MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 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...)

share|cite|improve this answer
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
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
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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