B.5.TU:5.1 - Discover whether one input is enough

A practitioner wants to use two available operations, f: X -> Y and g: X -> Z, on one input. The letters name distinct atomic types. The question is whether the chosen composition rules can produce both results from that input.

In a cartesian account, a copying operation Delta_X: X -> X x X supplies the pair of inputs. First copy, then apply the two operations: (f x g) after Delta_X sends x to (f(x), g(x)). The input of the parallel operation is now supplied.

Compare a resource-sensitive account generated only by f, g, identities, serial composition, tensoring and exchange of factors. Tensoring f with g takes X tensor X to Y tensor Z: it requires two inputs. Each given generator preserves the number of atomic factors, and serial composition, tensoring and exchange preserve that property. These rules therefore cannot construct X -> Y tensor Z. Supplying another input or adding an admissible copying operation changes what can be built. This is a result about the stated generated account.

Now apply the distinction. For a reusable data value interpreted through pure functions, both uses can receive that value. For a physical specimen consumed by an assay, obtain a physical way to supply what each assay needs. Dividing a specimen may supply those inputs if the assays admit the resulting portions. The word “copy” would leave that physical contribution unexplained.

The common method discovers the needed theory operation and returns its consequence to use. The cartesian construction and its contrast with tensor composition come from Baez and Stay, §2.3; the input-count argument makes the stated restricted case inspectable.