3
$\begingroup$

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.

$\endgroup$

1 Answer 1

6
$\begingroup$

$A\stackrel{\sim}{\hookrightarrow}B$? 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...)

$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$ Jan 12, 2010 at 3:41
  • 8
    $\begingroup$ 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. $\endgroup$ Jan 12, 2010 at 3:47
  • 5
    $\begingroup$ 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). $\endgroup$ Jan 12, 2010 at 3:50
  • $\begingroup$ 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. $\endgroup$ Jan 12, 2010 at 8:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.