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Sometimes I think that we have too much meta-discussion here. Nervetheless I feel the need to start the following one, apologies in advance. Let’s try to not turn it into an all too lengthy discussion.
Over in another thread here Andrew and myself both expressed dissatisfaction with the term smooth space as I myself have been using it in various entries. I think Andrew and me dislike the word for rather different reasons, but nevertheless it’s suboptimal. I was just reminded of that when editing variational calculus.
What would be a better term? I started to think I should instead say
consistently. How do you all feel about that?
Lawvere has sometimes used the phrase “smooth set”, which I personally think sounds nicer (“smooth 0-type” would have resonances for those immersed in homotopy type theory, but I’m not sure how many others). But if a plurality want to go with smooth 0-type, no complaints from me.
I like ’smooth set’. Even in hott, many of us usually say ’set’ instead of ’0-type’.
Does that mean we should also have ’topological sets’?
I like smooth set very much as it does remind one of shevaes of sets, as opposed to other, more special, categories of smooth spaces in the literature.
I still get often confused with the usage “Cartesian space” however, as it is never clear in $n$Lab community when it is means just the real $n$-dimensional space $\mathbb{R}^n$, and when one means the category of such equipped with a very specific Grothendieck topology (besides Urs is the only person from whom I heard word Cartesian space and for 2 years I was convinced that the term must include the Grothendieck topology to be called so as I learned the term in a text of him with such presentation; of course, I am very much used to a common phrase Cartesian coordinate system, and Cartesian coordinate). If one does not mean the specific Grothendieck topology, why not just saying $\mathbf{R}^n$, what even physicists understand.
I like ’smooth set’. Even in hott, many of us usually say ’set’ instead of ’0-type’.
But wouldn’t we need to say “smooth h-set” then?
It’s this “h-“prefixed terminology which I find a bit ugly and over which I prefer the nice “homotopy n-type” terminology.
Lawvere has sometimes used the phrase “smooth set”,
Do you have a reference for that? That would be nice to include in the entry.
(If I google for “Lawvere ’smooth set’” I get a single genuine hit… and that is to this very thread here.)
At IAS, I think we’ve all agreed that when working internally to HoTT, we can omit the “h-” since there is no other meaning of “set” in scope. The “h-” prefix is only necessary when speaking externally to HoTT, to distinguish them from (for example) the sets in a set theory within which we are constructing a model of HoTT.
All, right, good to know.
So I’d be happy with adopinting to say “smooth set”. Just one last thing:
“type” alludes to the fact that we do work in a topos. And so saying “smooth type” might have taken care of Andrew’s complaint, whereas “smooth set” probably does not: Andrew might complain that various subcategories of smooth sets also deserve to be called “smooth sets”. Not sure, maybe he won’t. :-)
Re #6: sadly, I don’t. I tried combing through my personal files yesterday (which are in some disarray) to find likely candidates among Lawvere’s articles, but didn’t find anything yet. Since there’s seemingly nothing online about this, I’d have to get to a proper library to really hunt it down (and of course my memory might have played a trick on me, as it does sometimes).
Todd, thank. Don’t worry, it’s not so important. I just thought if you have a reference sitting there on your desk, it would have been nice to link to it.
I know he writes “continuously varying set” for sheaves on a topological space. So possibly he’d write “smoothly varying set”. But I am not sure if I have seen it explicitly this way.
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