nLab
mate

Definition

Given a 2-category $K$, adjoint pairs $(\eta ,ϵ):f⊣u:b\to a$ and $(\eta \prime ,ϵ\prime ):f\prime ⊣u\prime :b\prime \to a\prime$ , and 1-cells $x:a\to a\prime$ and $y:b\to b\prime$, there is a bijection

$$K(a,b\prime )(f\prime x,yf)\cong K(b,a\prime )(xu,u\prime y)$$K(a,b')(f' x,y f) \cong K(b,a')(x u,u' y)

given by pasting with the unit of one adjunction and the counit of the other, i.e.

$$\begin{array}{ccc}a& \stackrel{x}{\to }& a\prime \\ f\downarrow & \lambda \Downarrow & \downarrow f\prime \\ b& \underset{y}{\to }& b\prime \end{array}\mapsto \begin{array}{ccccccc}b& \stackrel{u}{\to }& a& \stackrel{x}{\to }& a\prime & \stackrel{1}{\to }& a\prime \\ 1\downarrow & ϵ\Downarrow & f\downarrow & \lambda \Downarrow & \downarrow f\prime & \Downarrow \eta \prime & \downarrow 1\\ b& \underset{1}{\to }& b& \underset{y}{\to }& b\prime & \underset{u\prime }{\to }& a\prime \end{array}$$\array{ a & \overset{x}{\to} & a' \\ \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} \\ b & \underset{y}{\to} & b' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ b & \overset{u}{\to} & a & \overset{x}{\to} & a' & \overset{1}{\to} & a' \\ \mathllap{1} \downarrow & \mathllap{\epsilon} \Downarrow & \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} & \Downarrow \mathrlap{\eta'} & \downarrow \mathrlap{1} \\ b & \underset{1}{\to} & b & \underset{y}{\to} & b' & \underset{u'}{\to} & a' }

and

$$\begin{array}{ccc}b& \stackrel{y}{\to }& b\prime \\ u\downarrow & \mu \Uparrow & \downarrow u\prime \\ a& \underset{x}{\to }& a\prime \end{array}\mapsto \begin{array}{ccccccc}a& \stackrel{f}{\to }& b& \stackrel{y}{\to }& b\prime & \stackrel{1}{\to }& b\prime \\ 1\downarrow & \eta \Uparrow & u\downarrow & \mu \Uparrow & \downarrow u\prime & \Uparrow ϵ\prime & \downarrow 1\\ a& \underset{1}{\to }& a& \underset{x}{\to }& a\prime & \underset{f\prime }{\to }& b\prime \end{array}$$\array{ b & \overset{y}{\to} & b' \\ \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} \\ a & \underset{x}{\to} & a' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ a & \overset{f}{\to} & b & \overset{y}{\to} & b' & \overset{1}{\to} & b' \\ \mathllap{1} \downarrow & \mathllap{\eta} \Uparrow & \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} & \Uparrow \mathrlap{\epsilon'} & \downarrow \mathrlap{1} \\ a & \underset{1}{\to} & a & \underset{x}{\to} & a' & \underset{f'}{\to} & b' }

That this is a bijection follows easily from the triangle identities. The 2-cells $\lambda$ and $\mu$ are called mates (or sometimes conjugates) with respect to the adjunctions $f⊣u$ and $f\prime ⊣u\prime$ (and to the 1-cells $x$ and $y$).

Properties

Strict 2-functors preserve adjunctions and pasting diagrams, so that if $F:K\to J$ is a 2-functor and if $\lambda$ and $\mu$ are mates wrt $f⊣u$ and $f\prime ⊣u\prime$ in $K$, then $F\lambda$ and $F\mu$ are mates wrt $Ff⊣Fu$ and $Ff\prime ⊣Fu\prime$ in $J$.

If $\alpha :F\Rightarrow G$ is a 2-natural transformation?, then the naturality identities ${\alpha }_b\circ Ff=Gf\circ {\alpha }_a$ and ${\alpha }_a\circ Fu=Gu\circ {\alpha }_b$ are mates wrt $Ff⊣Fu$ and $Gf⊣Gu$.

Naturality

There are two double categories with objects those of $K$, vertical arrows adjoint pairs in $K$ and horizontal arrows 1-cells of $K$. In one the 2-cells are those of the form $\lambda$ above, while in the other they are those of the form $\mu$. It is easily shown, as in Kelly–Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any $K$ there is a double category $\mathrm{Adj}(K)$, defined up to isomorphism as above but with mate-pairs in $K$ as 2-cells.

What this means is that, for example, the mate of a square coming from a pasting diagram is given by pasting the mates of the individual 2-cells (whenever this makes sense).

In the double category $\mathrm{Adj}(K)$, every vertical arrow has both a companion (the left adjoint) and a conjoint (the right adjoint). (In fact, in some sense it is the universal double category cosntructed from $K$ with this property.) Therefore, it is equivalent to a 2-category equipped with proarrows. More explicitly, there is a forgetful functor $L:{\mathrm{Adj}}_V(K)\to K$ from the 2-category of objects, adjunctions and mate-pairs in $K$ to $K$ that sends an adjunction $f⊣u$ to $f$. It is locally fully faithful, and moreover every $Lf$ has a right adjoint in $K$ by definition; this gives the more traditional definition of a proarrow equipment.

Example

Let $F⊣U:D\to C$ be an adjunction in the 2-category $\mathrm{Cat}$, i.e. a pair of adjoint functors, and $A:*\to C$ and $X:*\to D$ be objects of $C$ and $D$ considered as functors out of the terminal category $*$. Then taking mates with respect to $1⊣1:*\to *$ and $F⊣U$ yields the familiar bijection

$$D(FA,X)\cong C(A,UX)$$D(F A,X) \cong C(A,U X)

and the pasting operations as above yield the usual definition of the isomorphism of adjunction by means of unit and counit. Moreover, the naturality of the mate correspondence yields naturality of the bijection.

References

• Max Kelly, Ross Street, Review of the elements of 2-categories, in Kelly (ed.), Category Seminar, LNM 420.

• Tom Leinster, Higher operads, higher categories, math.CT/0305049, Section 6.1