Chemists are secretly doing applied category theory! When chemists list a bunch of chemical reactions like

C + O₂ → CO₂

they are secretly describing a ‘category’.

That shouldn’t be surprising. A category is simply a collection of things called objects together with things called morphisms going from one object to another, often written

f: x → y

The rules of a category say:

1) we can compose a morphism f: x → y and another morphism g: y → z to get an arrow gf: x → z,

2) (hg)f = h(gf), so we don’t need to bother with parentheses when composing arrows,

3) every object x has an identity morphism 1ₓ: x → x that obeys 1ₓ f = f and f 1ₓ = f.

Whenever we have a bunch of things (objects) and processes (arrows) that take one thing to another, we’re likely to have a category. In chemistry, the objects are bunches of molecules and the arrows are chemical reactions. But we can ‘add’ bunches of molecules and also add reactions, so we have something more than a mere category: we have something called a symmetric monoidal category.

My talk here, part of a series, is an explanation of this viewpoint and how we can use it to take ideas from elementary particle physics and apply them to chemistry! For more details try this free book:

I am thinking, reading your notes, that the objects can be words in the natural language, and the function the binary operations of the boolean logic; is each theorem a categorical reasoning?
If I understand well, if there are two structures with a bijective function between categorical objects, then the spaces of the words can have space of the objects with the same properties.
I am thinking that the biological words can be the proteins, so that could be a bijection between some subspace of the words that have a representation in the proteins (hot, cold, bright, smell, nuiscance) with some subspace of the logical reasonings.

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I am thinking, reading your notes, that the objects can be words in the natural language, and the function the binary operations of the boolean logic; is each theorem a categorical reasoning?

If I understand well, if there are two structures with a bijective function between categorical objects, then the spaces of the words can have space of the objects with the same properties.

I am thinking that the biological words can be the proteins, so that could be a bijection between some subspace of the words that have a representation in the proteins (hot, cold, bright, smell, nuiscance) with some subspace of the logical reasonings.