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.



Notation for "the inclusion map is a homotopy equivalence"
$A\stackrel{\sim}{\hookrightarrow}B$? 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...)


