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

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*Related*

coming from the other direction. What is the mathematical structure of the olfactory sense?

Molecule-shape (or whatever a molecule-shape maps to in our/dogs’ mental maps of sense-able smells), like the space of consumer products or the space of organisms, is full of starkly different things but which are partitioned by Aristotelian categories. As far as I know we just put them in set-bags and not in informative categories.

———-

on your post. why would there be spring constants? and, isn’t this in a sense what the grothendieck project is about–fuzzing up the precise in order to see the broader structure?

Maybe I’m off base.

My gut sense is that the arena of odor and fragrances is not like -a vector space of different scents-.

The analogy is object:morphism::atom:bond

I think you’re right, it’s not a linear vector space