I would like to know whether a category theoretic "rationalization" of the mathematical theory of thermodynamics published by Elliot H. Lieb and Jakob Yngvason has been published or at least undertaken. Their axioms add structure to a preorder: \begin{description} \item [(A1) Reflexivity.] $X \stackrel {A}{\sim} X$. \item [(A2) Transitivity.] $X \prec Y$ and $Y \prec Z$ imply $X \prec Z$. \item [(A3) Consistency.] $X \prec X'$ and $Y \prec Y'$ imply $(X,Y) \prec (X',Y')$. \item [(A4) Scaling Invariance.] If $X \prec Y$, then $tX \prec tY$ for all $t>0$. \item [(A5) Splitting and recombination.] For $0 < t < 1$, \begin{center}$X\stackrel {A}{\sim} (tX,(1-t)X)$.\end{center} \item [(A6) Stability.] If, for some pair of states, $X$ and $Y$, \begin{center} $(X,\epsilon Z_0) \prec (Y, \epsilon Z_1)$\end{center} \noindent holds for a sequence of $\epsilon$'s tending to zero and some some states $Z_0, Z_1$, then $X \prec Y$. \item [(CH) Comparison hypothesis.] For any two states $X$ and $Y$ in the same state space, either $X \prec Y$ or $Y \prec X$. \end{description} REFERENCES Elliot H. Lieb, Jakob Yngvason, "A guide to entropy and the second law of thermodynamics," Notices of the AMS, May, 1998, pp. 571-581. Elliot H. Lieb, Jakob Yngvason, "The physics and mathematics of the second law of thermodynamics," Physics Reports, Volume 310, Issue 1, March 1999, pp. 1-96. (This has the proofs.) Elliot H. Lieb, Jakob Yngvason, "A Fresh look at entropy and the second law of thermodynamics," Physics Today, April 2000, pp. 32-37. (See also text only preprint at http://www.esi.ac.at/preprints/ESI-Preprints.html .) Respectfully yours, Ellis D. Cooper