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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 14th 2009
• (edited Sep 26th 2012)

In the process of expaning on n-truncated object in an (infinity,1)-topos I added some remarks along these lines to the beginning of homotopy n-type, thereby rewriting the first few sentences.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 26th 2012

am hereby moving the following old query box discussion from homotopy n-type to here:

[ begin forwarded discussion ]

+-- {: .query} Tim: When teaching homotopy theory I found blank looks from students if I used this idea as motivation as they felt it was too vague. I also do not like the idea of an $n$-type being a space as it does not allow one to say that two spaces 'have the same $n$-type.'

Toby: If you can explain the motivation that works here, then please do! But this doesn\'t say that a type is a kind of space, it says that it\'s a space 'up to ...', so two spaces have the same type if they\'re the same 'up to ...'. (In other words, we have a surjection Spaces → Types rather than an injection Types → Spaces, at least for purposes of motivation since of course both do exist).

Tim:As you sort of suggest, the problem is in the exact choice of words! At present you actually do say it is a space, at least as I read it, and that is my problem with the wording. Liberal use of inverted commas is not a good way around the difficulty. I will bounce a form of words past you to see how you like it first.

'In talking of a homotopy n-type, we are thinking of a space, or spaces, where the properties being considered are given by the homotopy groups $\pi_1$ up to $\pi_n$, so information recorded by the even higher homotopy groups is ignored.'

How about something along those lines?

Another question is how long should this entry on homotopy n-type be? I put in something on simplicial groups as an illustration, which I think you removed, but that section now reads strangely as after the first line (which mentions them) there does not seem to be any mention after that! As you and I seem, de facto, to be the main contributors on this entry (not exclusively) perhaps some discussion of the overall structure might be an idea.

Mike: The really precise way to say it is, of course, that a homotopy $n$-type is an object of the homotopy category (or $\infty$-category) of spaces where we invert the $n$-equivalences. This is analogous to defining a real number to be an equivalence class of Cauchy sequences of rationals; two spaces 'have the same $n$-type' in the same sense that two Cauchy sequences of rationals 'define (or converge to) the same real number'. Perhaps it would be useful to say this?

Toby: My only use of quotation marks was to provide direct quotations; of course we don\'t need scare quotes in the text. You are all saying fine things, so I will be bold and incorporate them now. (In the end, I tightened up the prose a bit, but change it if you don\'t like it.)

I didn\'t mean to remove anything about simplicial groups, just to generalise. But I specified how to interpret it for simplicial groups in the algebraic models below. But maybe none of that really depends on using simplicial groups? What you had before didn\'t read to me as an example so much as a prerequisite for the remainder to be correct.

Mike: I moved the discussion on nice spaces to nice topological space, and merged the two sections "Idea" and "Motivation" since I was confused about the distinction between them. =--

[ end forwarded discussion ]

• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeSep 26th 2012

Some comments that are not on your revision, Urs, but on the wording etc. of the entry as it stands:

(i) I am always a bit worried by an entry which has the phrase ’ the most important invariants’. That always seems to me to be dangerous. They are certainly very important, but definitive value judgements of this sort do worry me.

(ii) I mention also the concern that the ideas section has two related notions of homotopy n-type in it, although it is handled in the discussion further down the page.

I will adapt the entry very slightly hopefully to get around these worries.

• CommentRowNumber4.
• CommentAuthorronniegpd
• CommentTimeJun 23rd 2015
I have recently made somewhat more precise my general ideas on

"A philosophy of modelling and computing homotopy types"

and given a presentation at CT2015 on June 17 with that title. This is available on my preprint page. The following abstract shows the key ideas.

Abstract: This philosophy involves homotopically defined functors H from (Topological Data) to (Algebraic Data), and conversely "classifying space" functors B from (Algebraic Data) to (Topological Data). These should satisfy:

1. H is homotopically defined.
2. HB is naturally equivalent to 1.
3. The Topological Data has a notion of connected.
4. For all Algebraic Data A, we have BA is connected.
5. H preserves certain colimits of connected Topological Data.

The algebraic data splits into several equivalent kinds, ranging from "broad" to "narrow", related by Dold-Kan type equivalences. The broad data is used for conjecturing and proving theorems; the narrow data is used for calculations and relating to classical methods.

As examples of Algebraic Data we give groupoids, crossed modules and crossed squares. We give a sample computation, using crossed squares, of the homotopy 3-type of the mapping cone of the classifying space of a morphism of crossed modules.
--------------------------------------------------------------------------
The examples I gave nicely fitted with a previous lecture by George Janelidze which included the concept of "change of base".

This exposition is quite at variance with the current account of "higher homotopy van Kampen theorem" on the n-lab!

The Higher Homotopy Seifert-van Kampen Theorems with which I have been involved all involve "Structured Topological Spaces", in particular filtered spaces, n-cubes of spaces.

The supposed relation of these theorems with Lrie's work is discussed on
http://mathoverflow.net/questions/102295/generalisations-of-the-seifert-van-kampen-theorem

It appears to me that Lurie's results have nothing to do with the 2-dim and higher theorems applied in my CT2015 presentation.

Note also that the current n-lab entry on "higher versions of the van Kampen theorem" does not even mention the fundamental groupoid on a set of base points, which I published in 1967, A space with a set of base points is the initial form of "Topological Data".
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