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Examples
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Per Shang-Keng Ma, an entropy can be defined as the logarithm of the phase-space volume explored by the system over a given timescale; the states relevant for thermodynamics (mumble mumble microcanonical mumble mumble) would be the decategorification of the states used at the statistical-mechanical level.
Arrow of Time FAQ Sean 2007
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This 'forgetting' is so common that it has a name: decategorification.
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This is an interesting story, in that this knot invariant has given a purely algebraic definition by Mackaay, Stosic and Vaz, but they haven't given a proof of invariance, or that you get the right decategorification.
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Geordie and I have come up with a geometric description that allows you to prove invariance and decategorification.
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