Category Theory & Biology

Question

Category theory is becoming more and more used in the following fields (besides others)

(1) Quantum physics (e.g. C. Isham and B. Coecke et al)

(2) General relativity (e.g. A.K. Guts et al)

(3) Linguistics and natural language processing (e.g. J. Lambek and S. Abramsky et al)

(4) Computer science (anyway)

Question. Does someone know if there are any connections so far between category theory and biology (genetics or ecology perhaps or any other discipline)?

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Remark. I am aware that this question could also be posted on biology stack exchange. However my idea is that it could be better to ask it on the mathematics site, since it might rather be mathematicians that have some knowledge about such connections than biologists themselves.

Answer 1

You could have a look at the following two papers about genetics and category theory:

Category theory for genetics I: mutations and sequence alignments, https://arxiv.org/abs/1805.07002

Category theory for genetics II: genotype, phenotype and haplotype, https://arxiv.org/abs/1805.07004

These papers are revisions of the following one:

https://arxiv.org/abs/1708.05255

Answer 2

In evolution.

In neuroscience as well as other work in studying the brain via algebraic topology.

Answer 3

Gómez-Ramirez - A New Foundation for Representation in Cognitive and Brain Science; Category Theory and the Hippocampus (2014)

I did not read this book, but maybe it is an example of the application of category theory in biology / medicine.

Answer 4

Ecology is related! See this blog post, which describes algebraic operations on a topology of phylogenic trees with $n$ leaves.

The n-Category Cafe is a great casual source for applied category theory and compositionality.

Answer 5

I must apologize for posting an answer to an old question. My apology is not altogether sincere, since it appears from the answers posted previously that the question was not addressed. From the comments - with one exception - I gathered that commentators became somewhat confrontational. The excepted comment was by @Hans Engler and suggested to look at the work of Robert Rosen. One may have added a specific reference such as "Life Itself" to provide the OP with a more specific pointer. In there, Rosen lays out a specific construction by which certain aspects of that which may be life can be (a) successfully cast in the language of categories and (2) gainfully interpreted (to a certain extent) to benefit a view of biology from the perspective of structure. This, for me, is the quintessential aspect of connection - which is what OP was asking for.

One can now ask "Can I do this for something more specific?". My view of this was and is that any biologists who are caught using diagrams for their work, for example in immunology or, more generally, hematology, or neuro-biology, does some form of juggling with objects and arrows, where the arrows mean very specific biological operations.

For example: Blood formation (hematopoiesis), in its current model (sic!) states that all blood cell types (sic!) derive from members of the blood stem cell population. This may be boiled down to the statement that the abstractum blood decomposed into its cell types has an initial element. Furthermore, it is known that there are relationships $(s,t)$ between pairs of cell types and that these relationships are ordered and that source $s$ and target $t$ do not commute. In a sense, then, if we say $H$ for the collection of objects that make up the cell types, then there is a need for the product $H \times H$ to exist. Next, one can build a meaningful arrow $\to$ via $\to (s,t) \equiv s \to t$ as is well-known to mathematicians, but there is the less well-known necessary interpretation of differentiation. Thus, the arrow $\text{Stem} \to \text{preB}$ or any other arrow along these lines becomes biologically comprehensible language.

To summarize the points that I wish to make:

• When we ask for connections to category theory, then we must be aware that we will most likely have to build them on the basis of some reasonable understanding of the biology involved
• Phrase the constructs within the type theory (language) naturally existing in any category that one comes up with
• Be aware that category designs are unlikely to be cut-and-dry but, rather, require certain "extras" that one can only get at by looking into the objects

The sketchy design of the blood category above may serve as an example for the three points I just made. I have done more along those lines and written a Mathematica paclet to facilitate reasoning within this view of connections between category theory and cell biology, but I would like to stop here and see if more is actually wanted in this group.