Category Theoretic Framework for Chemistry I: a Tower of Chemistry

Abstract

Many laws of chemistry are exact within a limited scope and acquire a separate caveat outside it, and the caveats are usually treated as unrelated. This work argues that they share one cause. Each caveat marks a point where a question is asked of a description too coarse to answer it: the question belongs to a richer level of structure than the description carries. To make these levels explicit, the paper builds a tower of categories over the free symmetric monoidal category of a Petri net, the simplest categorical presentation of a reaction network. The levels, from the bottom up, are stoichiometry, thermochemistry, equilibrium, reaction kinetics, reaction mechanism, molecular geometry, and electronic structure. Each adds one kind of chemical content over the level below, and a forgetful functor runs back down. One question runs through the tower: what can a level express that the level below cannot? Answering it places each measurable quantity at the level where it lives, and identifies the content the levels below could record but not account for. The method is then turned on with two pieces of known chemistry. It recasts a classical criterion for when a reaction network has a unique stable equilibrium, the deficiency-zero theorem, as the rigidity of a single forgetful fibre. It also follows one familiar reaction up the tower: the ring opening that the Woodward-Hoffmann rules govern. At each level, from stoichiometry to electronic structure, the reaction becomes a distinct categorical object.