Fixed the typo with S^0.

]]>Sorry for my english (I’m french),

adufour. ]]>

Corrected the exactness properties. In particular, the join only preserves colimits when considered as an operation on augmented simplicial sets. The statement for simplicial sets is more complicated.

]]>I find this a confusing point, so I added a small warning about it to the relevant pages (simplex category, ordinal sum, join of simplicial sets) in case other beginners might feel the same.

]]>Right, thanks!

]]>The isomorphisms in (1) are either not natural or not order-preserving. (To get a natural bijection, swap the two halves of $[i] \oplus [j]$; to get an order isomorphism, do nothing.)

]]>This is embarrassing, but I must be missing something trivial. Which of the following statements is wrong?

- There are isomorphisms $B_{i,j}: [i] \oplus [j] \simeq [i+j+1] \simeq [j] \oplus [i]$ for $i,j \in \mathbf{\Delta}_+$.
- The isomorphisms $B_{i,j}$ are compatible with the associator morphisms, so that this monoidal structure is braided.
- The braiding isomorphisms satisfy $B_{i,j} \circ B_{j,i} = \mathrm{id}$, so that this monoidal structure is symmetric.
- The Day convolution takes symmetric monoidal structures to symmetric monoidal structures.

I do not have an electronic copy. The manuscript was produced at Hagen, not at Bangor. In any case it is an idea to check that the things you want to know and learn about are in it!!!! This is not obvious to me. Some of the things you want are there almost certainly, but there is not much about the move to higher groupoids etc. Have you glanced at the other notes that are available (S-cat notes etc.), on my personal n-lab page. Some of that material is a summary of what went into Kamps - Porter, so is a good place to look.

]]>Oh, sure! I knew about the google books version. I wonder if there is any way to get the publisher to sell an electronic version of it (or if you and Kamps could do it).

]]>No but a preview version is:

http://books.google.co.uk/books?id=7JYKxInRMdAC&printsec=frontcover&dq=Kamps+Porter&source=bl&ots=uuCkXpHhB5&sig=-S509JARITeXpXp9EtgXAGR3YDk&hl=en&ei=e2u0S8mtNNS04gaMtOjaDg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CBIQ6AEwBA#v=onepage&q=&f=false

UGH!

Find it by searching for Abstract homotopy and simple homotopy theory

]]>Wait, Kamps-Porter is available in an electronic version?

@David: I'm not confused about the higher homotopy groups. I mean that I've never studied/used them.

]]>Far be it for me to discourage buying that book, but I suspect the Menagerie (in one of its shorter versions, 10 chapters) would be sufficient for some of what Harry wants. Kamps - Porter is available in part on the web. Have a look at contents etc always before buying! We wrote that book as we felt there was a gap in the literature and the actual constuctible homotopies with cylinders that were functorial were all to often discarded as being non-fashionable compared with model categories.

]]>Presumably your book with Kamps. As I'm not sure what sort of homotopy theory this person Harry was talking to had in mind, I'm just suggesting the quick references that come to mind. I'm a bit nonplussed by Harry intently wanting to know about higher homotopy groups and seeming lost; as a formal object there's not much to them. Personally I think it's the operations that are more complicated and interesting, and the algebraic structres (crossed complexes, operations etc) that they form.

]]>Which book by me? Ronnie's book is available from his website in draft form.

I keep on trying to get people to make the link between the various n-groupoids (non-strict) and the homotopy operations (Whitehead products etc.) They must correspond to various bits of non-strictness (e.g. some indications and intuitions of interchange / Gray tensor product related to lowest order Whitehead product, but I would like to see formulae (that were moderately simple).

David Blanc has lots of work on Pi algebras and it is tantalising both for what it says and does not say! He has a very recent preprint with Simona Paoli that is very interesting. It is available from her web page http://math.aa.psu.edu/~simona/index_file/Page290.htm (last in the list).

]]>I love algebra though!

]]>The action of on the collection of s at various basepoints is good to know. Then there are various operations among the homotopy groups (giving rise to what is called a -algebra, see Blanc-Dwyer-Goerss in Topology in 2004). Other than that, they not too tricky. Going from groups to n-groupoids is of course the hard bit, unless you warm up via strict n-groupoids, say using crossed complexes. Then Ronnie Brown's book with Higgins and Severa on Nonabelian Algebraic Topology will come in handy. There's a lot of algebra there, but the main points should be extractable.

]]>No no I'm quite familiar with the fundamental groupoid. I meant the higher homotopy groups.

The fundamental groupoid is pretty easy to understand.

I'll buy that book this week after I finish these exercises from milnor.

]]>In that case, buy Ronnie Brown's book: it costs 5 pounds (USD 7.60) in electronic format - I have it and it is a very useful searchable resource. It only covers 1-dimensional homotopy groupoids, but it's a good intro to topology and the techniques of fundamental groupoids.

]]>I know nothing about based homotopy either! I mean, do you have any book suggestions for me to quickly get to all of this stuff with higher homotopy groupoids? I've heard that Tim Porter's book is good, but I'm broke at the moment, and I'm going to buy it over the summer. So until then, do you have any suggestions? I've looked at the beginning of Switzer, but everything he does is totally focused on based homotopy theory.

]]>Oh, okay -- I guess you mean wrapping your head around , , and so on -- and understanding what operations are involved. It's certainly true that the problem of understanding that is what drove me to my definition, and it's a very good exercise. You might try banging your head for a while against that, and seeing what you come up with. Then you can check against what other people (myself included) came up with -- that might pack a greater punch than just reading about it straight away.

It's also an interesting problem to try to whittle these big structures down to size by considering what their skeletons look like, more particularly how such skeletons are assembled from copies of , , etc. and operations between them. This leads quickly to things like k-invariants.

]]>No, I was talking to a graduate student at UIUC on IRC who suggested that to study homotopy groupoids rather than groups, I should learn at least one formalism for weak n-categories. He suggested that quasicategories are not optimal for this, because they're very hard to compute with.

]]>The 2004 notes by Eugenia Cheng and Aaron Lauda are very good (as are Tom's, but my memory is that Cheng-Lauda proceed at a more leisurely pace). Yes, you will need to know about operads, but you very well might already, and for which there is a ton of accounts -- my own first exposure to them was through reading Stasheff's seminal work on higher associativities and also J.P. May's Geometry of Iterated Loop Spaces, both of which I warmly recommend. What I wrote up on Trimble n-cats in the nLab is a slightly different approach, and the one I used in my original talk on the subject.

But I'm curious about that suggestion that you read up on this to "do homotopy theory", which sounds odd to my ears. Did someone write that down somewhere, so that can see the context where it was made?

]]>@Todd: It was suggested that I learn about the Trimble n-cat to do homotopy theory. Do you have a suggestion for reading about it, given that you're *the man himself*?