remark. Limits [leinster2016basic, ch. 5] [tt-0024]
✍️source

Adjointness is about the relationships between categories. Representability is a property of set-valued functors. Limits are about what goes on inside a category.

Whenever you meet a method for taking some objects and arrows in a category and constructing a new object out of them, there is a good chance that you are looking at either a limit or a colimit.

Context

limits [tt-0023]

remark. Limits [leinster2016basic, ch. 5] [tt-0024]
✍️source

Adjointness is about the relationships between categories. Representability is a property of set-valued functors. Limits are about what goes on inside a category.

Whenever you meet a method for taking some objects and arrows in a category and constructing a new object out of them, there is a good chance that you are looking at either a limit or a colimit.

definition. cone [leinster2016basic, 5.1.19] [tt-0028]

Let \({\cal C}\) be a category, \({\cal J}\) a small category, and \(\mathscr {D} : {\cal J} \to {\cal C}\) a diagram in \({\cal C}\) of shape \({\cal J}\).

A cone on \(\mathscr {D}\) is an object \(V \in {\cal C}\) (the vertex of the cone) together with a family \[ \left (V \xrightarrow {\pi _J} \mathscr {D}(J)\right )_{J \in {\cal J}} \] of arrows in \({\cal C}\) such that for all arrows \(J \to J'\) in \({\cal J}\), the diagram commutes.

The family of arrows are components of a natural transformation \(\pi : \Delta _V \to \mathscr {D}\), i.e. from the constant functor ( which assigns the same object \(V\) to any object \(J_i\) in \({\cal J}\)) to diagram functor \(\mathscr {D}\).

For simplicity, we refer to a cone by "a cone \((V, \pi )\) on \(\mathscr {D}\)".

example. a cone on a diagram [kostecki2011introduction, 4.9] [tt-0029]

The cone for a diagram is which indeed looks like a cone with the vertex \(V\).

definition. limit [kostecki2011introduction, 4.10] [tt-002A]

A cone \((V, \pi )\) on \(\mathscr {D} : {\cal C} \to {\cal J}\) is called a limit of \(\mathscr {D}\), denoted \[\lim \mathscr {D}\] if the diagram commutes for every cone \((V, \pi )\) on \(\mathscr {D}\).

The arrows \(\pi _J\) are called the projections of the limit.

Other possible terms of limit are limiting cone, universal cone.

example. limits [kostecki2011introduction, 4.11, example 1-4] [tt-002B]

The basic types of diagrams are actually examples of limits:

binary product [kostecki2011introduction, 2.18]:
pullback (fiber product) [kostecki2011introduction, 2.12]:
equalizer [leinster2016basic, 5.1.11]:
the limit of \(\mathscr {D} : \mathbf {\emptyset } \to {\cal C} \), where \(\mathbf {\emptyset }\) is an empty category: i.e. the terminal object \(\mathrm {1}\) in \({\cal C}\). In particular, for \({\cal C} = \mathbf {Set}\) we have \[\mathrm {-} \xrightarrow {!} V = \lim \mathscr {D} = \{∗\} \]

remark. cocone, colimit [kostecki2011introduction, 4.10] [tt-002E]

A cocone and a colimit are defined by dualization, that is, by reversing the arrows in cone [leinster2016basic, 5.1.19] and limit [kostecki2011introduction, 4.10].

In another word, given \(\mathscr {D}^{op} : {\cal J}^{op} \to {\cal C}^{op}\), a cocone on \(\mathscr {D}\) is a cone on \(\mathscr {D}^{op}\), a colimit of \(\mathscr {D}\) is a limit of \(\mathscr {D}^{op}\) [leinster2016basic, 5.2.1].

The arrows \(\pi _J\) are called the coprojections of the colimit.

In the same say, one can and show that coequaliser, coproduct, pushout and initial object are examples of colimits.

lemma. limits via products and equalizers [stacks2017stacks, 002N, 002P] [tt-0057]

If all products and equalizers exist, all limits exist.

Dually, if all coproducts and coequalizers exist, all colimits exist.

definition. has (finite) limits, (finitely) complete, left exact [kostecki2011introduction, 4.10] [tt-002F]

We say that a category \({\cal C}\) has (finite) limits or is (finitely) complete if every diagram \(\mathscr {D} : {\cal J} \to {\cal C}\), where \({\cal J}\) is a (finite) category, has a limit.

A category \({\cal C}\) is called left exact iff it is finitely complete.

remark. (finitely) cocomplete, right exact [kostecki2011introduction, 4.10] [tt-003C]

Dually to definition [tt-002F], when every diagram \(\mathscr {D} : {\cal J} \to {\cal C}\), where \({\cal J}\) is a (finite) category, has a colimit, it is said that the category \({\cal C}\) has (finite) colimits or is (finitely) cocomplete.

A category is called right exact iff it is finitely cocomplete.

lemma. (finitely) (co)complete category [kostecki2011introduction, 4.14] [tt-002J]

A category \({\cal C}\) is (finitely) complete if it has a terminal object, equalizers and (finite) products, or if it has a terminal object and (finite) pullbacks.

Dually, a category \({\cal C}\) is (finitely) cocomplete if it has an initial object, coequalizers and (finite) coproducts, or if it has an initial object and (finite) pushouts.

lemma. (co)complete functor category [kostecki2011introduction, 4.15] [tt-002I]

If category \({\cal D}\) is complete and category \({\cal C}\) is small, then the functor category \({\cal D}^{{\cal C}}\) is complete.

Dually, if category \({\cal D}\) is cocomplete and category \({\cal C}\) is small, then the functor category \({\cal D}^{{\cal C}}\) is cocomplete.

definition. preorder, partial order, total order [kostecki2011introduction, 1.2, example 9] [tt-002C]

Let \(P\) be a set. The properties

(reflexivity) \(\forall p \in P, p \leq p\)
(transitivity) \(\forall p, q, r \in P, p \leq q \wedge q \leq r \Rightarrow p \leq r\)

define a preorder \((P, \leq )\).

A partially ordered set (called a partial order, or a poset) is defined as a preorder \((P, \leq )\) for which

(antisymmetry) \(\forall p \in P, p \leq q \wedge q \leq p \Rightarrow p=q\)

holds.

A total order (or a linear order) is a partial order \((P, \leq )\) for which

(comparability) \(\forall p, q \in P, p \leq q \vee q \leq p\)

The category \(\mathbf {Preord}\) consists of objects which are preorders and of arrows which are orderpreserving functions.

The category \(\mathbf {Poset}\) consists of objects which are posets and of arrows which are order-preserving functions between posets, that is, the maps \(T: P \to P'\) such that \[ p \leq q \Rightarrow T(p) \leq T(q) \]

Any any preorder \((P, \leq )\) and poset \((P, \leq )\) can be considered as a category consisting of objects which are elements of a set \(P\) and arrows defined by \(p \to q \Longleftrightarrow p \leq q\).

definition. directed poset [rosiak2022sheaf, def. 285] [tt-0051]

A directed poset is a poset that is inhabited (nonempty) and for which every finite subset has an upper bound. Explicitly,

(directedness) \(\forall x,y \in P, \exists z \in P, x \leq z \land y \leq z\)

example. preorder, poset, directed poset [tt-0052]

An example of a preorder category which is not poset is:

An example of a poset category which is not a directed poset is [rosiak2022sheaf, example 3] :

An example that is a directed poset category but not a total order is: where each pair of nodes has a common upper bound (thus satisfying directedness), but there is no path between the two nodes on the center, thus violating comparability.

A more complicated example of a directed poset category which is not a total order is [spivak2013category, example 3.4.1.3]:

One can see immediately that this is a preorder because length=0 paths give reflexivity and concatenation of paths gives transitivity. To see that it is a partial order we only note that there are no loops.

To see that it is a poset, we note that every pair of nodes from one side or both sides has the central node as an upper bound, thus satisfying directedness.

But this partial order is not a total order because there is no path (in either direction) between some nodes, thus violating comparability.

definition. inverse limit, projective limit [kostecki2011introduction, 4.11] [tt-002D]

Let \({\cal J}\) be a directed poset and \(\mathscr {F} : {\cal J} \to {\cal C}\) be a contravariant functor. The limit of \(\mathscr {F}\) is called an inverse limit or projective limit, and is denoted \(\lim \limits _{\leftarrow {\cal J}} \mathscr {F}\) or simply \(\lim \limits _{\longleftarrow } \mathscr {F}\).

definition. direct limit, inductive limit [kostecki2011introduction, 4.12] [tt-002G]

Let \({\cal J}\) be a directed poset and \(\mathscr {F}: {\cal J} \to {\cal C}\) be a contravariant functor. The colimit of \(\mathscr {F}\) is called a direct limit (some called directed limit) or inductive limit, and is denoted \(\lim \limits _{\to {\cal J}} \mathscr {F}\), or simply \(\lim \limits _{\longrightarrow } \mathscr {F}\).

This is dual to inverse limit.

definition. preserves (all) (co)limits, left/right exact [kostecki2011introduction, 4.13] [tt-002H]

A functor \(\mathscr {F}: {\cal C} \to {\cal D}\) preserves (all) limits and is called left exact iff it sends all limits in \({\cal C}\) into limits in \({\cal D}\).

Dually, a functor \(\mathscr {F}: {\cal C} \to {\cal D}\) preserves (all) colimits and is called right exact iff it sends all colimits in \({\cal C}\) into colimits in \({\cal D}\).

remark. directions in (co)limits [tt-004Y]

	Limit	Colimit
diagram
arrows through the vertex	into the diagram	out of the diagram
on (co)shape P	pullback	pushout
categories have finite ...	left exact	right exact
functors preserve all ...	left exact	right exact
on directed poset	inverse/projective limit \(\lim \limits _{\longleftarrow } \mathscr {F}\)	direct/inductive limit \(\lim \limits _{\longrightarrow } \mathscr {F}\)

One can see from the table that, in general, limits have the direction "back" "into" (where "back", "left", "inverse" are directional consistent), and colimits have the opposite: "forward" "out of".

This might help to memorize the directions in these concepts without disorientation.

reference. Basic category theory [leinster2016basic]

@article{leinster2016basic,
 title = {Basic category theory},
 author = {Leinster, Tom},
 year = {2016},
 url = {https://arxiv.org/abs/1612.09375},
 journal = {arXiv preprint arXiv:1612.09375}
}

adjunctions [tt-001T]

definition. adjoint functor [kostecki2011introduction, 5.1] [tt-001Q]

Given functors we say \(\mathscr {L}\) and \(\mathscr {R}\) are a pair of adjoint functors, or together called an adjunction between them, \(\mathscr {L}\) is called left adjoint to \(\mathscr {R}\), and \(\mathscr {R}\) is called right adjoint to \(\mathscr {L}\), denoted \[\mathscr {L} \dashv \mathscr {R} : {\cal C} \rightleftarrows {\cal D}\] or iff there exists a natural isomorphism \(\sigma \) between the following two hom-bifunctors: \[ {\cal D}(\mathscr {L}(-), =) \cong {\cal C}(-, \mathscr {R}(=)) \] diagramatically,

The components of the natural isomorphism \(\sigma \) are isomorphisms \[ \sigma _{XY} : {\cal D}(\mathscr {L}(X), Y) \cong {\cal C}(X, \mathscr {R}(Y)) \]

remark. adjoint functor [kostecki2011introduction, 5.1] [tt-001R]

An adjunction \(\mathscr {L} \dashv \mathscr {R}\) means arrows \(\mathscr {L}(X) \to Y\) are essentially the same thing as arrows \(X \to \mathscr {R}(Y)\) for any \(X \in {\cal C}\) and \(Y \in {\cal D}\).

This means the diagram commutes for any arrows \(f: \mathscr {L}(X) \to Y\) in \({\cal D}\).

The above can also be diagramatically denoted by transposition diagram \[ \begin {array}{ccccccc} X' & \xrightarrow {x} & X & \xrightarrow {\sigma _{X Y}(f)} & \mathscr {R}(Y) & \xrightarrow {\mathscr {R}(y)} & \mathscr {R}\left (Y'\right ) \\ \hline \mathscr {L}\left (X'\right ) & \xrightarrow {\mathscr {L}(x)} & \mathscr {L}(X) & \xrightarrow {f} & Y & \xrightarrow {y} & Y' \end {array} \] or simply, \[ \frac {X \to \mathscr {R}(Y) \quad ({\cal C})}{\mathscr {L}(X) \to Y \quad ({\cal D})} \]

An adjunction is a concept that describes the relationship between two functors that are weakly inverse to each other [nakahira2023diagrammatic, sec. 4].

By "weakly inverse", we don't mean that applying one after the other gives the identity functor, but in a sense similar to eroding (i.e. enhancing holes) and dilating (i.e. filling holes) an image, applying them in different order yeilds upper/lower "bounds" of the original image [rosiak2022sheaf, sec. 7.1].

notation. string diagrams: adjunction [nakahira2023diagrammatic, sec. 3.1] [tt-001U]

Here we follow the string diagram style of [marsden2014category] and [sterling2023models], but with additional string diagram types inspired by [nakahira2023diagrammatic, eq. 3.1, 4.3].

The covariant hom-functor \({\cal C}(X, -)\), denoted \(-^X\), can be represented in string diagrams as where the dotted circle denotes any arrows with the domain \(X\) and codomain \(-\).
The contravariant hom-functor \({\cal C}(-, X)\), denoted \(X^-\), can be represented in a similar manner.
The hom-bifunctor \({\cal C}(-,=)\), also denoted \(=^-\), can be represented as where the dotted circle denotes any arrows with the domain \(-\) and codomain \(=\).
The natural isomorphism \[{\cal D}(\mathscr {L}(-), =) \cong {\cal C}(-, \mathscr {R}(=))\] in adjunction can be represented as

definition. transpose [kostecki2011introduction, 5.1] [tt-001X]

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), there exists \(f^{\sharp }\) and \(g^{\flat }\) such that the diagrams commute for any arrow \(f: X \to \mathscr {R}(Y)\) in \({\cal C}\), \(g: \mathscr {L}(X) \to Y\) in \({\cal D}\).

\(f^{\sharp }\) is called the left transpose of \(f\). \(g^{\flat }\) is called the right transpose of \(g\).

Other possible terms are left/right adjunct of each other, and mates [nlab2023adjunct].

remark. idempotent [zhang2021type, 5.30] [tt-0037]

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), we may obtain two endofunctors \( \mathscr {L} \mathbin {\bullet } \mathscr {R} : {\cal C} \to {\cal C}\) and \( \mathscr {R} \mathbin {\bullet } \mathscr {L} : {\cal D} \to {\cal D}\) that commute the diagram that means they are both idempotent, i.e. applying \(\mathscr {L} \mathbin {\bullet } \mathscr {R}\) any times yields the same result as applying it once, and similarly for \(\mathscr {R} \mathbin {\bullet } \mathscr {L}\).

definition. (co)unit [zhang2021type, 5.30] [tt-001V]

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), the natural transformation \[\eta : \mathit {1}_{{\cal C}} \to \mathscr {L} \mathbin {\bullet } \mathscr {R} \] is called the unit of the adjunction, and \[\epsilon : \mathscr {R} \mathbin {\bullet } \mathscr {L} \to \mathit {1}_{{\cal D}}\] is called the counit.

We call an arrow \[\eta _X: X \to (\mathscr {L} \mathbin {\bullet } \mathscr {R})(X)\] a unit over \(X\), and \[\epsilon _Y: (\mathscr {R} \mathbin {\bullet } \mathscr {L})(Y) \to Y\] a counit over \(Y\). They are components of the natural transformations \(\eta \) and \(\epsilon \), respectively.

Diagramatically, the diagrams commute.

lemma. universality of (co)unit [kostecki2011introduction, 5.3] [tt-001W]

The unit \(\eta \) and counit \(\epsilon \) of an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\) are universal, i.e. the diagram commutes for any \(f \in {\cal C}\), and the diagram commutes for any \(g \in {\cal D}\).

lemma. triangle identities [kostecki2011introduction, 5.4] [tt-001Z]

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), the diagrams commute.

Note that \(\mathscr {L}\) in \(\epsilon _\mathscr {L}\) is a subscript, meaning \(\epsilon _\mathscr {L}: {\cal D} \to {\cal D}, \mathscr {L}(X) \mapsto \epsilon _{\mathscr {L}(X)}\) for \(X \in {\cal C}\). Similar for \(\eta _\mathscr {R}\).

lemma. snake identities [nakahira2023diagrammatic, thm. 4.8] [tt-0030]

Continuing from notation [tt-001U], the triangle identities can be represented in string diagrams as follows, and called the snake identities (or zig-zag identities): where are the unit and counit of the adjunction, respectively.

notation. string diagram: snake identities [nakahira2023diagrammatic, thm. 4.8] [tt-0031]

Following notation [tt-001G], recall that a string diagram is composed from top to bottem, left to right, we can read the left snake in snake identities as \[ \mathscr {L} \xmapsto {(\eta \mathbin {\bullet } \mathscr {L}) \mathbin {\bullet } (\mathscr {L} \mathbin {\bullet } \epsilon )} \mathscr {L}\]

where each pair of parentheses corresponds to an overlay in the string diagram, and this is equivalent to

in triangle identities.

lemma. (co)unit and transposes [leinster2016basic, 2.2.4] [tt-001Y]

Given an adjunction with unit \(\eta \) and counit \(\epsilon \), the diagrams \[h=\eta _X \mathbin {\bullet } \mathscr {R}(h^{\sharp })\] and \[f^{\sharp }=\mathscr {L}(f) \mathbin {\bullet } \epsilon _Y\] commute.

notation. string diagrams: (co)unit and transposes [marsden2014category, lem. 3.6] [tt-0033]

In string diagrams, lemma [tt-001Y] can be represented as:

remark. topologically plausible [leinster2016basic, 2.2.9] [tt-0039]

The string diagrams in lemma [tt-0030] and notation [tt-0033] are topologically plausible equations, i.e. the equality can be obtained by simply pulling the string straight.

lemma. (co)unit and natural isomorphism [kostecki2011introduction, eq. 127] [tt-0020]

The natural transformation \(\sigma _{XY}\) and \(\tau _{XY}\) that are the components of the natural isomorphism in the adjunction \(\mathscr {L} \dashv \mathscr {R} : {\cal C} \rightleftarrows {\cal D}\) are related to the unit and counit of the adjunction: \[\begin {aligned} \sigma _{XY}(f) & = \eta _X \mathbin {\bullet } \mathscr {R}(f^{\sharp })\\ \tau _{XY}(g) & = \mathscr {L}(g^{\flat }) \mathbin {\bullet } \epsilon _Y \end {aligned} \] and they are reverse of each other \[\sigma _{XY} = \tau _{YX}^{-1}\]

proof.

This can be read out from the diagrams in universality of (co)unit [kostecki2011introduction, 5.3] and transpose [kostecki2011introduction, 5.1].

lemma. uniqueness of adjoints [kostecki2011introduction, 5.8] [tt-0021]

A left or right adjoint, if it exists, is unique up to natural isomorphism.

proof.

For the left adjoint, from the university of \(\eta \) it follows that there exists a unique, up to isomorphism, isomorphism between different left adjoints. It remains to show naturality of this isomorphim, which is left as an exercise. The proof for right adjoint follows by duality.

theorem. adjunction via (co)units [leinster2016basic, 2.2.5] [tt-0038]

Given categories and functors there is a one-to-one correspondence between the adjunction \(\mathscr {L} \dashv \mathscr {R}\) and the pairs of natural transformations \(\eta \) and \(\epsilon \) satisfying the the triangle identities.

proof.

From lemma [tt-001Z], it follows that every adjunction between \(\mathscr {L}\) and \(\mathscr {R}\) gives rise to a pair of transformations \(\eta \) and \(\epsilon \) satisfying the triangle identities.

To show that there exists a unique adjunction for \(\eta \) and \(\epsilon \), the uniqueness follows from lemma [tt-001Y], the existence can use the construction in the spirit of definition [tt-001X].

theorem. adjunction via initial objects [leinster2016basic, 2.3.6] [tt-003A]

Given categories and functors there is a one-to-one correspondence between:

the adjunction \(\mathscr {L} \dashv \mathscr {R}\)
natural transformations \(\eta : \mathit {1}_{{\cal C}} \to \mathscr {L} \mathbin {\bullet } \mathscr {R}\) such that \(\eta _X\) is initial in the comma category \(X \Rightarrow \mathscr {R}\) for every \(X \in {\cal C}\)

Diagramatically, where the functor \(X\) is the constant object functor.

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

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

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],

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]:

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 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]

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 \(B\).

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

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