Resource Theory

example annotation

Resource theories study how resources are exchanged in a given arena — “what buys what”, statically. 7 Sketches models a resource theory as a Symmetric Monoidal Preorder: $x\le y$ means ”$x$ can be converted into $y$”, $x\otimes y$ means ”$x$ and $y$ together”, $I$ means “nothing”. Wiring diagrams are recipes; the three questions are “is it possible?” ($\mathbf{Bool}$), “what is the minimum cost?” (Cost), and “what are the ways?” ($\mathbf{Set}$, monoidal categories).

Sources: 7 Sketches §2.1, §2.2.3, Examples 2.19, 2.86, Exercise 2.21; following [CFS16; Fri17].

Chemistry

Material collections such as ${\mathrm{H}}_2\mathrm{O}$, $\mathrm{NaCl}$, $2\mathrm{NaOH}$, $\mathrm{C}{\mathrm{H}}_4+3{\mathrm{O}}_2$ form the preorder $(\mathrm{Mat},\to ,0,+)$: $\to$ is the order (reaction), $+$ the monoidal product, $0$ the unit (7S Exercise 2.21). E.g. $2{\mathrm{H}}_2\mathrm{O}+2\mathrm{Na}\to 2\mathrm{NaOH}+{\mathrm{H}}_2$ (reactant $\to$ product).

Catalysis. From the reactions

$$y+k\to y^{\prime }+k^{\prime },x+y^{\prime }\to z^{\prime },z^{\prime }+k^{\prime }\to z+k(2.22)$$

the monoidal preorder laws give the composite $x+y+k\to x+y^{\prime }+k^{\prime }\to z^{\prime }+k^{\prime }\to z+k$ (2.23). Here $k$ is a catalyst: it appears on both sides, and $x+y\to z$ is not derivable without it. The wiring diagram (2.24) has three interior boxes and exterior box $x+y+k\to z+k$. Ovens “catalyze” baking pies: a means of production.

Chemistry is not closed (Example 2.86): closure would require a “substance” $2\mathrm{Na}⊸(2\mathrm{NaOH}+{\mathrm{H}}_2)$ obtainable from two water molecules — really a potential reaction unlocked by water.

Manufacturing: the discard axiom

“You can trash anything you want, and it disappears from view.” Adding the discard axiom $x\le I$ lets wires terminate (egg shells, butter wrappers) — valid in manufacturing but not chemistry (${\mathrm{H}}_2\mathrm{O}+\mathrm{NaCl}\to {\mathrm{H}}_2\mathrm{O}$ is not a legal equation: you cannot discard $10^{23}$ dissolved molecules).

Informatics: the copy axiom

Information can be copied (“one cup of butter never becomes two, but a single email can be sent to two people”): the copy axiom $x\le x\otimes x$ lets wires split. Copying a CD’s contents is possible; literally duplicating the disc “is magic”. Cell mitosis is a remarkable box $\mathsf{cell}\to \mathsf{cell}\otimes \mathsf{cell}$, not an axiom.

Resource theories are treated in much more depth in [CFS16; Fri17] (thermodynamics, communication channels, entanglement, asymptotic conversion rates). Database schemas are similarly “ad hoc” — formed for a particular purpose — and there is nothing wrong with that.