Discard and Copy Axioms

definition example

Two optional axioms on a Symmetric Monoidal Preorder $(X,\le ,I,\otimes )$ that change the Wiring Diagram style:

• (e) discard axiom: $x\le I$ for all $x\in X$ — every resource can be converted into nothing. New icon: a wire may terminate. Valid in manufacturing, not in chemistry.
• (f) copy axiom: $x\le x\otimes x$ for all $x\in X$ — every resource can be duplicated. New icon: a wire may split. Valid for (classical) information, not for physical objects.

Sources: 7 Sketches §2.2.3, Eqs. (2.25), (2.26).

Examples. With both axioms, the diagram “write email, copy it, send one copy to Alice and one to Bob” (2.26) makes sense. $\mathbf{Bool}=(\mathbb{B},\le ,\mathsf{true},\wedge )$ satisfies both ($x\le \mathsf{true}$ and $x\le x\wedge x$); so does any preorder whose $\otimes$ is the Meet and $I$ the top element (a cartesian monoidal preorder). Cost satisfies discard ($x\ge 0$) but not copy ($x\ge 2x$ fails for $0<x<\infty$).

Categorified. In a Monoidal Category discard and copy become morphisms ${\epsilon }_x:x\to I$ and ${\delta }_x:x\to x\otimes x$; when every object has them coherently (a comonoid structure, natural in $x$) the category is cartesian — this is Fox’s theorem, and it is why $\mathbf{Set}$ with $\times$ can duplicate and delete data while a category of quantum processes or linear resources cannot. In a Hypergraph Category every object carries a Frobenius structure: copy, discard, and their mirror images merge and create.