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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeNov 25th 2012
- PermaLink
Author: Urs
Format: MarkdownItexI have touched _[[H-space]]_, slightly expanded here and there and slightly reorganized it.

I have touched H-space, slightly expanded here and there and slightly reorganized it.
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeNov 25th 2012
- PermaLink
Author: Urs
Format: MarkdownItexA 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 $Ho(Top)$?

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>$ . 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>$ ?
- CommentRowNumber3.
- CommentAuthorMike Shulman
- CommentTimeNov 26th 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexHo(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?

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
- PermaLink
Author: DavidRoberts
Format: MarkdownItexOr 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.

Or perhaps a weak category object in the 2-category with underlying 1-category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>$ 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)
- PermaLink
Author: Urs
Format: MarkdownItexRight, 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).

Right, I didn’t say that well.

What I mean is the structure in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>$ given by a type $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo lspace="0.11111em">:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex">X_0 \colon Type</annotation></semantics></math>$ , a dependent type $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo lspace="0.11111em">:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>⊢</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo lspace="0.11111em">:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex">x_0,x_1 \colon X_0 \vdash X_1(x_0,x_1) \colon Type</annotation></semantics></math>$ , a function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo lspace="0.11111em">:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>⊢</mo><msub><mi>comp</mi> <mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mo lspace="0.11111em">:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_0,x_1,x_2 \colon X_0 \vdash comp_{x_0,x_1,x_2} \colon X_1(x_0,x_1) \times X_1(x_1,x_2) \to X_1(x_0,x_2)</annotation></semantics></math>$ such that there exists an equivalence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub><mo lspace="0.11111em">:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>⊢</mo><mo stretchy="false">(</mo><msub><mi>comp</mi> <mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub></mrow></msub><mo>∘</mo><msub><mi>comp</mi> <mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mo>≃</mo><msub><mi>comp</mi> <mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub></mrow></msub><mo>∘</mo><msub><mi>comp</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_0,x_1,x_2,x_3 \colon X_0 \vdash (comp_{x_0,x_2,x_3} \circ comp_{x_0,x_1,x_2} \simeq comp_{x_0,x_1, x_3} \circ comp_{x_1,x_2,x_3})</annotation></semantics></math>$ (and similarly for units).
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeNov 26th 2012
- (edited Nov 26th 2012)
- PermaLink
Author: Urs
Format: MarkdownItexOr 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.

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>$ -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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ -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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ -category” – in such a way that I can make “H-category” a pointer to an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ Lab entry whose existence and title receives a minimum of consensus.
- CommentRowNumber7.
- CommentAuthorMike Shulman
- CommentTimeNov 26th 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexWell, $A_\infty$ 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, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ 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)
- PermaLink
Author: Urs
Format: MarkdownItex> 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.

Well, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ category usually also refers to the enriched case.

You’d never say $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top)</annotation></semantics></math>$ -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
- PermaLink
Author: Mike Shulman
Format: MarkdownItexMy 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?

My point was that since $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>$ 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
- PermaLink
Author: jim_stasheff
Format: TextSeems 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
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
- PermaLink
Author: Urs
Format: MarkdownItex> 25 other lettes One day I'll invent an Umlaut-theory: _äu-cohomology_ for _äußerst interessante Kohomologie_.

25 other lettes

One day I’ll invent an Umlaut-theory: äu-cohomology for äußerst interessante Kohomologie.
- CommentRowNumber12.
- CommentAuthorMike Shulman
- CommentTimeDec 4th 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexWhat about something like "$A_3$-category"? (Or maybe $A_4$? I'm not sure how the indexing is aligned for $A_n$.)

What about something like “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">A_3</annotation></semantics></math>$ -category”? (Or maybe $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">A_4</annotation></semantics></math>$ ? I’m not sure how the indexing is aligned for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math>$ .)
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeDec 4th 2012
- PermaLink
Author: Urs
Format: MarkdownItexAh, that's a good pont! Yes. I'll try to get back to that later...

Ah, that’s a good pont! Yes. I’ll try to get back to that later…
- CommentRowNumber14.
- CommentAuthorjim_stasheff
- CommentTimeDec 5th 2012
- PermaLink
Author: jim_stasheff
Format: TextWhat 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...
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
- PermaLink
Author: Mike Shulman
Format: MarkdownItexThanks! Then "$A_3$-category" would be what I meant.

Thanks! Then “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">A_3</annotation></semantics></math>$ -category” would be what I meant.
- CommentRowNumber16.
- CommentAuthorUrs
- CommentTimeOct 10th 2019
- PermaLink
Author: Urs
Format: MarkdownItexadded brief pointer ([here](https://ncatlab.org/nlab/show/H-space#RingStructureOnHomology)) to the Pontrjagin ring. <a href="https://ncatlab.org/nlab/revision/diff/H-space/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/35">v35</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>

added brief pointer (here) to the Pontrjagin ring.

diff, v35, current
- CommentRowNumber17.
- CommentAuthornLab edit announcer
- CommentTimeJun 7th 2022
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexIn 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 <a href="https://ncatlab.org/nlab/revision/diff/H-space/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/46">v46</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>

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

diff, v46, current
- CommentRowNumber18.
- CommentAuthornLab edit announcer
- CommentTimeNov 2nd 2022
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexRemoved the link to the Arkowitz paper, since it was local filesystem link rather than a web url. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/H-space/48">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/48">v48</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>

Removed the link to the Arkowitz paper, since it was local filesystem link rather than a web url.

Anonymous

diff, v48, current
- CommentRowNumber19.
- CommentAuthorUrs
- CommentTimeNov 2nd 2022
- PermaLink
Author: Urs
Format: MarkdownItexThanks for the alert. But then let's just add a working link: * [[Martin Arkowitz]], *H-Spaces and Co-H-Spaces*, chapter 2 in: *Introduction to homotopy theory*, Springer (2011) 37-70 [[doi:10.1007/978-1-4419-7329-0_2](https://doi.org/10.1007/978-1-4419-7329-0_2)] <a href="https://ncatlab.org/nlab/revision/diff/H-space/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/49">v49</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>
Thanks for the alert. But then let’s just add a working link:
- Martin Arkowitz, H-Spaces and Co-H-Spaces, chapter 2 in: Introduction to homotopy theory, Springer (2011) 37-70 [doi:10.1007/978-1-4419-7329-0_2]
diff, v49, current
- CommentRowNumber20.
- CommentAuthorDavid_Corfield
- CommentTimeJan 9th 2023
- PermaLink
Author: David_Corfield
Format: MarkdownItexAdded the recent * [[Ulrik Buchholtz]], [[J. Daniel Christensen]], Jarl G. Taxerås Flaten, [[Egbert Rijke]], _Central H-spaces and banded types_ ([arXiv:2301.02636](https://arxiv.org/abs/2301.02636)) Will add to these authors' pages. <a href="https://ncatlab.org/nlab/revision/diff/H-space/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/50">v50</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>
Added the recent
- Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types (arXiv:2301.02636)
Will add to these authors’ pages.

diff, v50, current
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeJul 4th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[William Browder]], *Torsion in H-Spaces*, Annals of Mathematics, Second Series **74** 1 (1961) 24-51 [[jstor:1970305](https://www.jstor.org/stable/1970305)] <a href="https://ncatlab.org/nlab/revision/diff/H-space/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/52">v52</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>
added pointer to:
- William Browder, Torsion in H-Spaces, Annals of Mathematics, Second Series 74 1 (1961) 24-51 [jstor:1970305]
diff, v52, current
- CommentRowNumber22.
- CommentAuthorUrs
- CommentTimeJul 4th 2023
- PermaLink
Author: Urs
Format: MarkdownItexand pointer to: * [[Allen Hatcher]], §3.C in: *Algebraic Topology*, Cambridge University Press (2002) $[$[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)$]$ <a href="https://ncatlab.org/nlab/revision/diff/H-space/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/52">v52</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>
and pointer to:
- Allen Hatcher, §3.C in: Algebraic Topology, Cambridge University Press (2002) $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>$ ISBN:9780521795401, webpage $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>$
diff, v52, current
- CommentRowNumber23.
- CommentAuthorperezl.alonso
- CommentTimeDec 21st 2023
- PermaLink
Author: perezl.alonso
Format: MarkdownItexrational homotopy type of H-spaces <a href="https://ncatlab.org/nlab/revision/diff/H-space/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/53">v53</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>

rational homotopy type of H-spaces

diff, v53, current
- CommentRowNumber24.
- CommentAuthorUrs
- CommentTimeDec 21st 2023
- PermaLink
Author: Urs
Format: MarkdownItexThanks. This used to be mentioned in passing at *[[Sullivan model of loop space]]* but without a reference. I have expanded a little ([here](https://ncatlab.org/nlab/show/H-space#RationalHomotopy)) and cross-linked the two paragraphs. <a href="https://ncatlab.org/nlab/revision/diff/H-space/54">diff</a>, <a href="https://ncatlab.org/nlab/revision/H-space/54">v54</a>, <a href="https://ncatlab.org/nlab/show/H-space">current</a>

Thanks. This used to be mentioned in passing at Sullivan model of loop space but without a reference.

I have expanded a little (here) and cross-linked the two paragraphs.

diff, v54, current

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