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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have added a comment to your query .. BTW do sign in as Harry Gindi or whatever.
The point about the product is that in pointed spaces you need to squash the basepoint times I to a point to get things to work neatly. Alternatively, work within pairs of spaces (X,A), and use relative homotopy finishing off by restricting to A being a point.
Homotopies of based spaces use smash product . Which is equivalent to saying, the homotopy also has to keep the basepoint fixed.
I think it would probably be most helpful, pedagogically speaking, to begin with the even easier and even more classical example of unbased spaces. (-:
If we take the smash product, how do we get the two maps => to take the coequalizer? I only said what I did about bipointed spaces because this will induce two morphisms => .
How are we getting two canonical injections in the unbased version?
I only ask this because with simplicial sets, the interval has two vertices, so it naturally has two canonical injections into the product.
Also, LaTeX on the n-Forum? How?
Edit: I only ask this here because I don't have Tim's book, we're not learning it in my current AT class (which has thusfar avoided any categorical language), it's not in May's book, and Switzer's book uses it without exhibiting it as an end or coend or weighted colimit related by a dinatural transformation from the Hom functor..
Ah, thanks. Got it now.
In fact, the approach that Heiner Kamps used (based on the cubical idea of Kan before he simplicialised) is based on a cylinder functor. for which see
http://ncatlab.org/nlab/show/cylinder+functor
Our book is highly recommended!!!! some parts can be viewed online (including the pretty front cover!!!) try on google with Kamps Porter. You can decide that way if you want to spend your limited resources on it.
As the coequaliser construction is just quotienting by an equivalence relation, mentioning colimits seems a bit of overkill. Of course, you can use a cocylinder as an alternative.
I forgot to say that the lab entry on cylinder functor has a link to Kamps Porter.
I said I'm going to order your book when I have the money! You've already sold me on it! Enough with the shameless plugs ;).
I like being shameless!
I will put some surveys of stuff on the Lab, but that takes time.
Of course, what really ought to happen is that Harry buys Tim's book and works through it, putting stuff on the nLab as he does so. As he does so, we (and especially Tim) yell encouragement from the sidelines.
Once the summer starts, I could see putting lots of stuff on the nLab.
Thanks Andrew!
I have updated cylinder functor to have a link to a survey article of mine (first as lecture notes 1991, then survey paper 2003, still needing updates of references.) This may be useful. (If Harry wants to adapt some of the stuff for the lab, ..., come the summer!)
To save a it of time, here is the link directly: Abstract-Homotopy.pdf
@Tim: The link is redirecting to an empty page.
OOPS! I will try to fix it!
I will try this:
I am doing something wrong (too naive!!!!). It still OOPS!
Try
http://ncatlab.org/timporter/show/HomePage
That is correct but is not `active'. Perhaps someone can tell me what I should do, as I have clearly missed some point somewhere. It looks as if I had to use a different syntax to link to my personal homepage on the n-Lab.
I think you have to do something like timporter:blah or timporter/blah, if I remember correctly, but Toby or Andrew can probably tell you exactly.
The main point should be to glance at the paper!!!! Some of it was adapted for the section in Kamps-Porter.
I rewrote the Idea-section at homotopy category, trying to make it less verbose, more transparently to the point.
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