Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category formula_1 is a functor formula_2. If formula_1 is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.
A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on formula_1 into a category, and is an example of a functor category. It is often written as formula_5. A functor into formula_6 is sometimes called a profunctor.
A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–,"A") for some object "A" of C is called a representable presheaf.
Some authors refer to a functor formula_7 as a formula_8-valued presheaf.
Universal property.
The construction formula_29 is called the colimit completion of "C" because of the following universal property:
\longrightarrow D</math>
"Proof": Given a presheaf "F", by the density theorem, we can write formula_32 where formula_33 are objects in "C". Then let formula_34 which exists by assumption. Since formula_35 is functorial, this determines the functor formula_30. Succinctly, formula_37 is the left Kan extension of formula_31 along "y"; hence, the name "Yoneda extension". To see formula_37 commutes with small colimits, we show formula_37 is a left-adjoint (to some functor). Define formula_41 to be the functor given by: for each object "M" in "D" and each object "U" in "C",
Then, for each object "M" in "D", since formula_43 by the Yoneda lemma, we have:
which is to say formula_37 is a left-adjoint to formula_46. formula_47
The proposition yields several corollaries. For example, the proposition implies that the construction formula_48 is functorial: i.e., each functor formula_49 determines the functor formula_50.
Variants.
A presheaf of spaces on an ∞-category "C" is a contravariant functor from "C" to the ∞-category of spaces (for example, the nerve of the category of CW-complexes.) It is an ∞-category version of a presheaf of sets, as a "set" is replaced by a "space". The notion is used, among other things, in the ∞-category formulation of Yoneda's lemma that says: formula_51 is fully faithful (here "C" can be just a simplicial set.)