The n-Category Café

The category theorists here will probably welcome your opting for a topos.

Will you need anything homotopic? I recall Peter saying something on the possible advantages of derived categories for working ‘internally’ to make solid abelian groups into a condensed category, here:

The main reason is that the limit is not equal to the derived limit, and one should be taking the derived limit…I didn’t check this carefully, but it may be true that when passing to derived categories, things are again OK; but one will definitely lose some properties.

A lot of people here should be familiar with that: the same is sort-of true for the nLab. It is now recognized as an amazing resource, but did the people who spent hundreds of hours working on it get appropriate academic credit? I don’t think so.

Absolutely! That’s why I get snippy when people on social media find it amusing to mock it. Much has been written by people without a secure faculty position, some without any position.

I was wondering about the possibility of having my own small contribution to the nLab count for something in terms of ‘Impact’ for my department’s submission to the UK’s Research Excellence Framework. I listed a few endorsements here and here, but it would very difficult to establish.

Absolutely! That’s why I get snippy when people on social media find it amusing to mock it. Much has been written by people without a secure faculty position, some without any position.

That’s a very interesting point. I suspect you’d agree that the institutions and culture of academia are in many respects extremely conservative: theoretically 100% open to reasoned argument, but in practice sometimes absurdly reluctant to change. Occasionally that conservatism is a good thing, but mostly not. And one of the most harmful effects is the lack of proper recognition for so many activities that are obviously positive and beneficial. That can genuinely blight people’s lives.

There’s a secondary effect of this conservatism. It’s wholly understandable that you get snippy about nLab teasing, for the reason you state. But everything can benefit from good-faith criticism (and sometimes not-so-good-faith criticism), which is much harder to hear and learn from if one’s defensive guard is up through feeling underappreciated. (I’m speaking in generalities, not about you personally.) Or to put it another way, if one is feeling secure and respected, it’s easier to be open to criticism and use it to improve. When our systems and traditions deprive people of that security and respect, they also make life harder for them in this way.

If a constructive version of ultrafilter would help, I see from a while ago Andrej Bauer writes:

The proof of existence of the codensity monad really looks constructive, which is to be expected since it’s just general category theory. So this is quite interesting because it does indeed suggest a constructive alternative to ultrafilters.

The other remaks I wanted to make was that the codensity monad for the inclusion of finite sets into sets reminds me very strongly of Martin Escardo’s selection monad. This is worth looking into.

It thought I’d take a look at Andrej’s hint and started an $n$Lab page on the selection monad.

I guess the connection is something like seeing what David JM above calls the “condensate” of an ultrafilter as a kind of selection function for integration against the ultrafilter.

Thanks, good to know. Does this quotient have an interpretation as a moduli space (asking for a friend)?

Well, they literally have different consistency strengths. I expect pyknotic stuff to be conservative over condensed stuff when dealing with only the latter objects. But then mathematicians might as well use NBG as a foundation rather than ZFC, since it’s conservative for statements about sets. In fact I like this approach, so I’m not saying we shouldn’t do this kind of thing.

But try getting Johan de Jong to put pyknotic machinery into the Stacks Project. Based on the foundational predilections he has publicly stated, I imagine he’d only be happy with condensed mathematics.

Dustin wrote

I think it’s incorrect to say that condensed and pyknotic are pointing in somewhat different directions.

If, as seems likely, you’re referring to my comment in the post

My question on the comparison of condensedness/pyknoticity and cohesion generated an interesting discussion through which I think it became clear that the former is pointing in a somewhat different direction to the latter,

that’s not what I meant. I was contrasting the near synonymous pair condensedness/pyknoticity with cohesion.

Hi Richard,

Thanks for the questions!

1) Indeed, the unit interval is a quotient of the Cantor set $\{0,1\}^\mathbb{N}$ by the equivalence relation which “glues adjacent endpoints together”, or makes sure that $0.999999... = 1$ and the like. This induces a description as sheaf on extremally disconnected profinite sets S, since for such S mapping out of S commutes with quotients by compact Hausdorff equivalence relations.

I think there’s also nothing in principle wrong with describing the real numbers as the sheaf which assigns to a profinite set S the completion of the set of locally constant functions $S\rightarrow\mathbb{Q}$ with respect to the sup norm.

But another slightly more elaborate description of $\mathbb{R}$ has been crucial for us in the liquid story, writing $\mathbb{R}$ as a quotient of an overconvergent arithmetic Laurent series ring, as Peter describes in his challenge.

2) a,b) The category of condensed sets is compactly generated, but the compact objects themselves are not so nice. They are the quotients of compact Hausdorff spaces by equivalence relations which are generated, as equivalence relations, by some closed relation. There are many such equivalence relations which are not themselves closed, so compact condensed sets include lots of non-Hausdorff phenomena in addition to the usual compact Hausdorff spaces.

On the other hand there is another kind of “compactness” condition which comes from the world of topos theory. It’s called being “quasicompact and quasiseparated”. The quasicompact quasiseparated condensed sets are exactly the compact Hausdorff spaces.

One can also define connectedness in a topos-theoretic way. This unwinds to something very naive: a condensed set is connected if it’s nonempty and whenever you write it as a coproduct of two other condensed sets, one of them must be empty. For compactly generated weak Hausdorff spaces, which sit fully faithfully in condensed sets, it’s the same as connectedness for topological spaces.

c) Yes, you can see that every profinite set is quasicompact and quasiseparated because the site you chose is finitary, and then $[0,1]$ is the quotient of $\prod \{0,1\}$ by a closed equivalence relation, so it’s also quasicompact quasiseparated. Actually if you think about it this is basically the usual proof of compactness of $[0,1]$ by dyadic subdivision.

d) I haven’t thought about proving connectedness of the real numbers in a purely condensed way. I’d guess it will again be a version of some ordinary proof.

3) I’m not sure I follow the question here. I don’t see why you’d have to build Archimedeanness in so to speak or add it as an axiom when you can just construct $\mathbb{R}$ however you like and then work with it. Actually, the third description of $\mathbb{R}$ I mentioned above is the most useful one and it seems rather much more in the nature of direct construction than axiomatic specification.

Hi Dustin, thank you very much for the reply. This is very helpful. For now, I’ll just focus on the following; I’ll get back to my other questions/your replies to them at a later point.

Indeed, the unit interval is a quotient of the Cantor set $\left\{0,1\right\}^\mathbb{N}$ by the equivalence relation which “glues adjacent endpoints together”, or makes sure that $0.999999...=1$ and the like. This induces a description as sheaf on extremally disconnected profinite sets S, since for such S mapping out of S commutes with quotients by compact Hausdorff equivalence relations.

Thank you. The following re-telling of this, which avoids referring to topological spaces at all, was something like what I was looking for: motivating the definition of condensed sets from a standard category theoretic point of view.

The profinite set $\left\{0,1\right\}^{\mathbb{N}}$ is almost the unit interval. It would be the unit interval if we could identify things like $0.1$ and $0.0111111\ldots$. Thus we’d like to take a quotient (colimit). However, the category of profinite sets is not co-complete, and there is no reason for this quotient to exist as a profinite set. The obvious thing to do then is to pass to a co-completion of profinite sets. We can pass to the free co-completion, that is, take presheaves (I will ignore size matters), but then of course we destroy existing colimits in profinite sets; the remedy then is to work with sheaves with respect to some topology. I can’t say at this point why jointly surjective covers give exactly the right topology, but they are one of the most minimal topologies one could choose so that representables are sheaves and so that one retains a reasonable amount of information, so seem a reasonable choice to start with. Once we know that representables are sheaves, it is pretty obvious that the required colimit is a sheaf.

End of re-telling! The sort of thing I was getting at now was to see what we can do from this starting point, where we have not mentioned topological spaces yet; can we continue to proceed without mentioning them?

As I say, I will come back to some other aspects of your reply. For now, I’d like to try to formulate a statement of the liquidity theorem from this perspective. Before we can do that, though, there are some basic things to try to understand.

Firstly, we can of course construct the $p$-adic unit interval ($p$-adic integers) simply as the profinite set $\left\{ 0, \ldots, p-1 \right\}^{\mathbb{N}}$, and thus as a representable condensed set. Noting that we can replace the use of $\left\{ 0, 1 \right\}$ in the construction of the real numbers with $\left\{ 0, \ldots, p-1 \right\}$ for any $p$ and tweaking the quotient in the obvious way (i.e. we obtain an isomorphic condensed set in this way), the only difference between the real numbers and the p-adic integers for any $p$ comes from the quotient in the definition of the real numbers.

One can show that both the real and $p$-adic numbers constructed in this way can be equipped with the structure of condensed abelian groups (i.e. sheaf of abelian groups), again without referring to topological spaces. There is a perfectly good tensor product of condensed abelian groups. Here comes the first important point in the ‘liquid’ story: it is claimed that the tensor product of the real numbers and $p$-adic numbers in this sense is no good.

Before we go further, could we dwell on this a moment? What properties would we like this tensor product to have? Professor Scholze’s article is a bit vague on why the naive tensor product does not have these properties; can we spell this out?

PS - I hopped from talking about the unit interval to the real numbers. Once one has a unit interval, one can of course obtain the real numbers by glueing copies of the unit interval together end to end. No doubt one can proceed in other ways, e.g. in the construction of the unit interval that I outlined, one can probably replace $\left\{ 0, 1 \right\}^{\mathbb{N}}$ by $\mathbb{Z} \times \left\{ 0, 1 \right\}^{\mathbb{N}}$, where $\mathbb{Z}$ is regarded as a discrete profinite set.