From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sunday March 11, 2001 2:02PM How do cross-category results relate to category theory? In particular, is there a *deeper* theory than category theory which crosses different categories? I ask this question because I have been following the function p1(x,y) = 1 + y - x and its n-dimensional generalization pn(x,y) = 1 + y - Sn/n where Sn = x1 + x2 + ... + xn and where 0 < = y < = xi < = 1 for i = 1, 2, ..., n (although the inequality can be changed to just y < = xi in certain categories) not only across categories but branches of mathematics including probability, mathematical logic, mathematical physics, geometry, number theory, algebra including especially ring and module theory, geometric nonlinear functional analysis, etc. Even p1 alone appears to have remarkable importance across these fields and others (including functional differential and integrodifferential equations). The category object changes, but the function p1 and the inequality remains, and there is even a non-function (which can be made into a function with suitable restrictions) of type y/x --> 1 + y - x which often splits branches of mathematics down the middle with one branch using y/x and the other using 1 + y - x in a generalization of the notion of phase differences (solid versus liquid versus gas versus superfluid versus superconduction types generalized to rare versus common/frequent events, highly influencing or influenced versus low influencing/influenced events, zero curvature versus non-zero curvature events or objects, anomalous versus non-anomalous conditions, etc.). Osher Doctorow Ventura College, etc.