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As I'm constantly running across higher categories these days, I'm wondering what is a good starting point to get into the theory?

While I am aware of nLab and the n-Category Café, I am having a real hard time getting into the material provided. One reason for this is: afaik there is no generally favored way of defining higher categories, and different approaches lead to different properties / results / theories. This naturally leads to lots and lots of texts concerned with fundamental questions, such as comparison of these different approaches, which I am not interested in too much.

Lets be a little more precise with what I'm actually looking for: A straight forward introduction to one theory of weak higher categories, with applications in algebraic topology.

I hope someone here has already went through this search and is willing to post a guideline or at least some recommendations.

Update:

After a complete day of searching the web, I have finally found something very promising although it seems to only deal with strict $\omega$-groupoids: The still to be published book Nonabelian Algebraic Topology by Ronald Brown, Philip J. Higgins and Rafael Sivera. A draft is downloadable on this website too.

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Have you seen Lurie's HTT? I sat through the first few days of a seminar he was giving (after which it was over my head) where he described the notion of an $(\infty, 1)$-category (as a simplicial set satisfying a weak version of the Kan extension condition). –  Akhil Mathew Mar 24 '11 at 19:04
    
I will try to have look at it, as soon as I can get a copy, thanks. A lot of thinking about this topic suggests, that cubical theories seem to be the way to go if one wants to appropriately model homotopy. Is this common belief or just me? –  Alexander Thumm Mar 24 '11 at 20:28
    
Dear Alexander, Do you mean simplicial? You might see mathoverflow.net/questions/58497/… for some explanations. Note that HTT is available on Lurie's webpage. –  Akhil Mathew Mar 24 '11 at 23:27
    
Ok, thank you. No I did really mean cubical. Several constructions you would like to have will be significantly easier, or at least less technical. Note for example that $\Delta\times I$ is not a simplex. –  Alexander Thumm Mar 25 '11 at 8:15

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up vote 1 down vote accepted

I learned from Baez's article http://arxiv.org/abs/q-alg/9705009v1, which is also loaded with references.

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