In Peter Johnstone's 1979 paper On a topological topos, he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}_\infty$, the one-point compactification of the natural numbers, as a convenient big topos whose objects are a sort of space. He showed that this topos (now sometimes known as Johnstone's topological topos) has the following nice properties:
- It is a local topos, i.e. its global sections functor (whose left adjoint constructs "discrete spaces") has a fully faithful right adjoint, which constructs "indiscrete spaces". (But it is not locally connected, hence not "cohesive".)
- It contains the category of sequential topological spaces as a full subcategory closed under limits (and indeed reflective) — although sequential spaces are coreflective in all topological spaces, so limits in the former don't coincide with limits in the latter.
- The embedding of sequential spaces preserves colimits arising from open covers, closed covers, and quotients of sequentially closed equivalence relations. In particular, therefore, it preserves the construction of CW-complexes.
- The internally-constructed real numbers object coincides with the usual (sequential) space of real numbers with its usual topology.
- Geometric realization is the inverse image part of a geometric morphism from this topos to the topos of simplicial sets.
Much more recently, Barwick and Haine have introduced the topos of pyknotic sets with, it seems to me, a similar motivation. (The Scholze-Clausen category of condensed sets is related, but not a topos, so for purposes of this question I'm not as interested in it.) In some ways, pyknotic sets feel to me like an extension of Johnstone's topological topos to non-sequential notions of convergence. By definition, it consists of sheaves on a category of compact Hausdorff topological spaces, which includes $\mathbb{N}_\infty$. And the paper Pyknotic objects, I proves that the topos of pyknotic sets has analogues of properties 1 and 2 above, and partly 3:
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Pyknotic sets is a local topos.
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It contains the category of compactly generated spaces as a full subcategory closed under limits, while compactly generated spaces are coreflective in all topological spaces.
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This embedding preserves sequential colimits of compactly generated spaces whose colimit is $T_1$.
Thus, I wonder whether the topos of pyknotic sets shares any of the other nice properties of Johnstone's topological topos. Namely:
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Does the embedding of compactly generated spaces preserve any more colimits, such as those arising from open or closed covers? In particular, does it preserve the construction of CW-complexes?
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Does the real numbers object in pyknotic sets coincide with the usual (compactly generated) space of real numbers with its usual topology?
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Is geometric realization the inverse image part of a geometric morphism from pyknotic sets to simplicial sets?
Best Answer
Let me recall a little bit of the background. The question is about the relation between topological spaces and pyknotic sets, and properties of the topos of pyknotic sets. Recall that pyknotic sets are sheaves on the site of compact Hausdorff spaces, for the Grothendieck topology generated by finite families of jointly surjective maps. (So this includes, notably, finite covers by closed subsets; but because any open cover can be refined by a finite closed cover, open covers are also covers in this Grothendieck topology.) "Pyknotic" as opposed to "condensed" refers to one way of addressing the issue that the site is large; in this case, by fixing universes. The issue is inconsequential for the whole discussion, so let me not discuss it further.
For any topological space $X$, one can define a pyknotic set $\underline{X}$, whose $S$-valued points are the continuous maps from $S$ to $X$, for any compact Hausdorff space $S$.
On compactly generated topological spaces, this functor is fully faithful. By its definition, it also commutes with all limits. The first part of the question concerns the question to what extent it commutes with colimits.
The first type of colimits are geometric realizations. Assume you have a topological space $X$ that is written as the quotient of some topological space $Y$ by the equivalence relation $R=Y\times_X Y\subset Y\times Y$ (with induced topology). Under which conditions is the natural map $\underline{Y}/\underline{R}\to \underline{Y/R}=\underline{X}$ an equivalence?
We do not want this to be the case in all situations, as for "bad" equivalence relations (with dense orbits), the topological quotient space is quite bad, and the pyknotic one is actually much better. But often, the topological construction is nice, so we want it to be preserved.
To check whether a map of pyknotic sets is an isomorphism, it suffices to check that it is an isomorphism on $S$-valued points where $S$ is an extremally disconnected compact Hausdorf space -- this is because any compact Hausdorff space can be covered by an extremally disconnected one. Moreover, evaluation at extremally disconnected sets commutes with all sifted colimits, and in particular with quotients by equivalence relations. So we have to see whether for all such $S$, the map $$\mathrm{Cont}(S,Y)/\mathrm{Cont}(S,R)\to \mathrm{Cont}(S,X)$$ is bijective. Note that the map is necessarily injective -- if two continuous maps $f_1,f_2: S\to Y$ induce the same map to $X$, then they induce a continuous map to $R$. Thus, only surjectivity is at stake. Thus,
In particular, if $Y$ is a finite disjoint union of closed subsets of $X$ (corresponding to gluing $X$ from finitely many closed subsets), or if $Y$ is a disjoint union of open subsets of $X$ (corresponding to gluing $X$ from open subsets), then this holds. But there are much more general situations.
[Edit: Some details about why this is true for those two cases. If $Y$ is a finite disjoint union of closed subsets of $X$ (covering $X$), then after pullback along any continuous map $S\to X$, one gets a finite disjoint union of closed subsets of $S$ (covering $S$). This is itself a compact Hausdorff space, surjective over $S$, which hence admits a splitting (as $S$ is extremally disconnected). If $Y$ is a disjoint union of open subsets of $X$, then after pullback along any continuous map $S\to X$, one gets a disjoint union of open subsets of $S$, and as remarked above, this can be refined by a finite disjoint union of closed subsets of $S$ covering $S$, reducing us to the previous case.]
Another interesting type of colimit is a filtered colimit. Filtered colimits of topological spaces tend to be reasonable only when the transition maps are closed immersions, so let us assume that. Thus, consider a filtered index set $I$ and a diagram $X_i$, $i\in I$, of compactly generated topological spaces, with transition maps closed immersions, and let $X=\mathrm{colim}_i X_i$ be their colimit (in compactly generated topological spaces). Under what conditions is the natural map $$ \mathrm{colim}_i \underline{X_i}\to \underline{X} $$ an isomorphism?
Again, this can be checked on $S$-valued points for extremally disconnected $S$, and again injectivity is automatic, so the question is about surjectivity.
(The factorization is necessarily continuous, as $X_i\subset X$ is a closed subset.)
Unfortunately, this is not the case for all colimits, but it is often the case, as discussed in some detail in Proposition A.15 of Schwede's Global homotopy theory. In particular, it is always the case for sequential colimits.
A counterexample is given by writing $[0,1]$ as the filtered colimit of all of its countable closed subsets -- this is a colimit in (compactly generated) topological spaces, but it is not a colimit in pyknotic sets. Note that it would also be a colimit in Johnstone's topos, but I find it more realistic that $[0,1]$ is actually "larger" than this filtered colimit.
Now there are two questions about the pyknotic topos. First, whether the internally constructed reals agree with the externally defined object $\underline{\mathbb R}$. As a rule of thumb, such agreements tend to always hold -- internal constructions in pyknotic sets automatically acquire the "correct"/"expected" pyknotic structure. (This is one advantage of the pyknotic/condensed formalism over classical topological spaces -- usually, whenever you do some mathematical construction, you have to specify the topology by hand; but pyknotic structures just come along for the ride.)
For the internal real numbers, let me consider Cauchy reals first. This is the quotient of the group of Cauchy sequences of rational numbers (with given convergence) by the group of null sequences of rational numbers (with given decay), both constructed internally in pyknotic abelian groups. The internally constructed group of Cauchy sequences of rational numbers takes any extremally disconnected $S$ to sequences of locally constant functions $f_n: S\to \mathbb Q$ such that $|f_n-f_{n+1}|\leq 2^{-n}$; and in there one can consider the subspace of null sequences, where necessarily in addition $|f_n|\leq 2^{1-n}$. It is an elementary exercise to see that their quotient is precisely the group of continuous maps $S\to \mathbb R$ for the usual topology on $\mathbb R$. In fact, one is just looking at the completion of the normed abelian group $\mathrm{Cont}(S,\mathbb Q)$, which is indeed the Banach space $\mathrm{Cont}(S,\mathbb R)$.
[Edit: Some details about this verification. First, given any such sequence $f_n$, one can define a function $f_\infty: S\to \mathbb R$ by taking $s\in S$ to the limit of the Cauchy sequence $f_n(s)$. The function $f_\infty$ is continuous: If $s\in S$ is any element and $f_n$ is constant in the open neighborhood $U$ of $s$, then for all $s'\in U$, one has $$|f_\infty(s)-f_\infty(s')|\leq |f_\infty(s)-f_n(s)|+|f_n(s)-f_n(s')|+|f_\infty(s')-f_n(s')|\leq 2^{1-n} + 0 + 2^{1-n}\leq 2^{2-n}.$$ If $f_\infty=0$, then we must have $|f_n(s)|\leq 2^{1-n}$ for all $n$ and hence the $f_n$ lie in the subspace of null sequences. It remains to see that any continuous function $f: S\to \mathbb R$ can be written as a limit of locally constant functions $f_n: S\to \mathbb Q$ with the required convergence. It suffices to show that for any $n$ one can find a locally constant function $f_n: S\to \mathbb Q$ such that $|f-f_n|\leq 2^{-n-1}$. To do so, consider for any $s\in S$ the open subset $U_s\subset S$ of all $s'$ such that $|f(s)-f(s')|<2^{-n-1}$. These open sets cover, so there is a finite subcover. As $S$ is totally disconnected, this can be refined by a finite disjoint union of open and closed subsets. So there are finitely many open and closed subsets $U_i\subset S$ with points $s_i\in U_i$ such that for all $s'\in U_i$, one has $|f(s_i)-f(s')|<2^{-n-1}$. Now define $f_n$ to be equal to $f(s_i)$ on $U_i$.]
In the case of Dedekind reals, one would instead send any such $S$ to the Dedekind cuts over $S$, which essentially partition the locally constant functions $S\to \mathbb Q$ into a "left" and a "right" part. Again, it is an elementary exercise to see that all of them arise from some continuous map $S\to \mathbb R$.
[Edit: Some more details about this verification. Again, given such a Dedekind cut over $S$, one can define a function $f: S\to \mathbb R$ by taking any $s\in S$ to the value determined by the Dedekind cut in the fibre at $s$. To see that this is continuous, note that if $q<f(s)$ for some rational number $q$, then there must be some locally constant function $g: S\to \mathbb Q$ that is a lower bound for $f$, and such that $q<g(s)<f(s)$. As $g$ is locally constant, this implies that $q<g(s)=g(s')<f(s')$ for $s'$ in an open neighborhood $U$ of $s$. Similarly, upper bounds spread to neighborhoods. Conversely, note that any continuous function $f$ determines a unique Dedekind cut, as those locally constant functions to $\mathbb Q$ that lie below resp. above $f$.]
Finally, there was the question whether geometric realization defines a geometric morphism of topoi from pyknotic sets to simplicial sets. Taking away the jargon, this is just the question whether geometric realization commutes with finite limits. As it commutes with all colimits, and finite limits commute with filtered colimits, this reduces directly to the case of simplicial sets with finitely many nondegenerate simplices. But for such simplicial sets, the geometric realization is just a compact Hausdorff space, and the question of commutation with finite limits is independent of the context one works in (in fact, isomorphisms of compact Hausdorff spaces can be checked on underlying sets), and known to be true.