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• CommentRowNumber1.
• CommentAuthorexpixpi
• CommentTimeJan 30th 2014
Scale and entropy of an object can be defined within a category in a canonical way as Hom(S(X),e)=X and d(X)=S(Hom(X,X)) respectively.

Intuitively, in physical real systems, entropy is scale dependent in the sense that at small and big scales entropy seems to be maximum and arriving to a minimum at a certain characteristic scale. Do this fact some any fundament in categorical definitions ?
• CommentRowNumber2.
• CommentAuthorexpixpi
• CommentTimeJan 30th 2014
Only to clarify:

What is "the scale" of a categorical object X? It is defined as a Log-wise categorical construction. Namely, we have to fix a base object "e" of the category and, if exists (by construction or by universal property) is an object S(X) of the same category such that Hom(S,e) is isomorphic to X. Please, note some crutial log-wise properties:

1) Provided X and Y objects of the category, by universal properties, S(X x Y)=S(X) + S(Y) product and coproduct (in that category)
2) Also, provided an object X, then S(Hom(X,X)) = X x S(X) isomorphic

Then, entropy of an categorical object X may be defined as the SCALE of the object d(x)=Hom(X,X) within any category. Indeed is a functor with differential properties, since d(X x Y)=X x dY + dX x Y (cartesian product).

a) The Shanon entropy can be derived in category Set and e={0,1}. The scale is then its cardinality and absolute and relative entropy can be easily derived.
b) Using the Sierpinsky object in Top, and using properties regarding fiber bundles, Boltzman entropy can be easily derived for configuration topological spaces.
c) Regarding Hilbert spaces categories, the categorical entropy can be realized in quantum field entropies, but I am now working on this.
• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJan 31st 2014

You already said this in a different thread. See what Urs said there.