Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: adjoints and split idempotent transformations

Louis de Forcrand (Nov 16 2021 at 17:46):

There is an exercise in Mac Lane's Categories for the Working Mathematician, IV.1.4, attributed to Paré, which I copy here for reference.

(Paré.) Given functors $G\colon A \to X$ and $K\colon X \to A$ and natural transformations $\varepsilon\colon KG \to 1_A$, $\varrho\colon 1_X \to GK$ such that $G\varepsilon \cdot \varrho G = 1_G \colon G \to GKG \to G$, prove that $\varepsilon K \cdot K\varrho \colon K \to K$ is an idempotent in $A^X$ and that $G$ has a left adjoint if and only if this idempotent splits; explicitly if $\varepsilon K \cdot K\varrho$ splits as $\alpha\cdot\beta$ with $\beta\cdot\alpha = 1$ and $\beta\colon K \to F$, then $F$ is a left adjoint of $G$ with unit $G\beta \cdot \varrho$ and counit $\varepsilon \cdot \alpha G$.

I looked at the single reference in the book to a paper by Robert Paré, On absolute colimits, but didn't find anything concerning this result there. (Hopefully I didn't miss it.) I was wondering if anyone had anything more to say about this. (Seen it elsewhere, used it, has an interesting interpretation?) It looks interesting to me, but maybe it is just a fun exercise in string diagrams. Also I don't know much of anything about monads yet, maybe this is something well-known in that area?

John Baez (Nov 16 2021 at 18:14):

To understand math I usually have to turn it into words. This seems to be saying, first of all, that if $K$ is "not quite" the left adjoint of $G$ because just one of the zigzag equations holds, the other zigzag $\epsilon K \circ K \rho$, while not the identity, is at least an idempotent. That's easy to see using string diagrams: it follows from the zigzag equation that does hold!

Then it says $G$ will have a left adjoint iff this idempotent splits, and indeed you get a formula for its adjoint in terms of the splitting. I hadn't known that.

I haven't proved this part, but I feel you can prove this too using string diagrams... in which case this is not really a result about Cat, it's better: it's a general result that holds in any 2-category.

John Baez (Nov 16 2021 at 18:17):

Anyway, I don't know how or if this result is used anywhere... but it probably is: it may be hiding "between the lines" in that paper by Paré.

Nathanael Arkor (Nov 16 2021 at 18:19):

You could look in Paré's thesis, which is more comprehensive if I remember correctly.

John Baez (Nov 16 2021 at 18:22):

It would be nice to find an easy example of one of these "not quite" adjunctions. They're probably running around all over the place.

Patrick Nicodemus (Nov 16 2021 at 19:05):

I might be misunderstanding this, but shouldn't such a half adjunction be roughly equivalent to a weakly universal property? i.e., if F is not-quite left adjoint to G then for all X and f : X -> G(Y) there exists a not-necessarily-unique map h: F(X) -> Y with f = G(h) \circ eta. So the intensional equality rule for dependent type theory falls under this heading. But I haven't checked this.

Louis de Forcrand (Nov 17 2021 at 16:04):

John Baez said:

I haven't proved this part, but I feel you can prove this too using string diagrams... in which case this is not really a result about Cat, it's better: it's a general result that holds in any 2-category.

Using string diagrams it is not too hard; without it must be rather painful. I don't know anything about 2-categories yet either. One day I will.

Louis de Forcrand (Nov 17 2021 at 16:07):

Nathanael Arkor said:

You could look in Paré's thesis, which is more comprehensive if I remember correctly.

Like the paper, it is about absolute (co)limits; perhaps as John said this result is lurking somewhere in there, but I did not find it in his thesis either. There was something resembling it at the very end of Chapter 2, which I will have to read again to decipher. Thanks for the reference.

Louis de Forcrand (Nov 17 2021 at 16:07):

John Baez said:

It would be nice to find an easy example of one of these "not quite" adjunctions. They're probably running around all over the place.

Indeed!

Mike Shulman (Nov 17 2021 at 20:20):

I think a version of the adjoint functor theorem can be phrased using this result. Suppose $G:A\to X$ preserves limits. For any $x\in X$, let $Kx = \prod_{x \to G a} a$; this is a large product, but we can play games to reduce it to a small one as usual. Then we have product projections $K x \to a$ for any $x\to G a$, including in particular $\epsilon_a : K G a \to a$ if $x = G a$. And since $G$ preserves limits, we have $G K x = \prod_{x\to G a} G a$, so there is a tautological map $\rho_x : x \to G K x$ whose component at any $x\to G a$ is itself. It's easy to check the equation $G \epsilon \cdot \rho G = 1$, so by splitting the resulting idempotent on $K x$ we obtain a left adjoint to $G$.

Mike Shulman (Nov 17 2021 at 20:21):

I used something like this here to construct higher inductive types from an impredicative encoding.

Louis de Forcrand (Nov 18 2021 at 18:17):

This looks rather interesting. I'll think about it later and look at your post.