Timeline for What non-categorical applications are there of homotopical algebra?
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| Jun 8, 2014 at 1:20 | comment | added | HJRW | '[W]here do tools from category theory help to study the category of topological spaces or of commutative algebras?' Well, the whole point of this thread was to give examples where the homotopy-theoretic viewpoint is useful! And some of the answers, including your own, are very interesting. On the other hand, as has been pointed out elsewhere, some of the rhetoric about homotopy theory is rather grandiose; I'm afraid I think your point 3 is an example of this. | |
| Jun 7, 2014 at 20:36 | comment | added | Lennart Meier | ...Homotopy theory also suggests to look at mapping simplicial sets between groups (which might or not be useful - I am no expert). But do not hope for great revelations. In this generality, homotopy theory is rather formal. | |
| Jun 7, 2014 at 20:34 | comment | added | Lennart Meier | No, not in this form. But where do tools from category theory help to study the category of topological spaces or of commutative algebras? It is more that category theory provides common proofs of some (simple) lemmas and works as an organizing principle, which might suggest useful viewpoints. Likewise, as soon as one postulates that finite index subgroups are more or less the same as the whole group, homotopy theory suggests to look at commensurators (as isomorphisms in the homotopy category)... | |
| Jun 7, 2014 at 18:35 | comment | added | HJRW | "As soon as we say that two things are more or less the same, but not really isomorphic, we are doing secretly a kind of homotopy theory." This sounds like a sort of homotopy-theory imperialism, to me. Could you give an actual example where the tools of homotopy theory are useful in studying commensurability classes of groups, say? | |
| S Jun 7, 2014 at 8:21 | history | answered | Lennart Meier | CC BY-SA 3.0 | |
| S Jun 7, 2014 at 8:21 | history | made wiki | Post Made Community Wiki by Lennart Meier |