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I have touched H-space, slightly expanded here and there and slightly reorganized it.
A question on terminology:
in the spirit of the terms “H-monoid” and “H-group” it would be natural to say “H-category” for a category object internal to $Ho(Top)$. Are there any texts that say so, explicitly? Or else, is there any term which is semi-standard at least in some circles for categories internal to $Ho(Top)$?
Ho(Top) doesn’t have a lot of pullbacks, so I’m not exactly sure what you would want to mean by a category internal to it. Do you just mean a category-like thing internal to Top where the associativity and unitality only hold up to incoherent homotopy? I don’t think I remember ever seeing such a thing; do you have some example in mind?
Or perhaps a weak category object in the 2-category with underlying 1-category $Top$ and with equivalence classes of homotopies for 2-arrows? This probably only makes sense for cofibrant topological spaces, though.
Right, I didn’t say that well.
What I mean is the structure in $\infty Grpd$ given by a type $X_0 \colon Type$, a dependent type $x_0,x_1 \colon X_0 \vdash X_1(x_0,x_1) \colon Type$, a function $x_0,x_1,x_2 \colon X_0 \vdash comp_{x_0,x_1,x_2} \colon X_1(x_0,x_1) \times X_1(x_1,x_2) \to X_1(x_0,x_2)$ such that there exists an equivalence $x_0,x_1,x_2,x_3 \colon X_0 \vdash (comp_{x_0,x_2,x_3} \circ comp_{x_0,x_1,x_2} \simeq comp_{x_0,x_1, x_3} \circ comp_{x_1,x_2,x_3})$ (and similarly for units).
Or maybe that would rather deserve to be called an “H-Segal space” or the like. The term “H-category” would probably best fit for $Ho(Top)$-enriched categories. Any opinions?
I am asking because I am having discusison with somebody on how to say category internal to HoTT and I would like to make the point that the coherence is important. I kept saying: “notice that for the special 1-object case there is the crucual distinction between an H-monoid and an $A_\infty$-space” but I would like to be able to drop the “for the special case” and just say “there is the crucial distinction between an H-category and an $A_\infty$-category” – in such a way that I can make “H-category” a pointer to an $n$Lab entry whose existence and title receives a minimum of consensus.
Well, $A_\infty$ category usually also refers to the enriched case. I’m dubious of distinguishing any of these notions with a name like “H-category”, unless someone actually has a use for one of them. (Also, it’s a little unfortunate that in HoTT we sometimes prefix things with an “h-” without meaning homotopy incoherence, e.g. h-set and h-prop and h-level.)
Well, $A_\infty$ category usually also refers to the enriched case.
You’d never say $A_\infty$ for $Ho(Top)$-enrichment!
unless someone actually has a use for one of them.
It appears all over the place, for instance the H-category underlying a Segal space is one way to define what an equivalence in a Segal space is.
My point was that since $A_\infty$ refers to something enriched in spaces, if you wanted to say ’there is the crucial distinction…’ comparing them to H-categories, then H-categories would have to be enriched in, rather than ’internal’ to, Ho(Top).
What about: an H-category is a monad in the homotopy bicategory of spans?
25 other lettes
One day I’ll invent an Umlaut-theory: äu-cohomology for äußerst interessante Kohomologie.
What about something like “$A_3$-category”? (Or maybe $A_4$? I’m not sure how the indexing is aligned for $A_n$.)
Ah, that’s a good pont! Yes. I’ll try to get back to that later…
Thanks! Then “$A_3$-category” would be what I meant.
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