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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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Do you mean "products of copies of the unit interval"? – Charles Rezk Nov 12 '09 at 19:20
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
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

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