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

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

$\text{[math]}$

References

Reference. Category theory using string diagrams [marsden2014category]

2014
Daniel Marsden

@article{marsden2014category,
 title = {Category theory using string diagrams},
 author = {Marsden, Daniel},
 year = {2014},
 journal = {arXiv preprint arXiv:1401.7220}
}

Context

Adjunctions [tt-001T]

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

Given functors

$\text{[math]}$ 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 $\text{[math]}$ iff there exists a natural isomorphism \(\sigma \) between the following two hom-bifunctors: \[ {\cal D}(\mathscr {L}(-), =) \cong {\cal C}(-, \mathscr {R}(=)) \] diagramatically, $\text{[math]}$

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

$\text{[math]}$ 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 $\text{[math]}$ 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 $\text{[math]}$ 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 $\text{[math]}$

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

$\text{[math]}$ 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

$\text{[math]}$ 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

$\text{[math]}$ 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

$\text{[math]}$ commutes for any \(f \in {\cal C}\), and the diagram $\text{[math]}$ 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

$\text{[math]}$ 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):

$\text{[math]}$ $\text{[math]}$ where $\text{[math]}$ 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

$\text{[math]}$ 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 $\text{[math]}$ in triangle identities.

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

Given an adjunction

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

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

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

$\text{[math]}$

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

$\text{[math]}$ 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

$\text{[math]}$ 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,
$\text{[math]}$ where the functor \(X\) is the constant object functor.

Backlinks

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.

Related

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

Given an adjunction

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

Definition. string diagram [nakahira2023diagrammatic, table 1.1] [tt-003V]

A string diagram represents categories as surfaces (2-dimensional), functors as wires (1-dimensional), natural transformations as blocks or just dots (0-dimensional).

String diagrams has the advantage of being able to represent objects, arrows, functors, and natural transformations from multiple categories, and their vertical and horizontal composition, and has various topologically plausible calculational rules for proofs.