Limit and colimit of presheaves explained

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} = Fct (C ^{op} , Set ) .^[1]

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} 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 η : 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,η (\varinjlim f) \simeq \varinjlim η \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

• Book: Kashiwara. Masaki. Schapira. Pierre. Categories and sheaves. 2006. Masaki Kashiwara. Pierre Schapira (mathematician).

Notes and References

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.