Functor from a category's opposite category to Set
In
category theory, a branch of
mathematics, a presheaf on a
category
is a
functor
. If
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
into a category, and is an example of a
functor category. It is often written as
. A functor into
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
as a
-valued presheaf.
[1]
Examples
Properties
- When
is a
small category, the functor category
is
cartesian closed.
- The poset of
subobjects of
form a
Heyting algebra, whenever
is an object of
for small
.
- For any morphism
of
, the pullback functor of subobjects
has a
right adjoint, denoted
, and a left adjoint,
. These are the
universal and existential quantifiers.
- A locally small category
embeds
fully and faithfully into the category
of set-valued presheaves via the
Yoneda embedding which to every object
of
associates the
hom functor
.
- The category
admits small
limits and small
colimits.
[2] See
limit and colimit of presheaves for further discussion.
- The
density theorem states that every presheaf is a colimit of representable presheaves; in fact,
is the
colimit completion of
(see
#Universal property below.)
Universal property
The construction
is called the colimit completion of C because of the following
universal property:
Proof: Given a presheaf F, by the
density theorem, we can write
where
are objects in C. Then let
which exists by assumption. Since
is functorial, this determines the functor
. Succinctly,
is the left
Kan extension of
along y; hence, the name "Yoneda extension". To see
commutes with small colimits, we show
is a left-adjoint (to some functor). Define
to be the functor given by: for each object M in D and each object U in C,
![{\displaystyle {\mathcal {H}}om(\eta ,M)(U)=\operatorname {Hom} _{D}(\eta U,M).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bebe804c7f7fb459b5b819fe07f10830c59eb1f)
Then, for each object M in D, since
by the Yoneda lemma, we have:
![{\displaystyle {\begin{aligned}\operatorname {Hom} _{D}({\widetilde {\eta }}F,M)&=\operatorname {Hom} _{D}(\varinjlim \eta U_{i},M)=\varprojlim \operatorname {Hom} _{D}(\eta U_{i},M)=\varprojlim {\mathcal {H}}om(\eta ,M)(U_{i})\\&=\operatorname {Hom} _{\widehat {C}}(F,{\mathcal {H}}om(\eta ,M)),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de84b241f8cb5ecfd01412e7994d4daefffa8c36)
which is to say
is a left-adjoint to
.
The proposition yields several corollaries. For example, the proposition implies that the construction
is functorial: i.e., each functor
determines the functor
.
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.)
[4] 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:
is
fully faithful (here C can be just a
simplicial set.)
[5]
See also
Notes
References
Further reading