Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category ${\displaystyle C}$ is a functor ${\displaystyle F:C^{\mathrm{o}\mathrm{p}}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}}$. If ${\displaystyle C}$ 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 ${\displaystyle C}$ into a category, and is an example of a functor category. It is often written as ${\displaystyle \hat{C}={\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}}^{C^{\mathrm{o}\mathrm{p}}}}$. A functor into ${\displaystyle \hat{C}}$ 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 ${\displaystyle F:C^{\mathrm{o}\mathrm{p}}\to \mathbf{𝐕}}$ as a ${\displaystyle \mathbf{𝐕}}$-valued presheaf.^[1]

• A simplicial set is a Set-valued presheaf on the simplex category ${\displaystyle C=\mathrm{\Delta }}$.

• When ${\displaystyle C}$ is a small category, the functor category ${\displaystyle \hat{C}={\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}}^{C^{\mathrm{o}\mathrm{p}}}}$ is cartesian closed.
• The poset of subobjects of ${\displaystyle P}$ form a Heyting algebra, whenever ${\displaystyle P}$ is an object of ${\displaystyle \hat{C}={\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}}^{C^{\mathrm{o}\mathrm{p}}}}$ for small ${\displaystyle C}$.
• For any morphism ${\displaystyle f:X\to Y}$ of ${\displaystyle \hat{C}}$, the pullback functor of subobjects ${\displaystyle f^{*}:{\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(Y)\to {\mathrm{S}\mathrm{u}\mathrm{b}}_{\hat{C}}(X)}$ has a right adjoint, denoted ${\displaystyle {\mathrm{\forall }}_f}$, and a left adjoint, ${\displaystyle {\mathrm{\exists }}_f}$. These are the universal and existential quantifiers.
• A locally small category ${\displaystyle C}$ embeds fully and faithfully into the category ${\displaystyle \hat{C}}$ of set-valued presheaves via the Yoneda embedding which to every object ${\displaystyle A}$ of ${\displaystyle C}$ associates the hom functor ${\displaystyle C(-,A)}$.
• The category ${\displaystyle \hat{C}}$ 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, ${\displaystyle \hat{C}}$ is the colimit completion of ${\displaystyle C}$ (see #Universal property below.)

The construction ${\displaystyle C\mapsto \hat{C}=\mathbf{𝐅}\mathbf{𝐜}\mathbf{𝐭}(C^{\text{op}},\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭})}$ 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 ${\displaystyle F=\underrightarrow{\lim }y{U}_i}$ where ${\displaystyle U_i}$ are objects in C. Then let ${\displaystyle \tilde{\eta }F=\underrightarrow{\lim }\eta U_i,}$ which exists by assumption. Since ${\displaystyle \underrightarrow{\lim }-}$ is functorial, this determines the functor ${\displaystyle \tilde{\eta }:\hat{C}\to D}$. Succinctly, ${\displaystyle \tilde{\eta }}$ is the left Kan extension of ${\displaystyle \eta }$ along y; hence, the name "Yoneda extension". To see ${\displaystyle \tilde{\eta }}$ commutes with small colimits, we show ${\displaystyle \tilde{\eta }}$ is a left-adjoint (to some functor). Define ${\displaystyle {\mathscr{H}}om(\eta ,-):D\to \hat{C}}$ to be the functor given by: for each object M in D and each object U in C,

${\displaystyle {\mathscr{H}}om(\eta ,M)(U)={\mathrm{Hom}}_D(\eta U,M).}$

Then, for each object M in D, since ${\displaystyle {\mathscr{H}}om(\eta ,M)(U_i)=\mathrm{Hom}(y{U}_i,{\mathscr{H}}om(\eta ,M))}$ by the Yoneda lemma, we have:

${\displaystyle \begin{array}{rl}{\mathrm{Hom}}_D(\tilde{\eta }F,M)& ={\mathrm{Hom}}_D(\underrightarrow{\lim }\eta U_i,M)=\underleftarrow{\lim }{\mathrm{Hom}}_D(\eta U_i,M)=\underleftarrow{\lim }{\mathscr{H}}om(\eta ,M)(U_i)\\ & ={\mathrm{Hom}}_{\hat{C}}(F,{\mathscr{H}}om(\eta ,M)),\end{array}}$

which is to say ${\displaystyle \tilde{\eta }}$ is a left-adjoint to ${\displaystyle {\mathscr{H}}om(\eta ,-)}$. ${\displaystyle ◻}$

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

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: ${\displaystyle C\to PShv(C)}$ is fully faithful (here C can be just a simplicial set.)^[5]

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

• {{Cite book

| last1=Kashiwara | first1=Masaki | last2=Schapira | first2=Pierre | title=Categories and sheaves | year=2005 |url=https://books.google.com/books?id=mc5DAAAAQBAJ |publisher=Springer |isbn=978-3-540-27950-1 |volume=332 |series=Grundlehren der mathematischen Wissenschaften

• Lurie, J.. Higher Topos Theory. 
• Mac Lane, Saunders; Moerdijk, Ieke (1992). Sheaves in Geometry and Logic. Springer. ISBN 0-387-97710-4. 

• Presheaf in nLab
• Free cocompletion in nLab
• Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.

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