Notation. presheaf [tt-004B]
✍️source

We'll use \(\mathscr {F}\) to denote a presheaf since sheaf in French is faisceau.

Context

Representables [tt-002K]

Remark. representables [leinster2016basic, ch. 4, 4.1.15] [tt-002L]

A category is a world of objects, all looking at one another. Each sees the world from a different viewpoint.

We may ask: what objects see? Fix an object, this can be described by the arrows from it, this corresponds to the covariantly representable functor.

We can also ask the dual question: how objects are seen? Fix an object, this can be described by the arrows into it, this corresponds to the contravariantly representable functor.

Definition. set-valued [kostecki2011introduction, 4.4] [tt-001L]

A functor \(\mathscr {F} : {\cal C} \to \mathbf {Set}\) is called set-valued.

Definition. hom-functor [kostecki2011introduction, 3.1, example 10] [tt-001S]

For every locally small category \({\cal C}\), the covariant hom-functor, denoted \[{\cal C}(X,-): {\cal C} \to \mathbf {Set} \] is given by

$\text{[math]}$

Conversely, the contravariant hom-functor, denoted \[{\cal C}(-,X): {\cal C}^{op} \to \mathbf {Set} \] is given by

$\text{[math]}$

Further, the hom-bifunctor, denoted \[{\cal C}(-,=): {\cal C}^{op} \times {\cal C} \to \mathbf {Set} \] defined as a contravariant hom-functor at first argument and as a covariant hom-functor at second argument.

We see \(-\) and \(=\) as placeholders for any object and its "associated arrow" (whose domain/codomain is the object, respectively) in the corresponding category. And we use boxes to mark the placeholder objects in diagrams.

Diagramatically [leinster2016basic, 4.1.22],

$\text{[math]}$

Definition. representable functor [kostecki2011introduction, 4.4] [tt-001O]

A set-valued functor \(\mathscr {F}: {\cal C} \to \mathbf {Set}\) is called covariantly representable if for some \(X \in {\cal C}\), \[\tau : \mathscr {F} \cong {\cal C}(X,-)\] where \(\cong \) denotes a natural isomorphism.

Conversely, a set-valued functor \(\mathscr {G} : {\cal C}^{op} \to \mathbf {Set}\) is called contravariantly representable if for some \(X \in {\cal C}\), \[\tau : \mathscr {G} \cong {\cal C}(-, X)\]

Such an object \(X\) is called a representing object for the functor \(\mathscr {F}\) or \(\mathscr {G}\), respectively.

The pair \((\tau , X)\) is called a representation of the functor \(\mathscr {F}\) (respectively, \(\mathscr {G}\) ).

Remark. representation [leinster2016basic, 4.1.3, 4.1.17] [tt-002M]

A representation \((\tau , X)\) of a representable functor \(\mathscr {F}\) is a choice of an object \(X \in {\cal C}\) and an isomorphism \(\tau \) between the corresponding type of hom-functor and \(\mathscr {F}\).

Representable functors are sometimes just called representables. Only set-valued functors can be representable.

Definition. Yoneda embedding functors [leinster2016basic, 4.1.15] [tt-002T]

Let \({\cal C}\) be a locally small category. The covariant Yoneda embedding functor of \({\cal C}\) is the functor \[ \mathscr {H}^{\bullet }: {\cal C}^{op} \to [{\cal C}, \mathbf {Set}] \] defined on objects \(X\) by the covariant hom-functor on \(X\).

This functor embeds what every object in \({\cal C}\) sees the "world" of the category \({\cal C}\), i.e. arrows from each object.

Conversely, the (contravariant) Yoneda embedding functor of \({\cal C}\) is the functor \[ \mathscr {H}_{\bullet }: {\cal C} \to [{\cal C}^{op}, \mathbf {Set}] \] defined on objects \(X\) by the contravariant hom-functor on \(X\).

This functor embeds how every object in \({\cal C}\) is "seen", i.e. arrows to each object.

\(\bullet \) is a placeholder for an object. \(\mathscr {H}^X\) and \(\mathscr {H}_X\) denote the corresponding Yoneda embedding functors applied to \(X\), and are called covariant/contravariant Yoneda functors, respectively.

Diagramatically [rosiak2022sheaf, def. 161]:

$\text{[math]}$

When one speaks of the Yoneda (embedding) functor without specifying covariant or contravariant, it means the contravariant one, because it's the one used in the Yoneda lemma.

Lemma. Yoneda [leinster2016basic, 4.2.1] [tt-002N]

Let \({\cal C}\) be a locally small category. Then \[ \operatorname {Nat}(\mathscr {H}_X, \mathscr {F}) \cong \mathscr {F}(X) \] naturally in \(X \in {\cal C}\) and \(\mathscr {F} \in \left [{\cal C}^{\mathrm {op}}, \mathbf {Set} \right ]\), where \(\mathscr {H}_X\) is the (contravariant) Yoneda embedding functor on \(X\), and Nat denotes all the natural transformations between the two functors.

Notation. Yoneda lemma [leinster2016basic, 4.2.1] [tt-002U]

Diagramatically, \(\operatorname {Nat}(\mathscr {H}_X, \mathscr {F})\) is

$\text{[math]}$ and it's also denoted by \(\left [{\cal C}^{op}, \mathbf {Set}\right ]\left (\mathscr {H}_X, \mathscr {F}\right )\) in the sense of \(\operatorname {Hom}_{\left [{\cal C}^{op}, \mathbf {Set}\right ]}\left (\mathscr {H}_X, \mathscr {F}\right )\) where \(\left [{\cal C}^{op}, \mathbf {Set}\right ]\) is a functor category.

Remark. Yoneda philosophy [rosiak2022sheaf, sec. 6.6] [tt-002S]

The Yoneda lemma can be regarded as saying:

To understand an object it suffices to understand all its relationships with other things.

This is similar to the seventeenth-century philosopher Spinoza's idea that what a body is (its “essence”) is inseparable from all the ways that the body can affect (causally influence) and be affected (causally influenced) by other bodies.

The idea of Yoneda is that we can be assured that if a robot wants to learn whether some object \(X\) is the same thing as object \(Y\), it will suffice for it learn whether \[{\cal C}(-, X) \cong {\cal C}(-, Y)\] or, dually, \[{\cal C}(X, -) \cong {\cal C}(Y, -)\] i.e. whether all the ways of probing \(X\) with objects of its environment amount to the same as all the ways of probing \(Y\) with objects of its environment.

Lemma. full and faithful [kostecki2011introduction, 4.8] [tt-002X]

The Yoneda embedding functor \(\mathscr {H}_{\bullet }: {\cal C} \to [{\cal C}^{op}, \mathbf {Set}]\) is full and faithful.

Definition. Ob, Arr [leinster2016basic, 4.1.6] [tt-003U]

Given a small category \({\cal C}\), there is a functor \(\operatorname {Ob} : \mathbf {Cat} \to \mathbf {Set}\) that sends \({\cal C}\) to its set of objects where \(\mathbf {Cat}\) is the category of small categories. Thus, \[ \mathscr {H}^{\mathrm {1}}({\cal C}) \cong \operatorname {Ob}({\cal C}) \] where \(\mathscr {H}\) is a Yoneda embedding functor.

This isomorphism is natural in \({\cal C}\); hence \(\operatorname {Ob} \cong \mathbf {Cat}(\mathrm {1}, -)\) where \(\mathbf {Cat}(\mathrm {1}, -)\) is a covariant hom-functor.

Functor \(\operatorname {Ob}\) is representable. Similarly, the functor \(\operatorname {Arr} : \mathbf {Cat} \to \mathbf {Set}\) sending a small category to its set of arrows is representable.

Definition. presheaf [leinster2016basic, 1.2.15] [tt-002Q]

Let \({\cal C}\) be a category. A presheaf \(\mathscr {F}\) on \({\cal C}\) is a functor \({\cal C}^{op} \to \mathbf {Set}\).

It is called representable if \(\mathscr {F} \cong \mathscr {H}_X\) for some \(X\).

Notation. presheaf [tt-004B]
✍️source

We'll use \(\mathscr {F}\) to denote a presheaf since sheaf in French is faisceau.

Remark. presheaf and Yoneda lemma [leinster2016basic, 4.2.1] [tt-002V]

The Yoneda lemma says that for any \(X \in {\cal C}\) and presheaf \(\mathscr {F}\) on \({\cal C}\), a natural transformation \(\mathscr {H}_X \to \mathscr {F}\) is an element of \(\mathscr {F}(X)\) of shape \(\mathscr {H}_X\).

We may ask the question [chen2016infinitely, 68.6.4]:

What kind of presheaves are already "built in" to the category \({\cal C}\)?

The answer by the Yoneda lemma is, the Yoneda embedding \(\mathscr {H}_{\bullet }: {\cal C} \to [{\cal C}^{op}, \mathbf {Set}]\) embeds \({\cal C}\) into its own presheaf category.

In mathematics at large, the word "embedding" is used (sometimes informally) to mean a map \(i: X \to Y\) that makes \(X\) isomorphic to its image in \(Y\), i.e. \(X \cong i(X)\). [leinster2016basic, 1.3.19] tells us that in category theory, a full and faithful functor \(\mathscr {I}: X \to Y\) can reasonably be called an embedding, as it makes \(X\) equivalent to a full subcategory of \(Y\).

So, \({\cal C}\) is equivalent to the full subcategory of the presheaf category \([{\cal C}^{op}, \mathbf {Set}]\) whose objects are the representables.

Definition. The category of presheaves [kostecki2011introduction, 4.5, example 1] [tt-002R]

The functor category of contravariant set-valued functors \([{\cal C}^{op}, \mathbf {Set}]\), called the category of presheaves or varying sets, the objects of which are contravariant functors \({\cal C}^{op} \to \) Set. It may be regarded as a category of diagrams in \(\mathbf {Set}\) indexed contravariantly by the objects of \({\cal C}\).

By definition, objects of \({\cal C}\) play the role of stages, marking the "positions" (in passive view) or "movements" (in active view) of the varying set \(\mathscr {F}: {\cal C}^{op} \to \mathbf {Set}\). For every \(X\) in \({\cal C}^{op}\), the set \(\mathscr {F}(X)\) is a set of elements of \(\mathscr {F}\) at stage \(X\).

An arrow \(f: Y \to X\) between two objects in \({\cal C}^{op}\) induces a transition arrow \(\mathscr {F}(f): \mathscr {F}(X) \to \mathscr {F}(Y)\) between the varying set \(\mathscr {F}\) at stage \(A\) and the varying set \(\mathscr {F}\) at stage \(B\).

Related

Definition. presheaf [leinster2016basic, 1.2.15] [tt-002Q]

Let \({\cal C}\) be a category. A presheaf \(\mathscr {F}\) on \({\cal C}\) is a functor \({\cal C}^{op} \to \mathbf {Set}\).

It is called representable if \(\mathscr {F} \cong \mathscr {H}_X\) for some \(X\).