Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category $C$ is a functor $F\colon C^\mathrm\to\mathbf$. If $C$ 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 $C$ into a category, and is an example of a functor category. It is often written as $\widehat = \mathbf^$ and it is called the category of presheaves on $C$. A functor into $\widehat$ 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 $F\colon C^\mathrm\to\mathbf$ as a $\mathbf$-valued presheaf.

Examples

* A simplicial set is a Set-valued presheaf on the simplex category $C=\Delta$.
* A directed multigraph is a presheaf on the category with two elements and two parallel morphisms between them i.e. $C = (E \overset V)$.
* An arrow category is a presheaf on the category with two elements and one morphism between them. i.e. $C = (E \overset V)$.
* A right group action is a presheaf on the category created from a group $G$, i.e. a category with one element and invertible morphisms.

Properties

* When $C$ is a small category, the functor category $\widehat=\mathbf^$ is cartesian closed.
* The poset of subobjects of $P$ form a Heyting algebra, whenever $P$ is an object of $\widehat=\mathbf^$ for small $C$.
* For any morphism $f:X\to Y$ of $\widehat$, the pullback functor of subobjects $f^*:\mathrm_(Y)\to\mathrm_(X)$ has a right adjoint, denoted $\forall_f$, and a left adjoint, $\exists_f$. These are the universal and existential quantifiers.
* A locally small category $C$ embeds fully and faithfully into the category $\widehat$ of set-valued presheaves via the Yoneda embedding which to every object $A$ of $C$ associates the hom functor $C(-,A)$.
*The category $\widehat$ admits small limits and small colimits. See limit and colimit of presheaves for further discussion.
* The density theorem states that every presheaf is a colimit of representable presheaves; in fact, $\widehat$ is the colimit completion of $C$ (see #Universal property below.)

Universal property

The construction $C \mapsto \widehat = \mathbf(C^, \mathbf)$ 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 $F =\varinjlim y U_i$ where $U_i$ are objects in ''C''. Then let $\widetilde F = \varinjlim \eta U_i,$ which exists by assumption. Since $\varinjlim -$ is functorial, this determines the functor $\widetilde: \widehat \to D$. Succinctly, $\widetilde$ is the left Kan extension of $\eta$ along ''y''; hence, the name "Yoneda extension". To see $\widetilde$ commutes with small colimits, we show $\widetilde$ is a left-adjoint (to some functor). Define $\mathcalom(\eta, -): D \to \widehat$ to be the functor given by: for each object ''M'' in ''D'' and each object ''U'' in ''C'',
:$\mathcalom(\eta, M)(U) = \operatorname_D(\eta U, M).$
Then, for each object ''M'' in ''D'', since $\mathcalom(\eta, M)(U_i) = \operatorname(y U_i, \mathcalom(\eta, M))$ by the Yoneda lemma, we have:
:$\begin \operatorname_D(\widetilde F, M) &= \operatorname_D(\varinjlim \eta U_i, M) = \varprojlim \operatorname_D(\eta U_i, M) = \varprojlim \mathcalom(\eta, M)(U_i) \\ &= \operatorname_(F, \mathcalom(\eta, M)), \end$
which is to say $\widetilde$ is a left-adjoint to $\mathcalom(\eta, -)$. $\square$

The proposition yields several corollaries. For example, the proposition implies that the construction $C \mapsto \widehat$ is functorial: i.e., each functor $C \to D$ determines the functor $\widehat \to \widehat$.

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: $C \to \widehat$ is fully faithful (here ''C'' can be just a simplicial set.)

A copresheaf of a category ''C'' is a presheaf of ''C^op''. In other words, it is a covariant functor from ''C'' to ''Set''.

See also

* Topos
* Category of elements
* Simplicial presheaf (this notion is obtained by replacing "set" with "simplicial set")
* Presheaf with transfers

Notes

References

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Further reading

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*{{nlab, id=free+cocompletion, title=Free cocompletion
*Daniel Dugger
Sheaves and Homotopy Theory
th
pdf file
provided by nlab.

Functors
Sheaf theory
Topos theory