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Does anybody understand why Delta-generated spaces are locally presentable? This is of course claimed by Jeff Smith, and there is a paper by Fajstrup and Rosicky

A convenient category for directed homotopy

that proves it. But I can't understand the proof, and also it involves things that really should not be necessary from mathematical logic.

Note that Delta-generated spaces are just colimits of copies of the unit interval I, so they are the same as I-generated spaces. The general claim is that A-generated spaces are locally presentable for any A. The point must be that the topology in an A-generated space is determined by sets of a bounded size, depending on A. For example, in I-generated spaces, a point is in the closure of a subset if and only if you can get to the point by a convergent sequence. This has to be the key to the proof, but I have not been able to make this into a proof.

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No, I don't think so. Given an I-generated space X, form the diagram whose objects are maps from I to X and whose morphisms are commutative triangles. The colimit of this diagram (of copies of I, one for each map from I to X) is X if X is I-generated, and more generally is the X with the I-generated topology. Just like compactly generated. – Mark Hovey Nov 13 '09 at 0:36
OK, here is the simplest question I can't answer. Take a long colimit of injections $X_0 \Rightarrow X_1 \Rightarrow X_2 \Rightarrow $ of I-generated spaces indexed by an uncountable ordinal. Give the colimit X the weak topology (U is open if and only if U intersect each $X_i$ is so). Take a sequence in $X_0$ that converges in X. Prove it converges in one of the X_i. – Mark Hovey Nov 13 '09 at 13:06
But in Delta-generated spaces, we have all the simplices as generators, not just the unit interval I, right? (The higher simplices aren't I-generated, are they?) – Reid Barton Nov 29 '09 at 4:56
Yes, Reid, the higher simplices are Delta-generated. As Jeff told me: "space-filling curves". That is, a space-filling curve reveals Delta[n] to be a quotient of I (it is a closed surjection by compactness, so a quotient map). I-generated spaces are closed under quotients (and colimits in general). – Mark Hovey Nov 29 '09 at 18:36
It appears to me that indiscrete spaces of cardinality $\le \mathfrak{c}$, where $\mathfrak{c}$ is the cardinality of the continuum, can be obtained as quotients of $\Delta^1$. Thus we can construct an increasing sequence of ($\Delta$-generated) subspaces of cardinality $< \mathfrak{c}$ of a $\Delta$-generated space of cardinality $\mathfrak{c}$ whose union is the whole space. In particular, the presentability rank of a simplex is at least $\mathfrak{c}$. – Zhen Lin Jul 30 at 9:35

2 Answers 2

Here's a sketch of a more direct proof that $\Delta$-generated spaces (call this category $\Delta-\mathrm{Gen}$) are locally presentable. It's essentially a "compiling-out" of Fajstrup and Rosický's proof. The best estimate I'm able to extract for the accessibility rank is $(2^{2^{\aleph_0}})^+$, though from Zhen Lin's comment one should probably expect the true accessibility rank to be $(2^{\aleph_0})^+$.

First we show that the category $\aleph_0-\mathrm{Gen}$ of countably generated spaces -- those spaces which are to the countable topological spaces as $\Delta$-generated spaces are to simplices -- is locally presentable. Since $\Delta-\mathrm{Gen}$ is a full subcategory of $\aleph_0-\mathrm{Gen}$ which is closed under colimits, with a dense generator given by the simplices, it is also locally presentable.

(That argument might sound like it requires Vopenka's principle, but it doesn't: it just uses the characterization of locally presentable categories as those cocomplete categories with a dense generator [this hypothesis can be weakened to: strong generator] of presentable objects. If $\mathcal{K}$ is locally presentable then every object there is a cardinal $\lambda$ such that the object is $\lambda$-presentable [since for every object there is a $\lambda$ such that it is a $\lambda$-small colimit of canonical generators, and the $\lambda$-presentable objects are closed under $\lambda$-small colimits]. If $\mathcal{L}$ is a full subcategory closed under colimits, then all the objects of $\mathcal{L}$ are also presentable, so if $\mathcal{L}$ has a dense generator, it is locally presentable. Explicitly in this case, the simplices are continuum-sized colimits of countable spaces, so they are presentable. If the countable spaces were $(2^{\aleph_0})^+$-presentable, the simplices would be too; as it is though, I can only show that the countable spaces are $(2^{2^{\aleph_0}})^+$-presentable, so that's the best estimate I have for the simplices, too.)

The reason for bringing $\aleph_0-\mathrm{Gen}$ into the picture is that in $\aleph_0-\mathrm{Gen}$, it's easy to describe the topology on a colimit $X = \varinjlim X_i$ when the colimit is sufficiently filtered. Namely, $X$ has the topology where

a countably-supported ultrafilter $\mathcal{F} \in \beta_\omega X$ converges to a point $x \in X$ if and only if "$\mathcal{F}$ already converges to $x$ at some stage of the colimit", i.e. iff there exists an $X_i$ and an $x_i \in X_i$ mapping to $x$ and a countably-supported ultrafilter $\mathcal{F}_i \in \beta_\omega X_i$ which pushes forward to $\mathcal{F}$, such that $\mathcal{F}_i$ converges to $x_i$ in $X_i$.

Here we use the notion of ultrafilter convergence: an ultrafilter $\mathcal{F} \in \beta X$ is said to converge to a point $x \in X$ iff every neighborhood of $x$ is an element of $\mathcal{F}$. A countably-supported ultrafilter $\mathcal{F} \in \beta_\omega X$ is just an ultrafilter which contains a countable subset of $X$.

It's obvious from this description of a sufficiently-filtered colimit that spaces of sufficiently small cardinality are presentable, because a function between (countably-generated) topological spaces is continuous iff it sends convergent (countably-supported) ultrafilters to convergent ultrafilters. Since the countable spaces are dense in $\aleph_0-\mathrm{Gen}$, it follows that $\aleph_0-\mathrm{Gen}$ is locally presentable.

The subtlety, of course, comes in verifying that this description of ultrafilter convergence in a sufficiently-filtered colimit actually arises from a topology (and that this topology is countably-generated). Barr showed that a relation $R \subseteq \beta X \times X$ defined for all ultrafilters arises from a topological space if and only if $R$ is a lax algebra for the ultrafilter monad $\beta$, giving a (concrete) equivalence of categories between topological spaces and lax $\beta$-algebras. By replacing $R$ with a relation $R \subseteq \beta_\omega X \times X$ in this definition, we get a "lax-algebraic" description of $\aleph_0-\mathrm{Gen}$, which we can use to compute sufficiently-filtered colimits as above.

To be precise about the ultrafilter description of $\aleph_0-\mathrm{Gen}$, let me first review the ultrafilter description of general topological space. Consider a relation $ \beta X \overset{\pi_1}{\leftarrow} R \overset{\pi_2}{\to} X$, and write $\mathcal{F} \rightsquigarrow x$ if $(\mathcal{F},x) \in R$, i.e. $R= \{(\mathcal{F},x) \mid \mathcal{F} \rightsquigarrow x\}$. Then $R$ is the convergence relation for a topology on $X$ if and only if the following conditions hold:

  1. For every $x \in X$, $\mathrm{prin}(x) \rightsquigarrow x$, where $\mathrm{prin}(x)$ is the principal ultrafilter at $x$.

  2. If $\mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(\pi_2)_*(\mathcal{G}) \rightsquigarrow x$, then $\sum (\pi_1)_*(\mathcal{G}) \rightsquigarrow x$.

Here $()_*$ is the pushforward of ultrafilters, $f_*(\mathcal{F}) = \{A \mid f^{-1}(A) \in \mathcal{F}\}$ and $\sum: \beta \beta X \to \beta X$ is the sum of ultrafilters $\sum \mathcal{H} = \{A \mid \hat{A} \in \mathcal{H}\}$, where $\hat{A} = \{\mathcal{F} \mid A \in \mathcal{F}\}$.

Analogously, consider a relation $ \beta_\omega X \overset{\pi_1}{\leftarrow} R \overset{\pi_2}{\to} X$, with the notation $\mathcal{F} \rightsquigarrow x$ as before. Then $R$ is the convergence relation (restricted to countably-supported ultrafilters) for a countably-generated topology on $X$ if and only if the following conditions hold:

  1. For every $x \in X$, $\mathrm{prin}(x) \rightsquigarrow x$, where $\mathrm{prin}(x)$ is the principal ultrafilter at $x$.

  2. If $\mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(\pi_2)_*(\mathcal{G}) \in \beta_\omega X$ and $(\pi_2)_*(\mathcal{G}) \rightsquigarrow x$, and if $\sum (\pi_1)_*(\mathcal{G}) \in \beta_\omega X$, then $\sum (\pi_1)_*(\mathcal{G}) \rightsquigarrow x$.

Note that in (2), $\mathcal{G}$ is not required to be countably supported, nor is $(\pi_1)_*(\mathcal{G})$. So a countably-generated space is not apparently the same thing as a lax algebra for the monad $\beta_\omega$ of countably-supported ultrafilters -- it needs to satisfy a stronger associativity condition (2) which refers back to the full ultrafilter monad $\beta$. There ought to be some general 2-categorical or equipment-theoretic description of the relationship between these two monads and of this sort of "hybrid" lax algebra for them, but I haven't worked out what it should be.

I doubt that $\Delta-\mathrm{Gen}$ can be described directly in terms of a submonad of the ultrafilter monad $\beta$ -- this is the reason for bringing countably-generated spaces into the story.

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This really feels like the right idea. Especially, what you wrote in the gray box seems to be exactly what Mark thought had to be the key, and I agree with bringing in $\aleph_0$-generated spaces to leverage ultrafilters. I get a bit lost in the end. When you say "there ought to be some general 2-categorical..." is that a necessary part of the proof? It doesn't exactly feel like the sort of place Vopenka's principle might be hiding, but one has to be careful that this argument really does not require Vopenka anywhere. – David White 2 days ago
Oh, no -- that part isn't really part of the argument at all, just some musings on what sort of formal framework this alternative definition of countably-generated spaces might fit into. There's nothing like Vopenka in establishing the equivalence of these descriptions, just lots of uses of the ultrafilter principle, which is a weak form of choice. I've added a paragraph spelling out the other part of the argument. I'm working on writing up a detailed proof which I will try to post somewhere. – Tim Campion 2 days ago

I might be missing something, but it appears that the argument can be made as follows:

Denote by $I$ the full subcategory of topological spaces on the simplices. Then the inclusion $I \hookrightarrow Top_{\Delta}$ is dense (aka strongy generated) essentially by definition. This means that the canonical functor $$R:Top_{\Delta} \to Psh\left(I\right)$$ is fully faithful. I claim this functor has a left adjoint. It can be constructed very explicitly. Indeed, since $Top_{\Delta}$ is cocomplete, so we can produce a colimit-preserving functor $$L:Psh\left(I\right) \to Top_{\Delta}$$ as the left Kan extension of the inclusion $$I \hookrightarrow Top_{\Delta}$$ along the Yoneda embedding. It follows from the Yoneda lemma that $L$ is left adjoint to $R.$

Is the confusion in showing that $L$ is accessible?

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I'm assuming that $\Delta$-generated means that a space is a colimit of the canonical diagram of spaces in $I$ over $X,$ in analogue with what compactly generated means. – David Carchedi Nov 25 at 19:15
Isn't this the same as the argument in these notes? – Zhen Lin Nov 25 at 20:42
It looks like it. I mean, it is the natural thing to do, in my opinion. – David Carchedi Nov 25 at 21:32
Sure, but both suffer from the same problem – you still have to show that the reflector is accessible! – Zhen Lin Nov 25 at 22:00
Indeed, Dugger's notes only prove the theorem under the assumption of Vopenka's principle. I'd love to see if Tim's answer above avoids this need. – David White 2 days ago

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