The n-Category Café

Mates

Example (adjunct arrows). Taking the identity functors at the terminal category for the left-hand adjunction, mates are simply adjunct arrows.

$$\array{ \ast & \stackrel{a}{\to} & \mathcal{A} \\ \downarrow\dashv\uparrow & {}^{F a\to b}\swarrow\searrow^{a\to U b} & {}^F\downarrow\dashv\uparrow^U \\ \ast & \stackrel{b}{\to} & \mathcal{B}}$$

For instance the counit $\epsilon \colon F U \to 1$ is mates with the identity at $U$.

$$\array{ \mathcal{B} & \stackrel{U}{\to} & \mathcal{A} \\ \downarrow\dashv\uparrow & {}^{\epsilon}\swarrow\searrow^{1_U} & {}^F\downarrow\dashv\uparrow^U \\ \mathcal{B} & = & \mathcal{B}}$$

Example (adjunctions with a parameter). Suppose $\otimes \colon \mathcal{A} \times \mathcal{B} \to \mathcal{C}$ is a left-closed bifunctor, meaning that for each $a \in \mathcal{A}$ the functor $a \otimes -$ has a right adjoint that we’ll denote $hom(a,-)$. A morphism $f \colon a \to a'$ defines a natural transformation whose mate

$$\array{ \mathcal{B} & = & \mathcal{B} & \mathcal{B} & = & \mathcal{B} \\ {}^{a' \otimes -} \downarrow & {}^{f \otimes -}\swarrow & {}^{a \otimes -}\downarrow & \quad\uparrow^{\hom(a',-)} & \searrow^{\hom(f,-)} & \quad\uparrow^{\hom(a,-)} \\ \mathcal{C} & = & \mathcal{C} & \mathcal{C} & = & \mathcal{C}}$$

is the unique natural transformation compatible with the hom-set isomorphisms

$$\array { \mathcal{C}(a' \otimes b,c ) & \cong & \mathcal{B}(b,\hom(a',c)) \\ {}^{f\otimes-}\downarrow & & \downarrow^{\hom(f,-)} \\ \mathcal{C}(a \otimes b,c ) & \cong & \mathcal{B}(b,\hom(a,c))}$$

Using these morphisms, the pointwise right adjoints assemble into a bifunctor $hom \colon \mathcal{A}^{op} \times \mathcal{C} \to \mathcal{B}$. This is the content of Theorem IV.7.3 of MacLane which refers to such bifunctors as an adjunction with a parameter.

“Naturality” of the mates correspondence

You may have heard of some theorems whose proofs employ the theory of mates. One is that a right adjoint defines a (lax) monad morphism if and only if its left adjoint defines a colax monad morphism. Another is that an adjunction in a 2-category acted upon by a 2-monad $T$ lifts to the 2-category of $T$-algebras, lax $T$-algebra morphisms, and $T$-transformations if and only if the left adjoint is a strong $T$-algebra morphism. Both of these results are easy consequences of the following general theorem:

Theorem (Kelly-Street). Taking mates defines an “identity-on-dim$\lt 2$” double isomorphism between double categories $\mathbb{L}Adj$ and $\mathbb{R}Adj$ whose objects are categories, horizontal morphisms are functors, vertical morphisms are fully specified adjunctions in the direction of the left adjoints, and squares are natural transformations $\lambda$ resp. $\rho$ as above.

We might say the Kelly-Street theorem expresses the “naturality” of the mates correspondence — more precisely it describes the double functoriality of the mates correspondence. In practice it tells you precisely when a commutative diagram transforms into another commutative diagram by taking mates.

Mates between adjunctions with a parameter

To see how this theorem is used, let us consider an example that will be relevant to what follows.

Proposition. Suppose $\lambda$ is a natural transformation involving a pair of left-closed bifunctors $\otimes, \otimes'$. For each $a \in \mathcal{A}$, let $\rho_a$ denote the mate of $\lambda_a$. Then these natural transformations assemble into a natural transformation of two variables.

$$\array{ \mathcal{A} \times \mathcal{B} & \stackrel{A \times B}{\to} & \mathcal{A}' \times \mathcal{B}' & \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' \\ {}^\otimes\downarrow & {}^\lambda\swarrow & {}^{\otimes'}\downarrow & \uparrow^{hom} & \searrow^\rho & \uparrow^{hom'} \\ \mathcal{C} & \stackrel{C}{\to} & \mathcal{C}' & \mathcal{A}^{op} \times \mathcal{C} & \stackrel{A^{op} \times C}{\to} & \mathcal{A}'^{op} \times \mathcal{C}'}$$

Proof: Naturality of $\lambda$ in $\mathcal{A}$ can be expressed as an equation between two pasting diagrams in $\mathbb{L}Adj$.

$$\array{ \mathcal{B} & = & \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & & \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & = & \mathcal{B}' \\ {}^{a' \otimes -}\downarrow & {}^{f \otimes -}\swarrow & {}^{a \otimes -}\downarrow & {}^{\lambda_a}\swarrow & {}^{A a \otimes' -}\downarrow & = & {}^{a' \otimes -}\downarrow& {}^{\lambda_{a'}}\swarrow & {}^{A a' \otimes' -}\downarrow & {}^{A f \otimes' -}\swarrow & {}^{A a \otimes' -}\downarrow \\ \mathcal{C} & = & \mathcal{C}' & \stackrel{C}{\to} & \mathcal{C}' & & \mathcal{C} & \stackrel{C}{\to} & \mathcal{C}' & = & \mathcal{C}'}$$

Applying the Kelly-Street isomorphism transforms this to an equation in $\mathbb{R}Adj$ that expresses naturality of $\rho$ in $\mathcal{A}$. $\square$

We might say that $\rho$ is a pointwise mate of $\lambda$.

Now suppose $\otimes$ and $\otimes'$ are right closed as well as left closed. As above, the left and right closures assemble into bifunctors $hom_l \colon \mathcal{A}^{op} \times \mathcal{C} \to \mathcal{B}$ and $hom_r \colon \mathcal{B}^{op} \times \mathcal{C} \to \mathcal{A}$ so that there are compatible natural isomorphisms

$$\mathcal{C}(a \otimes b,c) \cong \mathcal{B}(b,hom_l(a,c)) \cong \mathcal{A}(a,hom_r(b,c)).$$

The triple $(\otimes, hom_l, hom_r) \colon \mathcal{A} \times \mathcal{B} \to \mathcal{C}$ is a two-variable adjunction. The pointwise mate of $\lambda$ with respect to $\mathcal{A}$ defines a natural transformation we might call $\rho_l$. Similarly, the pointwise mate of $\rho_l$ with respect to $\mathcal{C}$ defines a natural transformation we might call $\rho_r$. But we might also try to define $\rho_r$ to be the pointwise mate of $\lambda$ with respect to $\mathcal{B}$. It turns these definitions are equivalent and define a triple of parametrized mates.

$$\array{ \mathcal{A} \times \mathcal{B} & \stackrel{A \times B}{\to} & \mathcal{A}' \times \mathcal{B}' & \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' \\ {}^\otimes\downarrow & {}^\lambda\swarrow & {}^{\otimes'}\downarrow & \uparrow^{hom_l} & \searrow^{\rho_l} & \uparrow^{hom'_l} & \uparrow^{hom_r} & \searrow^{\rho_r} & \uparrow^{hom'_r} \\ \mathcal{C} & \stackrel{C}{\to} & \mathcal{C}' & \mathcal{A}^{op} \times \mathcal{C} & \stackrel{A^{op} \times C}{\to} & \mathcal{A}'^{op} \times \mathcal{C}' & \mathcal{B}^{op} \times \mathcal{C} & \stackrel{B^{op} \times C}{\to} & \mathcal{B}'^{op} \times \mathcal{C}'}$$

A previous blog post describes why I care about such natural transformations. Here, suffice it to say that the definition of an enriched model category involves two two-variable adjunctions.

“Naturality” of the parametrized mates correspondence

If you find yourself working with parametrized mates, a practical question immediately presents itself: when can a commutative diagram involving such natural transformations be transformed into another commutative diagram involving their parametrized mates, and what form does this new diagram take? Put another way:

Q. How can we express the “naturality” of the parametrized mates correspondence?

The answer, which I find really satisfying, is joint work with Eugenia Cheng and Nick Gurski.

The first step is to revisit the Kelly-Street theorem, which may be encoded in the observation that the double category $\mathbb{L}Adj$ admits an “involution” in the form of a double functor which exchanges the square on the left with the square on the right.

$$\array{ \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' & \mathcal{B}^{op} & \stackrel{B^{op}}{\to} & \mathcal{B}'^{op} \\ {}^F\downarrow\dashv\uparrow^U&{}^{\lambda}\swarrow &{}^{F'}\downarrow\dashv\uparrow^{U'} & {}^{U^{op}}\downarrow\dashv\uparrow^{F^{op}}&{}^{\rho^{op}}\swarrow &{}^{U'^{op}}\downarrow\dashv\uparrow^{F'^{op}} \\ \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & \mathcal{A}^{op} & \stackrel{A^{op}}{\to} & \mathcal{A}'^{op}}$$

In particular, each adjunction appears twice in $\mathbb{L}Adj$, once pointing in the direction of the left adjoint and once pointing in the direction of the (opposite of the) right adjoint. Note however that the way this data is encoded in the category of objects and vertical arrows in $\mathbb{L}Adj$ prohibits any ill-fated attempts to compose a left adjoint with a right adjoint.

This point is worth dwelling upon. In the real world, if we encounter a functor with codomain $\mathcal{A}$ and a functor with domain $\mathcal{A}^{op}$ we are undaunted: at the price of replacing one functor with its opposite, we regard this pair as composable. But in this context this sort of slight-of-hand is prohibited; as a consequence, composition and composability will be smoothly dealt with, as we shall soon see.

Cyclic (double) multicategories

Ignoring the mates for the moment, we have seen that the class of adjunctions has a $\mathbb{Z}/2$-action that exchanges a left adjoint with the opposite of its right adjoint. Similarly, the class of $n$-variable adjunctions has a $\mathbb{Z}/(n+1)$-action that replaces a functor of $n$-variables with the opposite of one of its $n$-variable adjoints.

To describe this action explicitly, it is convenient to force each functor to be contravariant: for the bifunctor $\otimes$ above, write $\mathcal{C}^\bullet$ for $\mathcal{C}^{op}$ and regard $\otimes$ as a contravariant functor $\otimes \colon \mathcal{A} \times \mathcal{B} \to \mathcal{C}^\bullet^{op}$. With this trick, we see that any $n$-variable adjunction may be regarded as a collection of functors

$$F_i \colon \mathcal{A}_{i+1} \times \cdots \times \mathcal{A}_n \times \mathcal{A}_0 \times \cdots \times \mathcal{A}_{i-1} \to \mathcal{A}_i^{op}$$

so that the hom-sets $\mathcal{A}(a_i, F_i(a_{i+1},\cdots,a_{i-1}))$ are compatibly isomorphic for all $i=0,\ldots, n$ and $a_j \in \mathcal{A}_j$. The collection $\{F_i\}$ represents an orbit of the $\mathbb{Z}/(n+1)$-action on $n$-variable adjunctions. Note that the inputs and outputs of the multi functors are permuted cyclically and that a category is “op”ed when it is moved from the domain to the codomain and vice versa.

Now of course a functor of $n$-variables and a functor of $m$-variables are composable just when the codomain of the former is among the categories in the domain of the latter or vice versa. The composite is then a functor of $(n+m-1)$-variables. Remarkably, if these functors represent an $n$-variable adjunction and an $m$-variable adjunction, then this naive composition is preserved by the cyclic actions producing an $(n+m-1)$-variable adjunction, provided we strictly adhere to our convention about “op”ing categories as they move between the domain and codomain and don’t allow ourselves to compose an output labeled $\mathcal{A}$ into an input labeled $\mathcal{A}^{op}$. This is the content of our first main theorem.

Theorem. Multivariable adjunctions assemble into a cyclic multicategory, i.e., into a multi category equipped with an involution “op” on objects and a $\mathbb{Z}/(n+1)$-action, of the form described above, on the class of $n$-array morphisms that is compatible with identities and composition.

The skeptical reader is encouraged to work through the following exercise.

Exercise. Suppose $\mathcal{A}$ has a closed monoidal structure $(\otimes,hom_l,hom_r) \colon \mathcal{A} \times \mathcal{A} \to \mathcal{A}^{op}$. Observe that the $\mathbb{Z}/3$-action has the form

$$\otimes \colon \mathcal{A} \times \mathcal{A} \to \mathcal{A} \quad \mapsto \quad hom_l \colon \mathcal{A} \times \mathcal{A}^{op} \to \mathcal{A}^{op} \quad \mapsto \quad hom_r \colon \mathcal{A}^{op} \times \mathcal{A} \to \mathcal{A}^{op}.$$

Convince yourself that any permitted composite of these bifunctors represents an adjunction of 3-variables.

The final punchline, of course, is that this structure extends to parametrized mates in any number of variables. Recall that a double category is a category object in Cat.

Theorem. Multivariable adjunctions and parametrized mates assemble into a cyclic double multicategory, i.e., a category object in cyclic multicategories.

Further details and lots of pictures can be found in the preprint or the slides referenced above.