displaystyle (\varinjlim f(i))(U)=\varinjlim f(i)(U).}$

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 the full subcategory of ${\displaystyle {\widehat {C}}}$. If ${\displaystyle \eta :C\to D}$ is a functor, if ${\displaystyle f:I\to C}$ is a functor from a small category I and if the colimit ${\displaystyle \varinjlim f}$ in ${\displaystyle {\widehat {C}}}$ is representable; i.e., isomorphic to an object in C, then,^[3] in D,

${\displaystyle \eta (\varinjlim f)\simeq \varinjlim \eta \circ f,}$

(in particular the colimit on the right exists in D.)

The density theorem states that every presheaf is a colimit of representable presheaves.

Notes

1. 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.
2. Kashiwara & Schapira 2006, Corollary 2.4.3.
3. Kashiwara & Schapira 2006, Proposition 2.6.4.

References

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