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9

Everyone has said very interesting and useful things, but the closest any of them have come to answering the question I wanted to ask (or at least being the sort of answer I was hoping for) is (my interpretation of) Lennart Meier's comment. So here I will expand that and post it as a cw answer. In 1-category theory, it's become common in certain circles to ...


3

There are many good answers here... I don't think Mike is asking about the localization of the category of spaces, but of localizing spaces as special spaces. (Though I could be wrong, and about the former I could say: it's plainly a cousin of Moore-Postnikov resolutions, once removed in categorification) And I think the best answer to why this is a ...


6

I'm not sure exactly what you are after, but here is an elementary discussion, surely well known to you. Rephrasing what Mark Hovey said, the first question you should ask yourself is probably why localize at all? There may be several reasons. Here are some: You have some a priori interest in only studying spaces up to a certain equivalence relation ...


9

Maybe this is too elementary an answer, but in my view, localization is a general phenomenon that you might want to do in any category C. You have a class of maps W and you want to form W^{-1}C, the category obtained by formally inverting the maps in W. This is what we do to form the homotopy category of a model category, this is what we do to localize ...


10

In particular, are there analogous constructions in ("cowardly old") algebra where we get something useful by formally inverting the maps inverted by some other functor? Sure. Localize the category of presheaves on a space with respect to stalkwise isomorphisms. The result is the category of sheaves, which is more interesting than just looking at ...


4

$\textbf{This is a point of view which means that it is only one side of the story}$. Let me start with Gel'fand theorem. The (opposite) category of compact Hausdorff spaces is equivalent to the category of commutative $\mathbb{C}-^{\star}$algebras. Roughly speaking the functor with associate to any space $X$ its algebra of continuous complex vaulted ...


0

THIS IS NOT AN ANSWER, rather an additional question I always wanted to know how the following purely abstract-nonsensical (category-theoretic) constructions fit into the particular setup of stable homotopy. Any object $E$ of any closed monoidal category $(\mathscr S,S,\bigwedge,[\_,\_])$ determines an adjoint pair of functors $E\bigwedge\_\dashv[E,\_]$ ...


15

A student of mine asking for a motivation unmotivated by applications? Haven't I taught you anything, Mike? (Joking of course.) However, perhaps one way to avoid talking about future applications is to reflect on implicit past applications and the explicitly stated original motivations. This is not to take away from your answer, Craig, you know I agree ...


12

This is perhaps orthogonal to your desire for an a priori reason to motivate localization. Indeed it's intrinsically a posteriori. I still think it's a good reason to care about localization that is deeper than the fact that the assignment $X \mapsto E_*(X)$ is a somewhat computable invariant. If one has a category of things (like spaces) and some sort of ...



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