Let Ξ be some mathematical entity — be it topos or category or whatnot — with morphisms.
To each morphism of Ξ, assign a spring constant k (in practice, I’m hoping there are more or less canonical spring constants for each Ξ)
We could obtain a vibrational spectrum of Ξ (this is done in practice with small molecules, in which the vibrational spectrum is determined by the application of group theory) , and if Luca Turin’s theory of olfaction is correct
(http://en.wikipedia.org/wiki/Vibration_theory_of_olfaction), one could with the right
olfactory prosthesis (which is well beyond current technology) , smell the difference between FinSet, Grp, and nCob.
More to the point, the whole enterprise would no longer have the sensory qualia of a child in the 1950s playing with an erector set which contains morphisms and objects, and some flashlights (functors)
When you read about the olfactory sense of dogs, http://www.pbs.org/wgbh/nova/nature/dogs-sense-of-smell.html, I wonder just what exotic structures might exist that we could only get a whiff of if we could olfactorize Ξ and its congeners