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Ellis D. Cooper (Jun 27 2021 at 19:30):What do you call it when: (1) there is a finite set of equations involving constants, and variables whose values may be hyperreal numbers (or equivalent), and random variables with real values; (2) the usual arithmetic operations extended to hyperreal numbers (so to include ratios of infinitesimals, hence derivatives, including partial derivatives); (3) "equation bonds" which are predicates that are defined by identifying a variable appearing in (a side of) one equation with a variable appearing in (a side of) another equation. Such equations include ordinary and partial differential equations, stochastic and probabilistic equations, discrete versions of all of the above, and all initial and boundary conditions. For example, if dx/dt(t)=f(x(t)) and y(0)=y_0, then identifying x with y converts the two previously unrelated equations into a single initial-value problem by "bonding" them along the identification of x with y. In other words, an equation bond constrains solutions of otherwise entirely independent equations. A solution to an equation bond consists of solutions to the equations which also satisfy the constraint. My question is, how would a categorist best explain "equation bonds" in general?
For philosophical and biological context, see http://www.cognocity.org/circularchy/2021-CONTENTS-ZD.pdf
Jade Master (Jun 28 2021 at 19:16):I think you could use pushouts. The idea is that you first define an "open" equation to be a cospan of finite sets equipped with an equation e with variables in X. The idea is that I and O are input and output sets pointing to elements of X which may identified with variables of other open equations.
Jade Master (Jun 28 2021 at 19:19):Then to bond equations you could take another such open equation with equation e', take the pushout of these cospans to join the output variables of the first with the input variables of the second, then pushforward e and e' to this new pushout.
Jade Master (Jun 28 2021 at 19:20):There is something similar in @John Baez and @Blake Pollard 's paper "A compositional framework for reaction networks". Look at the category Dynam in section 3 I think.
John Baez (Jun 28 2021 at 19:27):Yes, we study systems of first-order ordinary differential equations, which we call dynamical systems... but the same methodology should work for equations of any kind.
John Baez (Jun 28 2021 at 19:31):Jade explained it pretty well so I won't say more!