Contents

Idea

Chemistry is the branch of physical science that studies the properties of condensed matter, from the scale of atoms upwards.

A central part of chemistry is the identification of reaction processes – chemical reactions – by which a variety of chemical substances (species of atoms or molecules?) interact to form new combinations. Such processes are typically denoted by formulas of the form

$a A + b B \leftrightarrow c C + d D$

where the capital letters denote species of atoms/molecules?, while the lower case letters are natural numbers. This indicates that $a$ molecules of type $A$ may react with $b$ molecules of type $B$ to $c$ molecules of type $C$ together with $d$ molecules of type $D$.

In thermodynamic equilibrium? such reactions occur in both directions, whence the double arrow. A reaction of the form

$A \to c C + d D$

would be called a decay process. Notice that analogous processes appear elsewhere, for instance there are nuclear reactions? in particle physics described by the same kind of reaction diagrams. Also D-branes for the topological string undergo reaction processes of this form, see below.

One might be tempted to recognize morphisms as in category theory in this notation, but care needs to be exercised to arrive at a sensible concept.

Formalization of reaction processes

It is unlikely that any formalization of reaction processes (say in category theory) will depend on the specific nature of the chemical elements. Among formalizations that do model reaction processes akin to those seen in chemistry are stochastic Petri nets and triangulated categories with stability conditions:

Stochastic Petri nets or Chemical reaction networks

Given a set $S$ of ‘species’, a complex of those species is a function $C : S \to \mathbb{N}$. A reaction network is a set of species together with a directed multigraph whose vertices are labelled by complexes of those species. Edges of the multigraph correspond to transitions.

Any reaction network gives rise to a Petri net, and vice versa. In either case, we can label each transition by a rate constant in $(0, \infty)$. This gives stochastic versions of the nets and networks. (See BaezNet.)

Triangulated categories with stability conditions

One successful formalization in category theory of the idea of reaction processes appears in the context of D-branes for the B-model topological string. These are (hypothetical) physical objects that appear in different species labeled by objects in a triangulated derived category (of quasicoherent sheaves on some Calabi-Yau variety). There are processes

$A \leftrightarrow B \oplus C$

by which a brane of type $A$ may decay into a brane of type B and a brane of type C.

While these are not reactions of chemical elements, clearly the general form of reaction processes is similar.

And for the case of these D-branes, there happens to be a well-established and useful mathematical formalization of these reaction processes via category theory:

Namely every reaction process as above corresponds to a homotopy fiber sequence (a “distinguished triangle”) in the triangulated category, of the form

$B \longrightarrow A \longrightarrow C \,.$

(See Aspinwall 04, search the document for the keyword “decay”.)

Mathematically such a homotopy fiber sequence in a triangulated category precisely expresses the fact that $A$ is a “twisted direct sum” of $B$ and $C$ (extension, semidirect product), hence much like the plain direct sum, but with some “interaction” included.

In addition there are Bridgeland stability conditions on such triangulated categories of topological D-branes. These determine which of these reaction processes lead to stable compounds, i.e. whether, in the above example, the brane of type $A$ will really decay into branes of type B and C, or if conversely the latter will fuse. (See again Aspinwall 04, search the document for the keyword “stability”.)

After its motivation from D-brane reaction processes, the study of triangulated categories with Bridgeland stability conditions has become a rich and active area of pure mathematics in itself.

References

Last revised on December 15, 2016 at 04:49:55. See the history of this page for a list of all contributions to it.