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It's sometimes convenient to have different notations for "$A$ is a subset of $B$" depending on what the inclusion map does:

  1. If it's non-surjective, $A\subsetneq B$ or $A\subset B$, depending on your religion
  2. If it's surjective, $A=B$ :)
  3. If the image is a precompact set, $A\Subset B$

Does there exist notation to indicate that the inclusion $A\hookrightarrow B$ is a homotopy equivalence? I'd like to use something similar to 1-3.

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up vote 4 down vote accepted

$A\stackrel{\sim}{\hookrightarrow}B$? Alternatively, using Oberdiek's stackrel.sty you could say something like

A \mathrel{\raisebox{2pt}{$\stackrel[\raisebox{1pt}{$\sim$}]{}\subset$}} B

and play a little with the raiseboxes so that this aligns more or less correctly (this depends on your final font, and your publisher's typographer is not going to love you for this...)

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The symbol \hookrightarrow usually denotes a split monomorphism. This generalizes the notion of an embedding. Using Mariano's notation, this means that it's a map that is split mono in the homotopy category, which means that it admits a (I always forget if it's left or right) homotopy inverse. –  Harry Gindi Jan 12 '10 at 3:41
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The symbol denotes whatever the author tells you it will denote in his comments about notation, and there is a special place in hell for users of unexplained notation. I have never used the hooked arrow to mean anything but an inclusion map. –  Mariano Suárez-Alvarez Jan 12 '10 at 3:47
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Homotopy theorists are likely to interpret the hooked arrow with a tilde as "acyclic cofibration", which in general neither implies nor is implied by "inclusion which is a homotopy equivalence" (though it's certainly a similar notion; for example they agree for inclusions of a subcomplex of a CW complex). –  Reid Barton Jan 12 '10 at 3:50
    
Dear Mariano, you write "The symbol denotes whatever the author tells you it will denote".This is practically Humpty Dumpty's reply to Alice in Through The Looking-Glass "When I use a word [...] it means just what I choose it to mean". I am sure it is your ever grinning Cheshire cat who whispered that in your ear. –  Georges Elencwajg Jan 12 '10 at 8:29
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