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The category $\mathbf{Top}$ of topological spaces and continuous maps is infamously not cartesian closed. However, it is weakly cartesian closed (weakly = no uniqueness condition in universal property of exponential). Moreover, the category of spaces and homotopy classes of maps $\mathbf{hTop}$ is cartesian closed.

My question is: is $\mathbf{hTop}$ locally cartesian closed?

ssp8
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  • Without knowing the definitions, I can't be sure, but often something being "locally _____" follows from it being _____ globally. Why do you doubt this here? – Mike Pierce Mar 10 '21 at 17:25
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    @MikePierce Locally cartesian closed means all slices are cartesian closed. A category can be cartesian closed but not locally cartesian closed (eg. $\mathbf{Cat}$). – ssp8 Mar 10 '21 at 17:28
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    I think I have just realised the answer: $\mathbf{hTop}$ is not locally cartesian closed because the slices don't even have products, for products in slices are pullbacks in the base category and $\mathbf{hTop}$ does not have pullbacks (see https://math.stackexchange.com/questions/4018667/pushouts-in-category-of-topological-spaces-modulo-homotopy?rq=1). – ssp8 Mar 10 '21 at 17:44
  • And there I was worrying about exponentials! – ssp8 Mar 10 '21 at 17:52
  • Could you give a definition of weakly cartesian closed? – Tyrone Mar 10 '21 at 20:23
  • Maybe delete the question or change it then? – Henno Brandsma Mar 10 '21 at 22:28
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    I think you're really going to have to go all the way up to the $\infty$-category of spaces being locally cartesian closed. And please note, especially since I leave this implicit in my old answer at your link, that it's the category of topological spaces with weak homotopy equivalences inverted that's cartesian closed, not the category of topological spaces and homotopy classes of maps. This hTop is also the category of CW complexes and homotopy classes of maps. – Kevin Carlson Mar 11 '21 at 15:49
  • @Tyrone Exactly the same as cartesian closed (which you can easily find) but omit "unique" in the universal property of exponentials. – ssp8 Mar 12 '21 at 09:56
  • @HennoBrandsma Happy to delete if that is the consensus, but at the moment I think there is still some discussion/clarification to be had. What would you suggest as a change? – ssp8 Mar 12 '21 at 09:58
  • Well you know now it’s not locally Cartesian closed. So what’s left of the question? Any outstanding issues? Put that in the question and change the title. – Henno Brandsma Mar 12 '21 at 10:00
  • @KevinArlin Right, and presumably you means locally cartesian closed in the $\infty$ sense? Ah yes, thanks for pointing that out -- I had glossed over it. So is it not true that you get cartesian closure if you do not restrict to CW complexes (with homotopy classes of maps)? – ssp8 Mar 12 '21 at 10:01
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    @ssp8 Right on both points. – Kevin Carlson Mar 12 '21 at 12:28
  • @KevinArlin Just to clarify, do you know/expect the $\infty$-category of spaces to be locally cartesian closed, or are you just saying that to have any chance you must go to $\infty$? I guess the issue with pullbacks resolves in the $\infty$ case, but what about right adjoints to these? It's not clear to me, but then again I'm not too savvy with $\infty$-stuff. – ssp8 Mar 13 '21 at 10:24
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    It’s true. The $\infty$-category of spaces is even the archetypal $\infty$-topos. But this is not at all something verifies in a few minutes. It takes four or five chapters of Cisinski’s book, which I think is the quickest treatment available. – Kevin Carlson Mar 13 '21 at 13:28
  • @KevinArlin btw, it's just the uniqueness part of the universal property of exponentials that holds up to homotopy, right? The relevant diagram commutes on the nose, as $\mathbf{Top}$ is weakly cartesian closed? – ssp8 Apr 03 '21 at 11:07

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