nLab mate

Context

2-Category theory

2-category theory

Contents

Definition

Proposition

Given a 2-category $K$, adjoint pairs $(\eta,\epsilon) \colon f \dashv u \colon b \to a$ and $(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a'$ , and 1-cells $x \colon a \to a'$ and $y \colon b \to b'$, there is a bijection

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

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

$\array{ a & \overset{x}{\to} & a' \\ \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} \\ b & \underset{y}{\to} & b' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ b & \overset{u}{\to} & a & \overset{x}{\to} & a' & \overset{1}{\to} & a' \\ \mathllap{1} \downarrow & \mathllap{\epsilon} \Downarrow & \mathllap{f} \downarrow & \mathllap{\lambda} \Downarrow & \downarrow \mathrlap{f'} & \Downarrow \mathrlap{\eta'} & \downarrow \mathrlap{1} \\ b & \underset{1}{\to} & b & \underset{y}{\to} & b' & \underset{u'}{\to} & a' }$

and

$\array{ b & \overset{y}{\to} & b' \\ \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} \\ a & \underset{x}{\to} & a' } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ a & \overset{f}{\to} & b & \overset{y}{\to} & b' & \overset{1}{\to} & b' \\ \mathllap{1} \downarrow & \mathllap{\eta} \Uparrow & \mathllap{u} \downarrow & \mathllap{\mu} \Uparrow & \downarrow \mathrlap{u'} & \Uparrow \mathrlap{\epsilon'} & \downarrow \mathrlap{1} \\ a & \underset{1}{\to} & a & \underset{x}{\to} & a' & \underset{f'}{\to} & b' }$
Proof

That this is a bijection follows easily from the triangle identities.

Definition

The 2-cells $\lambda$ and $\mu$ in prop. 1 are called mates (or sometimes conjugates) with respect to the adjunctions $f \dashv u$ and $f' \dashv u'$ (and to the 1-cells $x$ and $y$).

Properties

General

Proposition

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

Proposition

If $\alpha \colon F \Rightarrow G$ is a 2-natural transformation?, then the naturality identities $\alpha_b \circ F f = G f \circ \alpha_a$ and $\alpha_a \circ F u = G u \circ \alpha_b$ are mates wrt $F f \dashv F u$ and $G f \dashv G u$.

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 $Adj(K)$, 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 $Adj(K)$, 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 $L \colon Adj_V(K) \to K$ from the 2-category of objects, adjunctions and mate-pairs in $K$ to $K$ that sends an adjunction $f \dashv u$ to $f$. It is locally fully faithful, and moreover every $L f$ has a right adjoint in $K$ by definition; this gives the more traditional definition of a proarrow equipment.

Examples

Example

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

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

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.

Example

If the ambient 2-category $K$ is the delooping of a monoidal category $(\mathcal{C}, \otimes)$ in that

$K \simeq \mathbf{B}_\otimes \mathcal{C}$

then an adjunction in $K$ is a pair of dual objects and the mate-construction is the construction of dual morphisms between dualizable objects.

Example

Suppose given a commutative square (up to isomorphism) of functors:

$\array{ & \overset{f^*}{\to} & \\ ^{g^*}\downarrow && \downarrow^{k^*}\\ & \underset{h^*}{\to} & }$

in which $f^*$ and $h^*$ have left adjoints $f_!$ and $h_!$, respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute

$k^* f^* \to h^* g^*$

has a mate

$h_! k^* \to g^* f_!$

defined as the composite

$h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_! \,.$

Ones says that the original square satisfies the Beck-Chevalley condition if this mate is an equivalence.

Multi-variable mates

There is a version of the mate correspondence that applies to two-variable adjunctions and $n$-variable adjunctions; see Cheng-Gurski-Riehl.

References

Revised on March 9, 2014 20:09:10 by Mark Gomer? (68.54.30.66)