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I have been editing the entry homotopy equivalence to include a brief discussion of strong homotopy equivalences and Vogt's lemma. In so doing, I have followed my nose and found various other entries to edit. For instance that for Hans Baues, that for cylinder functor, etc. I am thinking that the general area of Henry Whitehead's idea of algebraic homotopy, may be a useful intermediate one between the infinity category ideas (which could be seen as just a 'souped up' version of Kan complexes), (I am not saying they are just that a cynic might make them out to be!) and the algebraic topologists desire to perform calculations. Note the quotes at algebraic homotopy. Of course, they d not say what 'compute' means in this context. (Note we do not have an entry on Whitehead as yet.)
I have created an entry for J.H.C.Whitehead. Please check to see if I have done him justice!
I have asked a technical question at homotopy equivalence, that may be of interest. I do not have a complete answer although I have some ideas.
removed query box
+–{.query} Tim: I do not know if there is a neat formulation of the full homotopy coherent version of this, nor exactly in what settings the analogous abstract versions of Vogt’s lemma go across
Mike Shulman: I think there’s a “full coherentification” version for quasicategories in HTT somewhere. =–
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