The n-Category Café

Yes, thanks! I’ll fix that.

Something about this is bothering me, though: the question is then whether bad $X,T,Y$ can arise (… without $X'$ knowing how to fix things) when $T$ is also pullback for two cospans with a common foot.

So we have a span $X' \leftarrow T' \rightarrow Y'$. And for every $x' : X'$ we have a set $P x'$ and for every $y' : Y'$ a set $Q y'$; and the pullback of all-the-$P$s over $T'$ is the set of pairs

while the pullback of all-the-$Q$s over $T'$ is the set

But we want these to be equivalent in a way commuting with the map forgetting $p$ or $q$, which means we want, for every $t$, An Isomorphism $\varphi_t : P o' t \sim Q i' t$.

The pushout $X +_T Y$ then looks like this mess:

which maps to $X' +_{T'} Y'$ by $(x,p) \mapsto x$ and $(y,q) \mapsto y$…

I’m still rather expecting that the answer should be “no”, in general, but now it’s because of $X',T',Y'$ that look like this: $$\array{ t_1 & \to & x \\ \downarrow & & \uparrow \\ y & \leftarrow & t_2 }$$ and the fact that the $\varphi_{t_j}$ don’t have to be related in any way.

Presumably, it’s something like:

You answer this problem for me, and I’ll bow down before you, expressing my deferential respect.

To me the first one looks like someone holding up their hands to say “stop”. Or maybe pretending to be moose antlers. And there’s another one at the end that I can’t interpret at all. Partly because they’re both too small for me to see clearly.

I’m sorry if this comes across as whiny or Luddite, and I don’t mean to pick on John particularly here. But I really preferred the Internet back before large sections of it regressed to logographic writing. Can I enter a plea that people go back to using words to say exactly what they mean?

You can right-click on the image and view image properties. This shows the image file name, which in this case indicates that the emoticons are “bowing down” and “roll eyes”.

But I know that misses your point. Personally, emoticons and emojis don’t bother me, though I don’t always understand what they mean and sometimes have to resort to manoeuvres like the one just described. I don’t use them myself beyond the primitive :-) and variants, which I think are very useful in internet communications where tone can be hard to convey.

Here’s what they mean:

Bowing down to thank anyone who helps me solve this problem, in humorously exaggerated praise:

Nervously contemplating my weakness as a category theorist when it comes to reasoning with mixtures of limits and colimits:

Their meaning, like facial expressions, is context-dependent. If you don’t get them, you’re only missing a decorative ‘frosting’ on the cake of my post. I don’t use emojis to convey crucial steps in a mathematical argument.

Communication has always changed with the technology that enables it, and that’s not going to stop. Communication has always been ambiguous, too, especially to those not “in the know” (whatever that happens to mean). The subtle academic in-jokes of category theorists are just as hard for nonexperts to interpret as emoticons.

Everyone gets grumpy when they feel they’re not “in the know”, and most people, past a certain age, feel that the communication habits of young people are sloppy and standards are declining.

We all know category theorists who consider the practice of discussing math on blogs and forums to be dangerously decadent. Going back much further, in the Phaedrus Socrates warned of the dangers of writing as opposed to dialogue: the reader can’t ask questions of the text, and the text is addressed to everyone, not crafted to the individual. The interesting thing about blogs and smart phones is that they turn writing back into a form of dialogue. Emoticons add a primitive version of facial expressions… which will probably become more sophisticated as time goes by.

Nice, good catch! Thanks Mike, Tom, and Jesse, and anyone else who put at least an ounce of thought into this.

Here’s how I came up with this, by the way. This statement is true without any assumptions of monicity or epicity for homotopy pushouts and pullbacks of homotopy spaces (or more generally in any $\infty$-topos). So, I thought, let’s find an example of such a cube where $X,Y,T,X',Y',T'$ are all sets but the (homotopy) pushouts are not; then probably it will no longer work if we replace the homotopy pushouts by their set-truncations (the pushouts in $Set$).

One of the simplest (perhaps the simplest) pushout of sets that is not a set is

$$\array { 2 & \to & 1 \\ \downarrow && \downarrow \\ 1 & \to & S^1 }$$

exhibiting $S^1$ as the suspension of $2 = S^0$. Let’s put that on the bottom as $X',Y',T'$. Now we need a map $P\to S^1$ whose pullback over $1\to S^1$ is a set. The obvious choice is $P=1$, in which case this pullback computes the loop space $\Omega S^1 = \mathbb{Z}$. This leads to a counterexample with $X=Y=T=\mathbb{Z}$, $g(t) = t \mod 2$, $i_2(t) = \lfloor \frac{t}{2}\rfloor$, and $o_1(t) = \lceil \frac{t}{2}\rceil$.

But this doesn’t live in finite sets, so instead we can let $P=S^1$ and $P\to S^1$ be the double covering map (the first example being equivalently the universal cover $R\to S^1$). This leads to the example I described above.

Wow — thanks, Mike! It’s especially cool how you got the counterexample by thinking homotopically.

This makes me much more comfortable about demanding that the legs of our cospans

be monic. Kenny and I are working on a theory where these cospans, equipped with some extra structure, are ‘open Markov processes’: sets of states equipped with time-dependent probability distributions where probability can flow in or out of the cospans’ feet. We’re building a symmetric monoidal double category where the horizontal 1-cells are open Markov processes and the 2-cells are ‘coarse-grainings’, diagrams like this:

again equipped with extra structure. The idea of a coarse-graining is that it maps an open Markov process to a simpler one. Coarse-grainings are already widely studied for ordinary Markov processes, but we need to define them for open Markov processes.

We don’t need the cospans’ legs to be monic when working solely with the horizontal 1-cells. But we need the squares in our 2-cells to be pullbacks to get certain things to work… and thanks to you, we now know that for this property to be preserved by horizontal composition, we need the legs of our cospans to be monic.

This is a mild imposition, but I’m happier knowing that we’re making it because we need it.

In your honor, I will not express my gratitude with an emoticon.

When you see something like this, you feel pity for the category of sets.

(My remark was to the counter example.)

So the true statement that I wound up needing to use in my paper is this:

Given a cube in the category of finite sets:

where the top and bottom faces are pushouts, the two left vertical faces are pullbacks, and the arrows $o_1'$ and $i_2'$ are monic, then the two right vertical faces are pullbacks.

Actually a stronger statement is known: if either $o_1'$ or $i_2'$ is a monic, the bottom square is a pushout, and the two left vertical faces are pullbacks, then the top face is a pushout iff the two right vertical faces are pullbacks.

I suppose some people think the proof of this is obvious, but it doesn’t feel obvious to me. And it seems to me that the best proofs sneak up on this statement through some abstract ideas, rather than ‘bashing it out’.