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I am in the middle of editing some work for a monograph and have to answer some points raised by referees. They are extolling homotopical algebra as a paradigm and I have been stressing more algebraic homotopy. This raises in my mind the question to what if any is the difference between them. My feeling is homotopical algebra is top-down, global, about the whole category being studied, whilst algebraic homotopy is sort of more bottom-up, and about algebraic models for the objects. This is more or less what our current entry on homotopical algebra implies, but I really would appreciate the opinion of others.
I agree. Homotopical algebra is essentially a nonabelian version of homological algebra (I believe Quillen and Grothendieck thought of it like this). Certainly Quillen’s and others’ model categories (category of fibrant objects, Waldhausen categories etc) are about how the whole category of objects works, and how this models HOT or variants. Algebraic homotopy goes back to Whitehead, as you know, and is about models for homotopy types. To draw a revisionist distinction, homotopical algebra is about large (oo,1)-categories, and algebraic homotopy is about small (n,0)-categories (usually for small n, but not always)
PS Is this monograph the one on profinite homotopy?
Is this monograph the one on profinite homotopy?
Yes.
I noted a nice quote form Baues: ’The fascinating task of homotopy theory is… the investigation of ’algebraic principles’ hidden in homotopy types. We may be confident that such principles are of importance in mathematics far beyond the scope of topology… ’. That is from his ’homotopy types’ article in the Handbook of Algebraic Topology. That article is very nice for motivating the problem and suggesting the approach. (Thoroughly recommended, at least the first few sections).
This relates in fact to several of the discussion on the Lab and the forum. We have some idea on how to define Whitehead products etc. in Algebraic Homotopy situations, we have lots of ’coherence strictness obstructions’ in infty-cats and clearly they are closely related, but it is not always clear what the relations are. There is a paper: ‘Higher Homotopy Operations and Cohomology’ by David Blanc, Mark Johnson and James Turner, (Journal of K-theory: Volume 5, Issue 01 , pp 167 -200 ) which is very interesting an uses lots of nice constructions to getting some links between homotopy operations and things well known to the denizens of the nLab basement!
@David Why do you say ’revisionist’?
Another point of contact between this and other more or less recent discussions in the forum is on the invariants of directed homotopy types. The fundamental category etc. are more directed algebraic homotopy than directed homotopical algebra. In all this it seems to me that we need both attitudes / approaches. The paper I refereed to previously attempts in part to bridge the gap (which is not really there, and is an illusion) in the non-directed context. Perhaps the homotopy hypothesis is also really about saying that there is no gap!
I say revisionist only because Whitehead and Quillen (and their contemporaries at the time) would not have used those words!
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