## November 1, 2012

### Parametrized Mates and Multivariable Adjunctions

#### Posted by Tom Leinster Guest post by Emily Riehl

(Note: the Café went down for a few days in early November 2012, and when Jacques got it back up again, some of the comments had been lost. I’ve tried to recreate them manually from my records, but I might have got some of the threading wrong.)

The mates correspondence provides a means to transfer information from a diagram involving left adjoints to a diagram involving right adjoints. The basic observation is that in the presence of functors and adjunctions, arranged in the manner displayed below, there is a bijective correspondence between natural transformations $\lambda$ and $\rho$. $\array{ \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' \\ {}^F\downarrow\dashv\uparrow^U& &{}^{F'}\downarrow\dashv\uparrow^{U'} \\ \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' }$

$\array{ \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' & & \mathcal{A} & \stackrel{A}{\to} & \mathcal{A}' \\ {}^F \downarrow & {}^{\lambda}\swarrow & \downarrow^{F'} &      & {}^U\uparrow &\searrow^{\rho} & \uparrow^{U'} \\ \mathcal{B} & \stackrel{B}{\to} & \mathcal{B}' & & \mathcal{B} &\stackrel{B}{\to} & \mathcal{B}' }$ Corresponding $\lambda$ and $\rho$ are called mates and are related by the pasting diagrams displayed here. Today I’d like to report on the preprint Multivariable adjunctions and mates, joint with Eugenia Cheng and Nick Gurski, which extends this notion to adjunctions with parameters. Our main theorem, also described in these slides describes the categorical structure that precisely captures the “naturality” of the parametrized mates correspondence.

But first let’s acquaint ourselves with some examples of ordinary mates.

### 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.

Posted at November 1, 2012 11:53 PM UTC

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### Re: Parametrized Mates and Multivariable Adjunctions

Great post! I think you may have mixed the notation up a little in the paragraph beginning “to describe this action explicitly…”, e.g. did you mean to write $\mathcal{C}^\bullet{}^{op}$? Maybe you did, but in that case I’m a little confused.

(And for anyone who thinks this looks familiar, I mentioned it briefly after Eugenia’s talk at the PSSL back in April.)

And a question that I think I’ve brought up before, but it deserves to be asked again: Kelly and Street’s theorem applies not just to categories, functors, and adjunctions, but to internal adjunctions in any 2-category. Have you thought about building a cyclic double multicategory out of some sort of generalization of a 2-category?

(Originally posted at 12:56:08 on 3 Nov 2012)

Posted by: Mike Shulman on November 11, 2012 11:00 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Yes, I did. Since I’d already specified the domain and codomain of $\otimes$, it only becomes a contravariant functor if we replace its codomain by the opposite of its opposite category.

The reason I expressed this in a rather silly way, as $\mathcal{C}^{\bullet}^{op}$, it to make the following point. If you name the domain and codomain categories of your bifunctor so that it’s contravariant from the get go - so here they’d be the categories $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}^\bullet$ (but now treat $\mathcal{C}^\bullet$ as just some category; forget that the bullet once signified something) - then all of the adjoints are contravariant in the same way. Here $\otimes$ goes from $\mathcal{A} \times \mathcal{B}$ to $\mathcal{C}^\bullet^{op}$, $hom_l$ goes from $\mathcal{A} \times \mathcal{C}^\bullet$ to $\mathcal{B}^{op}$, and $hom_r$ goes from $\mathcal{C}^\bullet \times \mathcal{B}$ to $\mathcal{A}^{op}$.

Does this make any sense?

(Originally posted at 20:41:26 on 3 Nov 2012)

Posted by: Emily Riehl on November 11, 2012 11:01 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Yes, it does, thanks. I think partly what confused me was the word “contravariant”. Usually when I see the phrase “a contravariant functor from $A$ to $B$” it means a functor $A^{op}\to B$ or $A\to B^{op}$, so that “a contravariant functor from $A$ to $B^{op}$” would actually mean a functor from $A$ to $B$.

(Generally I try to avoid calling anything a “contravariant functor” because of this potential for confusion, just saying “functor” and putting all the op’s in the right place.)

(Originally posted at 12:06:41 on 4 Nov 2012)

Posted by: Mike Shulman on November 11, 2012 11:03 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Posted by: Emily Riehl on November 11, 2012 11:04 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

With this cyclical symmetry, is there any scope for representations via planar algebras and their higher dimensional versions, such as described at the Secret Blogging Seminar here? I think it was canopolises that have the cyclindrical shape that could represent maps between cyclically represented things.

(Originally posted at 12:48:01 on 5 Nov 2012)

Posted by: David Corfield on November 11, 2012 11:05 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

For those of you who like string diagrams and don’t already know it, there’s a nice way to picture the mates correspondence.

There is the correspondence of natural transformations:

$\lambda \colon F'\circ A \Rightarrow B\circ F \qquad \leftrightarrow \qquad \rho \colon A\circ U \Rightarrow U'\circ B.$

Pictorially, this is represented as [My convention for string diagrams is right to left and bottom to top.]

The actual map is given, using the unit of the $F'$, $U'$ adjunction and counit of the $F$, $U$ adjunction, as (Originally posted at 16:40:01 on 5 Nov 2012)

Posted by: Simon Willerton on November 11, 2012 11:09 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Good. That’s more or less what I sketched out on a piece of paper, though yours is neater.

(Originally posted at 09:17:55 on 6 Nov 2012)

Posted by: David Corfield on November 11, 2012 11:12 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

David, thanks for pointing me to the secret blogging seminar blog, and Simon, thanks for the pictures. (I have no idea how you drew those.) I’m personally not very well acquainted with the various graphical calculi so I feel ill-equipped to try and answer this question.

It seems that an important idea that should be captured by this picture is that one should be able to represent a multivariable adjunction by any of its constituent functors, but somehow the only allowable composites should be those that produce a new adjunction (eg, you can’t compose a “left” adjoint with a “right” adjoint).

A naive idea would be to represent a multi-functor as a “spider” - a circle bearing the name of the functor and legs equipped with arrows pointing in or out (exactly one of which points out) and labeled with the names of the appropriate objects. The rule is that you can swap the label of the spider with any of its cyclical conjugates by also reversing the direction of the outward arrow and one of the inward arrows, but at the same time you must replace both labels by their opposite. Then two spiders compose just when one has an inward pointing leg and another has an outward pointing leg with the same label.

One might imagine you could extend this to parametrized mates by adding an extra dimension. But now this is starting to feel a bit more like pasting diagrams (with which I’m more comfortable) and thus isn’t quite in the spirit of string diagrams and their ilk.

(Originally posted at 18:10:35 on 6 Nov 2012)

Posted by: Emily Riehl on November 11, 2012 11:15 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Of course, there’s an obvious “profunctorial” simplification. Encode a multivariable adjunction as a “spider” - a circle with legs that are undirected this time and labeled in the following manner. Looking at “the left adjoint” for the adjunction, there is a leg labeled by each of the input objects and also a leg labeled by the opposite of the output object.

(We can say this another way: an n-variable adjunction is a profunctor on (n+1) categories that admits (the obvious) (n+1) different representations. If this profunctor has the form $A_0 \times \cdots \times A_n \to$ Set then the legs of the spider should be labeled by $A_0^{op}, A_1^{op},\ldots, A_n^{op}$. Or you can adopt the opposite convention and label the legs by $A_0,\ldots, A_n$; this corresponds to using labeling the legs with the inputs and the opposite of the output of one of “the right adjoints”.)

The composition rule is that two spiders can be composed along any pair of legs whose labels are opposite. The composite spider is produced by gluing the bodies of the spiders together and erasing the legs along which the composition occurred (no change in labels is necessary).

If you wanted to lift this composition to the setting I described previously, with directed inputs and outputs, there are two ways to do so: each multifunctor could be composed into the other. The fact that these composites agree is one of the axioms of a cyclic multicategory.

(Originally posted at 19:44:04 on 6 Nov 2012)

Posted by: Emily Riehl on November 11, 2012 11:17 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

The usual way to add a dimension to string diagrams is to make them into surface diagrams. We had some discussion about surface diagrams for extranatural transformations in the comments to this post (with some of the discussion spilling over to this post).

Here is a first attempt at depicting parametrized mates with surface diagrams. The first page shows three categories $A,B,C$ as surfaces, three 2-variable functors $F,G,H$, and two of the counits for a 2-variable adjunction between them. (In my surface diagrams, products of categories go from back to front, functors go from right to left, and 2-cells go from top to bottom. The involution $(-)^\bullet$ flips diagrams top-to-bottom.)

The second page shows one of the triangle identities for a 2-variable adjunction. Note that the equality of domains in these two diagrams uses another “triangle identity”, happening at the level of “labeling graphs for extranaturals” (or perhaps profunctors).

The third page is a parametrized mate. I used slightly darker shades of red, blue, and green for the categories $A',B',C'$ in the second 2-variable adjunction. I’m not at all confident that this picture will make sense to anyone else, though.

Unfortunately, all the nice cyclicity that happens at the level of the multicategory seems to go out the window at the level of the surface diagrams, with surfaces having to pass through each other everywhere rather than remaining in some nice cyclic order. Maybe I just wasn’t clever enough to see the cyclicity — I certainly hope so! Can someone else do better?

(Originally posted at 04:02:58 on 7 Nov 2012)

Posted by: Mike Shulman on November 11, 2012 11:18 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Woah…

Posted by: Emily Riehl on November 12, 2012 3:16 AM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

I should add that Eugenia and I informally talked about these 2-cells as rolodexes. Her infamous black books that contain some of our attempted calculations involve pictures where we try to stick rolodexes together, flip the page, then stick some more on, etc. While a fun game, the rolodex pictures never quite made the cut in terms of utility.

Posted by: Nick Gurski on November 12, 2012 9:35 AM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

Cor! What kind of surface is that, Mike? Has anyone done a typology of surfaces? There are branched ones, and I remember John Baez was telling me about fake ones back here.

Is there anything to the Secret Blogging Seminar idea that if you’re dealing with some levels of duality, then better to keep as circular as you can, rather than rectangular?

Posted by: David Corfield on November 12, 2012 3:18 PM | Permalink | Reply to this

### Re: Parametrized Mates and Multivariable Adjunctions

I would call this sort a stratified (singular) surface.

Circular pictures work when all objects and morphisms have a particular kind of dual. Here, I don’t think that’s the case — for instance, not all functors have adjoints.

Posted by: Mike Shulman on November 13, 2012 6:42 PM | Permalink | Reply to this

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