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Stream: learning: questions

Topic: What is this 2-cell in Prof called

James Gilles (Jul 22 2026 at 03:02):

I have this invertible 2-cell in the symmetric (I think?) monoidal proarrow equipment (is that the current terminology?) of profunctors. Is there a name for this cell? It seems like a handy gadget for turning things around, something related to adjunctions?

Nathanael Arkor (Jul 22 2026 at 04:53):

Could you spell out what this cell is, for those unfamiliar with the surface diagrams?

James Gilles (Jul 22 2026 at 16:56):

Oh whoops, I had that in the picture but re-drew it and forgot to re-add it.

The top edge is a profunctor $P : C^{op} \nrightarrow D$.

The bends on the bottom are profunctors:

$\eta_D : 1^{op} \nrightarrow D^{op} \times D$

and

$\mu_C : (C \times C^{op})^{op} \nrightarrow 1$

Given by shuffling variables around on the hom functor, so you have e.g. $\eta_D(., (d_1, d_2)) = Hom_D(d_1, d_2)$.

The full bottom of the diagram is the composite of 3 profunctors, from left to right:

$Hom_C \times \eta_D : C \nrightarrow C \times D^{op} \times D$

$Hom_C \times P^{op} \times Hom_D : C \times D^{op} \times D \nrightarrow C \times C^{op} \times D$

$\mu_C \times Hom_D : C \times C^{op} \times D \nrightarrow D$

Call this composite $Bend : C \nrightarrow D$.

The hom functor is an identity for profunctor composition, and I believe this extends to the bent versions of it with shuffled variables.

So the 2-cell should be a natural isomorphism $Bend(c, d) \cong P(c, d)$.

The left and right edges of the cell are -- the identity functors on C and D, if you're thinking of Prof as a double category; if you're thinking of it as a bicategory, they don't mean anything.

Nathanael Arkor (Jul 23 2026 at 06:14):

My impression is that people tend not to name invertible 2-cells unless they are definitionally part of a structure (e.g. the associator in a monoidal category). So I would not expect anyone to have named this 2-cell.

John Baez (Jul 23 2026 at 12:15):

Some people call this 2-cell the zigzagurator. There are really two zigzagurators of which this is one: they appear whenever we have a biadjunction, and they generalize of the zigzag equations that hold for the unit and counit of an adjunction. Here I explained how these 2-morphisms describe beta-reduction and eta-reduction in the lambda calculus.

If you like commutative diagrams more than string diagrams, you're likely to speak of the 'triangle identities' for an adjunction rather than the 'zigzag equations'. In that case you can call the zigzagurators the 'triangulators', which sounds a bit more dignified.

Nathanael Arkor (Jul 23 2026 at 15:27):

John Baez said:

Some people call this 2-cell the zigzagurator.

Who, and where? A Google search for "zigzagurator" turns up no results.

John Baez (Jul 23 2026 at 16:30):

I guess just people I talk to! I guess none of them have the gall to put this in print. On page 21 of Compact closed bicategories, @Mike Stay called them "zig-zag isomorphisms".

James Gilles (Jul 23 2026 at 17:18):

@John Baez woah, those are cool pictures.

What is the biadjunction formally in this case? Do we need to think of Prof with monoidal product as a 3-category to see it?

John Baez (Jul 23 2026 at 17:20):

The biadjunction, which arises from every object in Prof having a "dual" (biadjoint), can be formalized by thinking of Prof as a compact monoidal bicategory. Mike Stay defines monoidal bicategories in exquisite detail, but they amount to almost the same thing as one-object tricategories.

Mike Stay discusses Prof and why it's a compact monoidal bicategory, so you'd probably enjoy searching the paper for "Prof" and reading about that, and looking up some references. But he doesn't go into a lot of detail about Prof, because he's mainly studying another example of a compact monoidal bicategory.

Mike Stay (Jul 23 2026 at 18:30):

zigzagurator

Or even a zag-ziggurator, because the zig-zags are sort of shaped like a ziggurat's silhouette.

James Gilles (Jul 24 2026 at 20:03):

Followup 1-categorical question:

The string diagrams in your linked notes @John Baez are diagrams in a cartesian closed category.

The bubbles for lambda abstraction/evaluation and the shaded region between input and output variables are necessary in this context.

If the relevant cartesian closed category was Set, we could embed it in Rel, which is a compact closed category, and redraw the diagrams without the bubbles. This works because the natural isomorphism

$Set(X \times Y, Z) \cong Set(X, [Y, Z])$

is sent to

$Rel(X \times Y, Z) \cong Rel(X, Y \times Z)$

by the embedding -- i.e. by sending the a map

$c : X \rightarrow [Y, Z]$

to the relation

$\{(x, (y, z)) | c(x)(y) = z\}$.

Is this an instance of a more general construction?

Mike Stay (Jul 24 2026 at 22:10):

Maybe this? https://johncarlosbaez.wordpress.com/2020/04/21/star-autonomous-envelopes/

John Baez (Jul 26 2026 at 07:47):

Is Rel the star-autonomous envelope of Set? I don't know. But I agree that Mike Shulman's construction is good to think about, for this issue.

Mike Shulman (Jul 27 2026 at 15:25):

The s-a envelope is a giant construction like the Yoneda or Isbell embedding; it's almost never going to coincide with something you already had. In addition, the whole point of the envelope is that the embedding into it is full and closed monoidal, which $Set \to Rel$ is not.