The n-Category Café

Along the same lines as Emily Riehl’s comment, one should remark that the condensed sets/abelian groups/etc already show up in Bhatt-Scholze (under a different name), as does the the important idea of using Stone-Cech compactifications to obtain a nice basis for the topos. The new contribution of Clausen-Scholze is the introduction of the idea of solidity (and later liquidity) and its many amazing properties.

Thanks, I added a note to my post on your first point. Do these other good features – solidity and liquidity – carry over to pyknotic sets?

I see Barwick and Haine write

As emphasised by Scholze, however, the distinction between pyknotic and condensed does have some consequences beyond philosophical matters. For example, the indiscrete topological space $\{0,1\}$, viewed as a sheaf on the site of compacta, is pyknotic but not condensed (relative to any universe). By allowing the presence of such pathological objects into the category of pyknotic sets, we guarantee that it is a topos, which is not true for the category of condensed sets.

It would be too glib to assert that the pyknotic approach values the niceness of the category over the niceness of its objects, while the condensed approach does the opposite. However, it seems that the pyknotic objects that one will encounter in serious applications will usually be condensed, and the majority of the good properties of the category of condensed objects will usually be inherited from the category of pyknotic objects. (BarHai 19, 0.3)

It should be noted, however, that in Mike’s Real Cohesion, the further right adjoint modality $\sharp$ is not used in the definition of the middle modality $\flat$, nor in the definition of the left modality $\pi_{\infty}$. (It is used to prove some facts about $\flat$.)

So long as the map $f$ from the pro-etale site of $X$ to the pro-etale site of the point point has a fully faithful inverse image $f^{\ast}$, I believe that the $\flat$ portion of Mike’s Cohesive Type Theory could be used.

I think much of this can be done, already from what I said in the previous message.

(Regarding terminology: Clausen and I call $\infty$-groupoids “anima” (plural: anima). We regard the name $\infty$-groupoid as too unwieldy – the name suggests a semblance to groups, which is not really a good first approximation. Lurie’s choice of “spaces” does not work in the setting of condensed mathematics: Condensed anima have two directions, a homotopy-theoretic direction (anima) and an actual topological space direction (condensed), and it is the condensed direction that feels like the space direction, so the term condensed spaces would be a complete misnomer. On the other hand, see my paper with Cesnavicius (on purity for flat cohomology) for some discussion of anima and animation, including some justification for that name.)

1) Yes. This follows formally from adjointness.

2) Yes, it is the condensed anima $BG$ where $G=\mathrm{Gal}(\overline{k}/k)$ is the absolute Galois group of $k$. There is no extra information.

3) If $X$ is a scheme over a field $k$, you can consider $f: X_{\mathrm{proet}}\rightarrow \mathrm{Spec}\, k_{\mathrm{proet}}$. Sheaves on $\mathrm{Spec}\, k_{\mathrm{proet}}$ are equivalently condensed anima with $G$-action (where $G$ is the absolute Galois group as above), by descent. Applying the left adjoint to $X$ then produces a $G$-action on $\pi_\infty(X_{\overline{k}})$.

4) Yes, you can take that Hom internally on $\mathrm{Spec}\, k_{\mathrm{proet}}$.

5) I presume one can get $\ell$-adic Tate twists (but not motivic Tate twists!) in some way out of $\pi_\infty(\mathbb{P}^1)$.

However, I should also say that I am rather sure none of this has any relevance to Beilinson-Soule.

Confirmed my suspicion: This is not true.

A related fact is that if $X$ is a scheme over an algebraically closed field $k$, and $k'/k$ is an extension of algebraically closed fields, with $X'$ the base change of $X$ to $k'$, then $\pi_\infty(X')\to \pi_\infty(X)$ is not an isomorphism, in general.

Some approximations to such statements are true, for example invariance of $\ell$-adic cohomology under change of algebraically closed base field, if $\ell$ is different from the characteristic of $k$. But, for example, this fails when $\ell$ equals the characteristic of $k$.

Regarding concreteness: Let’s take $X=\mathrm{Spec}\, k$ with $k$ algebraically closed as our base. There are many schemes $Y$ over $k$ with points $Y(k)$ empty: For example, $Y=\mathrm{Spec}\, k'$ for some algebraically closed field $k'/k$, $k'\neq k$. In particular, this implies that any concrete sheaf $F$ satisfies $F(Y)=\ast$ for all such $Y$. But such $F$ are really awkward – in practice $F(k)$ will always inject into $F(k')$. Not even the disjoint union of two points is concrete!!

So in this setting concrete sheaves are a strange concept. In algebraic geometry, schemes are not built from $k$-valued points.

As has been said, not all of cohesion is needed to generate interesting consequences. One important part of the story is how the differential cohomology diagram provides an account of differential cohomology. This doesn’t need the right adjoint to $f_{\ast}$. Urs suggests that $f_!$ doesn’t need to preserve finite products either, so generalized Chern characters, etc., should follow in this pro-étale situation.

Also, I would not over-emphasize concreteness. It’s true that $\sharp$ can be used to formalize concreteness, but one of the whole points of cohesion is that not all interesting objects are concrete (in fact, non-concreteness is another way of describing the solution to the problem of (co)kernels), so it’s unclear that this is of much real value. As David Jaz Myers pointed out, $\flat$ and $ʃ$ are of at least as much real use in doing mathematics cohesively. The lack of $ʃ$ is, I think, what leads to a significantly different flavor more than the lack of $\sharp$.

$\sharp$ is having a hard time on this thread, but in case anyone hasn’t seen this, Lawvere’s account of cohesion involved the idea of overcoming a fundamental opposition between the idempotent comonad constant on the initial object and the idempotent comonad constant on the terminal object, sketched here.

Thanks, this is very helpful! As I said, I was/am struggling to see why a cohesive topos is defined like it is, but I am beginning to see some light.

I guess the above example, that the big pro-etale topos of schemes over a fixed algebraically closed field $k$ is, almost, cohesive over pyknotic sets, is fitting this philosophical framework then?

I guess the above example, that the big pro-etale topos of schemes over a fixed algebraically closed field $k$ is, almost, cohesive over pyknotic sets, is fitting this philosophical framework then?

Yes.

conceptual crux

conceptual crutch? It is was a conceptual crux you couldn’t just dismiss it! ;-)

Thanks, I overlooked that yesterday late, but such size restrictions seem to be common when using topological sites. Escardo and Xu restricted to the separable case.

How crucial is it to work with large objects in the theory of pyknotic sets?

About cohesion, Mike already remarked that the topological topos is not cohesive. A similar argument seems to work here.

Mike, you made the connection to quasi-topological spaces in the nlab. I was about to ask whether this connection was known in the literature, but I see that the issue was already raised in the nforum.

What is the difference then? Is it only the choice of whether to bound the size of objects in the site at some “small” inacessible or to use a large site?

Just a quick couple of remarks. Firstly, unless there are some unexpected differences between the étale and pro-étale worlds, ‘after $l$-completion’ it should be true that $\pi_{\infty}(-)$ preserves products. I.e. probably one could work not with condensed anima but with $l$-completed condensed anima everywhere if one needed this.

As a second remark, one should get product preservation in some cases I think even without $l$-completion, e.g. when one of the schemes involved is $\mathbb{A}^{1}$ ($\mathbb{A}^{1}$-homotopy invariance). This is related to some of my earlier comments, e.g. this one. May I ask whether you saw/got a chance to think about this one by the way, Professor Scholze?

One further remark that occurs to me, by the way, about $\pi_{\infty}(\mathbb{A}^{1})$-homotopy as defined in my earlier comment, is that, in contrast to the na¯ve version of $\mathbb{A}^{1}$-homotopy, the na¯ve version of $\pi_{\infty}(\mathbb{A}^{1})$-homotopy equivalence is I believe already an equivalence relation, because one can use the $\infty$-groupoid structure to ‘reverse’ and ‘compose’ $\pi_{\infty}(\mathbb{A}^{1})$-homotopies.

Morevoer, the cohomology theory $Hom(\pi_{\infty}(X), K(A,n))$ for some $X$ and some $A$ in this $\pi_{\infty}(\mathbb{A}^{1})$-local $(\infty,1)$-category, which I suggest must be similar to algebraic singular cohomology, is definitely $\pi_{\infty}(\mathbb{A}^{1})$-local (this follows from some adjunction juggling). In other words, using $\pi_{\infty}(\mathbb{A}^{1})$ instead of $\mathbb{A}^{1}$ seems to erase the distinction between na¯ve and non-na¯ve $\mathbb{A}^{1}$-homotopy theory. This is an important technical point I believe.

From Peter’s Lectures on Analytic Geometry, p. 76

there are two natural functors from CW complexes to condensed anima: A fully faithful functor, via the full subcategory of condensed sets; and another non-faithful functor, factoring over the full-∞-subcategory of anima. Let us (abusively) denote the first functor by $X \mapsto X$, and the second by $X \mapsto |X|$. Then $S^1$ is a physical circle, while $|S^1|$ is some ghostly appearance of a point with an internal automorphism (the anima $B \mathbb{Z}).$

That kind of phenomenon is very much modal HoTT territory, with the distinction between the topological 0-type, the circle $\mathbb{S}^1$, and the (discrete) higher inductive type, $S^1$ (using the notation of Mike’s paper). The ‘shape’ of the former is equal to the latter.

In a sense, I suppose, this is arising from the cohesive relation between topological ∞-groupoids and ∞-groupoids, before inserting topological ∞-groupoids into condensed ones.

Hello, Clark! Nice to run into you, as always, even if only online. Though I guess that’s how people run into each other these days anyway.

Yes, there is such a reasonable functor from condensed anima to $\infty$-topoi. It sends a condensed anima $X$ to the $\infty$-topos $X_{et}$ of “etale sheaves” over X (c.f. Scholze’s diamonds paper for the analogous construction there). There are two nice descriptions of this. One is the one you suggest: take the usual $\infty$-category of sheaves of anima on the (extr. disc.) profinite sets, and Kan extend from there to general $X$. Thus, an etale sheaf on a general $X$ is a compatible family of etale sheaves on all profinites mapping to $X$. Another description is that it identifies with a certain full subcategory of $\operatorname{CondAn}_{/X}$ closed under finite limits and colimits (and pullbacks), namely the one spanned by the “etale” maps to $X$. I’ll skip the precise definition of “etale”.

In the examples I’ve had occasion to think about, this association $X\mapsto X_{et}$ is quite reasonable and not very destructive. For a (T_1) topological space $T$, there is a pullback map from the usual $\infty$-topos of sheaves of anima on $T$ to the $\infty$-topos of etale sheaves on the associated condensed anima. For locally compact Hausdorff space, or a CW-complex, this pullback map identifies the target with the Postnikov completion of source. Similarly, for BG, the classifying anima of a profinite group, the $\infty$-topos of etale sheaves identifies with the Postnikov completion of the “usual” $\infty$-topos of sheaves on finite continuous $G$-sets with the canonical Grothendieck topology.

On the other hand, I expect that in general $X \mapsto X_{et}$ is not fully faithful. One test would be to see whether $X_{et}$ knows the difference between the filtered colimit of all compact at most countable subspaces of $[0,1]$ and $[0,1]$ itself. Actually, this is a question that probably many people here at the cafe could weigh in on:

Question: Is the pullback functor from sheaves of anima on the interval $[0,1]$ to compatible families of sheaves of anima on all compact at most countable subspaces of $[0,1]$ an equivalence?

The answer to the analogous question on the level of $0$-topoi (i.e. locales) or on the level of topological spaces is negative, basically because the topology on $[0,1]$ is sequential. But it’s not clear to me what’s going on for $\infty$-topoi or even $1$-topoi. I found it kind of tricky to analyze, which is one of several reasons why a few years ago I abandoned $\infty$-topoi for this kind of thing and switched to the condensed perspective. Of course I’ve also learned many times over the years that just because I find something tricky or difficult doesn’t necessarily mean that it really is so…

Finally, regarding your question about shape theory, note that on the “discrete” condensed anima $X$, meaning the usual anima, $X_{et}$ is the usual slice topos of anima over $X$. Thus, returning to general $X$, for formal reasons we get a comparison map of pro-anima $\operatorname{Shape}(X_{et})\rightarrow \vert X\vert$. Again this is an equivalence for CW complexes, compact Hausdorff spaces, etc., but probably not in general.

One last remark. Since $X\mapsto X_{et}$, viewed as a functor to $\infty$-topoi with geometric morphisms as morphisms, is left Kan extended from the very simple profinite sets, it feels like it “wants” to be a left adjoint. But there is no right adjoint, for reasons like the one Peter (Scholze) mentioned: if your $\infty$-topos is at all “category-like”, e.g. admits non-trivial specializations between points, then the associated condensed anima won’t exist and the associated pyknotic space will likely be pathological.

It seems that the difference in nature between condensed anima and $\infty$-topoi is twofold:

1. $\infty$-topoi can be “category-like” whereas condensed anima can only be “space-like”;

2. $\infty$-topoi are based on abstracting “second-order” structure: open subsets of topological spaces. Condensed anima are based on abstracting “first-order” structure: points of topological spaces.

I guess it is fundamentally these two differences which give rise to all the other differences. For example the relative simplicity of the $\infty$-category of condensed anima vs. the $\infty$-category of $\infty$-topoi is heavily influenced both by 1 and 2. And the fact that condensed anima are good for mixing algebraic and topological structures (whereas $\infty$-topoi are not) is clearly due to point 2.

I also believe (though do not have honest evidence to support this belief, so take it with a grain of salt) that these differences mean that comparisons between natural constructions associated to condensed anima and $\infty$-topoi, such as their associated pro-anima as discussed above, can only work out in certain nice special cases and not in general. Thankfully, however, the nice special cases do include many objects of general interest.

I’m pretty sure that these properties also hold for condensed sets. In some sense condensed sets are the version of Johnstone’s topos that replaces convergence along sequences by convergence along ultrafilters; the difference should not be relevant here. (Let me instead point out that Johnstone’s topos does not have enough projective objects, and in a way condensed sets are the minimal (up to cardinality questions) way to make them have enough projective objects.)

Something’s off here regarding the interpretation of the above functor as a pushforward in a map of topoi. The left adjoint would send $A$ to $\beta A$, but this does not commute with finite products as $\beta(A\times B)\neq \beta A\times \beta B$.

On the other hand, one can regard the above functor as a pullback in a map of topoi, so giving a map from the bornological topos to the condensed topos. (I’m ignoring set theory here, so either pass to the pyknotic setup, or replace “topos” by “macrotopos”.) This is induced by the map of sites taking any profinite set $S$ to the bare set $S$.

This functor evidently loses a lot of information, but it is conservative: If $f: X\to Y$ is a map of condensed sets that is an isomorphism on associated bornological sets, then in particular $X(\beta A)\cong Y(\beta A)$ for all sets $A$, and the Stone-Cech compactifications of discrete sets form a basis for the pro-etale site of a point.

The drive to turn some or other branch of mathematics into “algebraic geometry” is interesting. Breiner, in the work I’ve referred to a couple of times already, writes

Although contemporary model theory has been called “algebraic geometry minus fields”, the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings.

Whereas Lurie’s approach to this model-theoretic topic in Ultracategories uses ultraproducts to manage convergence, Breiner has his models form a topological groupoid.

I wonder if there’s a place for condensed mathematics in model theory.

Peter wrote

I think the comparison to the condensed/pyknotic setup is cleaner if one simply takes the bornological site to consist of all sets, and covers finite families of jointly surjective maps (and resolves set-theoretic issues in the same way as before).

In a subsequently published preprint

• F. William Lawvere, Section 2 of Toposes generated by codiscrete objects in combinatorial topology and functional analysis, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, pdf,

Lawvere considers on p. 11 the variant for all sets of cardinality $\leq \lambda$ for a large cardinal $\lambda$.

Hi Yemon,

Thanks for your comments and questions. I find the interaction with functional analysis fascinating. Let me start with some quick responses to your questions/musings; if you’d like me to say more about any of these points I’d be very happy to expand. I believe they’re also all discussed in Peter’s notes on analytic geometry.

1) Smith spaces are the same as Waelbrock dual spaces. Smith’s work came much earlier; it proved the main theorem on them (the dualilty with Banach spaces), but did not specifically isolate the concept. We got the name from Akbarov.

2) Smith spaces are more natural than Banach spaces from the condensed perspective because they are controlled by a nice compact subset instead of a nice open subset (the unit ball) like in the Banach setting. Also, Banach spaces are filtered colimits of Smith spaces while Smith spaces are filtered limits of Banach spaces; filtered colimits are nicer in their algebraic and homological properties so it makes sense to take Smith spaces as fundamental.

3) But they are not fundamental enough! Smith spaces embed fully faithfully in condensed R-vector spaces; this is good, but Smith spaces are not an abelian category, and we also care about things like Ext’s. There is an induced functor from Waelbrock’s enlarged (abelian!) cateogry of quotients of Smith spaces, but that functor is not fully faithful!

4) The lack of full faithfuness in 2) is not accidental; it arises because Waelbrock’s rather formally defined category does not “see” certain natural functional analytic maps and constructions, specifically those like Ribe’s famous extension which are based on the entropy functional. These extensions take place outside the familiar comforting context of locally convex spaces, so it is tempting to exclude them “by fiat” as basically everyone does implicitly or explicitly, but from the condensed perspective they are simply “there” and must be reckoned with. Besides, their structre is fascinating and leads to a one-parameter family of deformations of the real numbers, as described in Lecture VI of Peter’s notes.

5) The conclusion is that if you want a naturally ocurring abelian category suitable for functional analysis you absolutely need to allow non-locally convex spaces in addition to the familiar locally convex ones.

6) Fortunately such an abelian category exists; in fact there is a one-parameter family of such categories. These are the “p-liquid R-modules”. They sit naturally in condensed R-modules, have excellent categorical and homological properties, include essentially all examples of functional analytic spaces people use, and calculations such as tensor products, hom’s, ext’s etc. work out nicely with them.

6) More specifically, p-liquid R-modules are an example of what Peter and I define as an “analytic ring”. This is an axiomatic set-up consisting of a condensed ring plus a full subcategory of condensed modules over that condensed ring, which one can to first approximation think of as the “completete” modules. The axioms are very strong and force the category of complete modules to behave much in the same way as ordinary modules over an ordinary ring. All the honest work lies in producing examples of such analytic rings.

7) Proving that the p-liquid theory “exists” (satisfies the axioms of an analytic ring) is very difficult and requires a delicate mixture of homological algebra and functional analysis, with implicit constants several layer deep floating around everywhere. See lectures VIII and IX of Peter’s notes. But once you have it you can take it as a black box and many results in complex analytic geometry prove themselves “by hand” with categorical techniques, plus simple calcuations in key cases.

Best regards Dustin

Dear Yemon,

thanks for your question, and for allowing yourself to be silly! In fact, I’m sorry for this totally over-the-top first paragraph or two of my lectures, but I wanted to make the ambition clear. And I actually think that the theory developed in the first 9 lectures is something of real value even to actual functional analysts, and is actually going some way towards the ambition.

As Dustin has explained in his long comment, proving our results about liquid R-vector spaces requires very serious efforts (it might be the most difficult theorem I’ve (co-)proved), both conceptually but also just the bare estimates. But once you have it you can easily do a lot of things in complex analysis and potentially even in other situations like Atiyah-Singer where you usually run into difficult technical issues of doing homological algebra with topological vector spaces. The machinery we developed is supposed to gracefully handle such problems. We’ve mostly worked out some things for complex-analytic spaces so far, but there’s nothing stopping this approach from working for smooth manifolds and the like.

I was just looking through some of Waelbrock’s work, and indeed it seems he’s also done other things which go in a similar direction as our work. I see that he and Buchwalter defined a notion of “compactological space” which turns out to be equivalent to the notion of “quasi-separated condensed set”. (Quasi-separatedness is a general topos-theoretic concept; here it unwinds to the following: a condensed set X is quasi-separated iff the diagonal map X –> X x X is a closed inclusion, in the sense that every base-change to a profinite set identifies with a closed inclusion of profinite sets. It is the natural separation axiom to impose in the condensed context.) Indeed, both are equivalent to the full subcategory of Ind(compact hausdorf spaces) spanned by the Ind-objects which can be represented using only closed inclusions for transition maps.

Thanks for mentioning our approach. I also think that the Ind(Ban) approach can give a completely unified approach to algebraic geometry and analytic geometry. But we have only completed a few steps out of what seems like millions.

I think I can prove at least one of the statements above. Give me time to write down what I have in mind.

I think I can prove at least one of the statements above. Give me time to write down what I have in mind.

If I understand well what you’re after, you want to show that a ultracategory is just a $T$-algebra for a suitable monad.

My claim is that the monad acts as follows on objects:

$$T : \mathcal{A} \mapsto \int^X \beta(X)\times \mathcal{A}^X$$

I am basing this conjecture on the fact that an ultracategory consists of maps

$$i_X : \beta(X)\times \mathcal{A}^X \to \mathcal{A}$$

that if this conjecture is correct amount exactly to a functor $T\mathcal{A} \to \mathcal{A}$ (by the universal property of the coend, the $i_X$ forming a cowedge in $X$).

(Sure, we could debate about where exactly I’m taking the coend in question… $\beta(X)$ is a topological space, and $\mathcal{A}^X = \prod_{x\in X} \mathcal{A}$ is a category; I see different, inequivalent ways to give a meaning to the product $\beta(X) \times \mathcal{A}^X$…

A deeper problem is: strictly speaking $T$ does not exist, because as $X$ ranges over all sets the coend I’m taking is on too large a diagram. I kindly ask you to set aside size issues and follow a bit of coend-nonsense: so, just for today $T$ exists and defines a functor.

1. First, the initial cowedge $i_X : \beta(X) \times \mathcal{A}^X \to T A$ has a component at $X=1$, the terminal category; this gives $T$ the structure of a pointed functor (of course, $i$ depends naturally on $\mathcal{A}$, thus we have a natural transformation $\eta : 1 \Rightarrow T$).

2. A candidate multiplication map $T T\mathcal{A} \to T\mathcal{A}$ can be obtained as follows:

$$\begin{array}{rl} T T\mathcal{A} &= \displaystyle \int^Y \beta(Y) \times T\mathcal{A}^Y \\ &\displaystyle = \int^Y \beta(Y) \times \left( \int^X \beta(X)\times \mathcal{A}^X \right)^Y \\ &\displaystyle \to \int^Y\int^X \beta(Y) \times \beta(X)^Y \times \mathcal{A}^{X\times Y} \\ &\displaystyle \to \int^Y\int^X \beta(Y) \times \beta\beta(X)^{\beta Y} \times \mathcal{A}^{X\times Y} \\ &\displaystyle \overset{\epsilon}\to \int^Y\int^X \beta\beta(X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \overset{\mu^\beta}\to \int^Y\int^X \beta (X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \to \text{colim}_Y\int^X \beta (X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \cong \int^X \beta (X) \times \mathcal{A}^{X} \end{array}$$

All this is pretty standard ($\epsilon$ is the evaluation, counit of the cartesian closed structure on… wherever $\beta$ takes values; $\mu^\beta$ is the multiplication of the monad $\beta$); the last step is “colimits over a category with a terminal object are evaluation at that object”.

Of course, there’s still some work to do now: the map $\mu : T T\mathcal{A} \to T\mathcal{A}$ is associative and has $\eta$ as a unit. I just wanted to provide quantitative support to your initial guess.

David, I hadn’t commented before because I’ve only given “Ultracategories” the briefest skim, but given our conversation over here, I’ll do a bit of thinking out loud.

In short, I don’t get the idea. First let’s do what you suggest but with 1- rather than 2-monads. The codensity monad of your inclusion $FinSet \hookrightarrow Cat$ sends a category $A$ to the discrete category on the set of ultrafilters on $\pi_0(A)$. Here $\pi_0(A)$ denotes the set of connected-components of $A$. (This follows from the adjunction $\pi_0 \dashv D$.) And presumably that’s not the right thing.

So if your proposal is right, something significant must happen when we step up from 1-monads to 2-monads. Can you explain what that is?

Yes, thank you, David! That’s a really nice characterization. I guess I was implicitly using it, or a variant of it which says that [0,1] is the only compact Hausdorff space $I$ equipped with two endpoints and a gluing isomorphism from two copies of itself glued along the endpoints to itself, such that a point in $I$ is uniquely determined by a sequence of letters “L” and “R” saying in which half of the decomposition the point lies, which half of that half, etc. etc. Not nearly as crisp as Freyd’s characterization, but it’s what the proof uses :)

Dustin, there’s a topological version of Freyd’s characterization that might fit what you’re doing. It goes like this.

Let’s say a bipointed space is a topological space equipped with an ordered pair of distinct, closed points. Given bipointed spaces $(X, x^-, x^+)$ and $(Y, y^-, y^+)$, one can form their wedge $X \vee Y$ in the usual way (take their disjoint union, identify $x^+$ with $y^-$, and take $x^-$ and $y^+$ to be the new basepoints). Let $C$ be the category in which an object is a bipointed space $X = (X, x^-, x^+)$ equipped with a continuous, basepoint-preserving map $X \to X \vee X$. For example, $([0, 1], 0, 1)$ together with the doubling map is an object of $C$.

Theorem The terminal object of $C$ is $[0, 1]$, with its standard (Euclidean) topology.

(Of course, if you stick to Hausdorff spaces then you can drop the condition that the basepoints are closed.)

The nice thing about this theorem is that you don’t get some trivial topology on $[0, 1]$. You get the Euclidean topology. And the result can be understood as saying that $[0, 1]$ is uniquely suitable as a parametrizing space for homotopy theory, as explained here.

This theorem isn’t mentioned in the nLab page that David linked to. It follows from Theorem 2.2 of my paper arXiv:1010.4474. There, the result is embedded in a much wider theory of self-similarity, but it’s also not hard to prove directly.

…. and in the above post: “monomorphism” instead of “isomorphism”. Sorry all! My brain’s a bit fried these days.

The solidification of spectra is completely parallel to the story of solid abelian groups discussed above. It is an idempotent endofunctor, and is the left adjoint to the inclusion of solid spectra into condensed spectra. I’m always getting confused about signs, so I don’t know whether this makes it a reflector or coreflector.

About passing to finite level: If $S=\lim_i S_i$ is a profinite set, written as a (co)filtered limit of finite sets, then for any discrete anima $X$, considered as a condensed anima via the fully faithful embedding, one has $X(S)=\colim_i X(S_i)$. This property in fact characterizes discrete objects: A condensed anima $X$ is discrete if and only if for all profinite $S$ as above, $X(S)=\colim_i X(S_i)$.

I’m also a little confused about the precise inverse limit Dustin wants to take, but here is a variant that definitely works.

For any $n$, let $I_n$ be the disjoint union of the intervals $[i/2^n,(i+1)/2^n]$ for all $i=0,\ldots,2^n-1$. Then $f_n: I_n\to I=[0,1]$ is surjective, and one can recover $I$ as the geometric realization of the Cech nerve $I_n^{\bullet/I}$ of $f_n$.

For varying $n$, there are natural transition maps, and so let $I_\infty = \lim_n I_n$. Sequential limits of surjections are still surjections in condensed sets – this uses the existence of compact projectives. So $f_\infty: I_\infty\to I$ is still a surjection, and one can recover $I$ as the geometric realization of the Cech nerve $I_\infty^{\bullet/I}$.

But now observe that $I_\infty$ (and more generally all terms of the Cech nerve) is profinite; in fact, the natural map

$I_\infty\to \lim_n I_n\to \lim_n \pi_0(I_n)$

is an isomorphism. This is because all the intervals are shrinking in size. In fact, $I_\infty$ gets naturally identified with binary expansions of elements $[0,1]$, without identifications such as $0.011111... = 0.10000...$, so $I_\infty = \prod_{\mathbb{N}} \{0,1\}$.

Similarly,

$I_\infty^{\bullet/I}\to \lim_n I_n^{\bullet/I}\to \lim_n \pi_0(I_n^{\bullet/I})$

is an isomorphism.

Now let $X$ be any discrete anima; for simplicity truncated (in order to commute some limit with a sequential colimit in a second). Then one can compute $\Hom(I,X)$ as the limit of the cosimplicial object $\Hom(I_\infty^{\bullet/I},X)$. As each term is profinite, each term is the sequential colimit of

$\Hom(\pi_0(I_n^{\bullet/I}),X)$.

This sequential colimit can be exchanged with the limit, so it is enough to compute for each $n$ the limit of the cosimplicial object

$\Hom(\pi_0(I_n^{\bullet/I}),X)$.

But it is easy to understand the combinatorics of the connected components of $I_n^{\bullet/I}$, and hence compute this limit to be $X$ itself. This shows that the map $X\to \Hom(I,X)$ is an isomorphism for truncated anima $X$, but then also for all $X$ via a Postnikov limit.

What have we proved? That for all disctete anima $X$, the map $X\to \Hom(I,X)$ is an isomorphism of anima, where the right-hand side is the (external) Hom in condensed anima. But in fact, it is an isomorphism of condensed anima (i.e., is also true when using the internal Hom), and reading closely, the whole proof works internally. The one thing has to check is that discrete anima $X$ satisfy the equation $X(S)=\colim_i X(S_i)$, i.e. $\Hom(S,X) = \colim_i \Hom(S_i,X)$, internally, which is again true.