Australian Category Seminar

A function $\mathbb{R} \to \mathbb{R}$ may encode a system whose output at a given time depends only on its input at that time. A natural generalisation involves shift-invariant operators $\ell^2(\mathbb{R}) \to \ell^2(\mathbb{R})$, which may encode systems whose output at a given time depends on its input at all preceding times. One can reasonably ask what happens to the differential calculus in this situation. Well: ordinary derivatives are replaced by Frechet derivatives; and Taylor series are replaced by so-called Volterra series.

This talk will sketch the general theory with a vagueness that would make a functional analyst wince; the point really being to provide grist for a category theorist who fancied turning the whole business into combinatorics or profunctor calculus.