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    $\begingroup$ @Peter: Please don't take this as asserting disagreement with what you wrote there. I'm genuinely curious: do you intend this as an answer to the question, or more like commentary on the question? If it is meant as an answer to the actual question (whatever it means exactly), then my guess is that your answer is 'no'. But then, with Tom Leinster above, I don't know what you'd be saying no to, exactly. $\endgroup$ Commented Dec 27, 2011 at 16:25
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    $\begingroup$ Except that he perhaps exaggerates the utility of $\infty$-categories, I agree with Dylan's answer above. His point (1) applies to certain contexts, and the all'' in his (2) is way too strong. But his point is right: that is not what $\infty$ categories (or categories for that matter) are there for. One can of course specialize either to more calculable contexts (as in Ronnie's answer) but that misses the thrust of the question, I think. It would be nice if there were more young mathematicians who felt at home with both explicit computations'' and abstract theory. $\endgroup$ Commented Dec 27, 2011 at 19:41
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    $\begingroup$ Professor May -- I realize you've said this multiple times now, but I'm unsure: could you give an example of a serious "explicit computation"? Most of what I've come across in algebraic topology has been either very classical computations (i.e. 1950s computation of cohomology rings) or abstract theoretical material (from then onwards) such as model categories or infinite loop spaces. $\endgroup$ Commented Dec 28, 2011 at 14:37
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    $\begingroup$ For explicit computations see for example Ellis, G.J., About HAP: Third Homotopy Groups Of Suspensions Of Classifying Spaces: hamilton.nuigalway.ie/Hap/www/SideLinks/About/… Here the homotopy 3-type of the suspension of a classifying space of a group $G$ is determined by the nonabelian tensor product $G \otimes G$ and a "commutator" morphism from that to $G$, whose kernel is the 3rd homotopy group; and one often needs a computer to do the computations. The theory does not use cohomology or model categories. $\endgroup$ Commented Jan 15, 2012 at 15:46