# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 25th 2012

I have touched H-space, slightly expanded here and there and slightly reorganized it.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 25th 2012

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 $Ho(Top)$?

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2012

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?

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeNov 26th 2012

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 26th 2012
• (edited Nov 26th 2012)

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).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 26th 2012
• (edited Nov 26th 2012)

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.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2012

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.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 26th 2012
• (edited Nov 26th 2012)

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.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeNov 27th 2012

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?

• CommentRowNumber10.
• CommentAuthorjim_stasheff
• CommentTimeNov 27th 2012
Seems to me we have an h-virus and an H-virus.
Last time I checked, there were 25 other lettes, cf. why they are called
the J-homomorphism and K-theory
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 27th 2012

25 other lettes

One day I’ll invent an Umlaut-theory: äu-cohomology for äußerst interessante Kohomologie.

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeDec 4th 2012

What about something like “$A_3$-category”? (Or maybe $A_4$? I’m not sure how the indexing is aligned for $A_n$.)

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeDec 4th 2012

Ah, that’s a good pont! Yes. I’ll try to get back to that later…

• CommentRowNumber14.
• CommentAuthorjim_stasheff
• CommentTimeDec 5th 2012
What about something like “A_3-category”? (Or maybe A_4? I’m not sure how the indexing is aligned for A_n.)

If the reference is to A_n-spaces, the n refers to the number of variables:
A_2 is an H-space, A_3 has just an associating homotopy...
• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeDec 5th 2012

Thanks! Then “$A_3$-category” would be what I meant.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 10th 2019

added brief pointer (here) to the Pontrjagin ring.

1. In general, internalization is different from enrichment, and the definitions above for i.e. H-monoid, H-group, H-ring made it fairly clear that the definition of H-category considered here are category objects internal to Ho(Top), rather than Ho(Top)-enriched categories.

Anonymous