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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)?


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.

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  • $\begingroup$ You could have a look whether it's applied in biochemistry, there's a lot of overlap between that field and quantum phys, comp sci and language processing. $\endgroup$ – Jam Sep 7 '18 at 12:17
  • $\begingroup$ Look at work by Robert Rosen. $\endgroup$ – Hans Engler Sep 7 '18 at 13:15
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    $\begingroup$ @HansEngler Thanks, however e.g. Robert Rosen.The representation of biological systems from the standpoint of the theory of categories reads like a short introduction to some basic categorical definitions. I miss an application of categories to biology there. Do you have some other publications in mind? $\endgroup$ – FWE Sep 7 '18 at 14:22
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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

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  • $\begingroup$ @ Remy - Thanks for the references, very interesting papers. What is the motivation in this case for using Lawvere theories? Is it correct, that since the functorial models have values in Set this could also be done e.g. using universal algebra? $\endgroup$ – FWE Oct 19 '18 at 16:30
  • $\begingroup$ btw. I am totally fine with the approach using Lawvere theories (or sketches, forms or similar) - I would just like to know, if there is a particular reason. E.g. in case one would not work in Set but in some other model category, the approach would seem somehow mandatory. $\endgroup$ – FWE Oct 19 '18 at 17:38
  • $\begingroup$ The reason for using the limit-preserving functor formalism is that it makes it easier for possible analytics and it is also easier to implement with a programming language. At the end of the day, the goal is to use category theory to make computational biology methods more formal. On the other hand, monads are formal objects that hide the information. These may be one day useful, but it is hard to see how they could be used right now. $\endgroup$ – Remy Oct 22 '18 at 20:56
  • $\begingroup$ Note that the first paper uses models in Set while the second one uses models in idempotent commutative monoids. I am working on another project that requires models in the category of partitions. The functor approach has so far allowed me to jump from formalism to practice and vice versa quite easily. Maybe in the future, this approach will lead to a more formal and compact one. Also, note that the types of limit sketches that I use are quite specific, because we deal with a particular theory: genetics. Going more formal would probably prevent us from actually 'doing genetics'. $\endgroup$ – Remy Oct 22 '18 at 21:05
  • $\begingroup$ Finally, regarding your question about using universal algebra and Lawvere theories, note that the limits used in the previous papers are not restricted to products (this is the type of limits used in universal algebra). I actually use a lot of wide pullbacks to integrate pieces of information that have a non-empty intersection -- this often happens in computational biology where a piece of data needs to connect with other pieces of data in a coherent way (networks, databases, etc). $\endgroup$ – Remy Oct 22 '18 at 21:24
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In evolution.

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

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  • $\begingroup$ Sorry but where are the applications of the first link to actual evolution problems? The closest part I can see is the one mentioning Markov models on trees, which are quite classical and not based on operads at all. $\endgroup$ – Did Sep 8 '18 at 23:44
  • $\begingroup$ I’m not aware of concrete applications. OP was asking for connections between genetics and categories. Hence the link. $\endgroup$ – Ittay Weiss Sep 9 '18 at 19:17
  • $\begingroup$ Surely this is my fault but I fail to see the connections, except if "connections" is understood un a rather vague sense. Maybe you could expand on the nature and extent of these "connections", supposedly specific to category theory (and for this, it does not suffice to make the obvious point that classical mathematical notions useful in the theory of evolution, such as Markov chains, can be reformulated into a categorical language). $\endgroup$ – Did Sep 9 '18 at 19:43
  • $\begingroup$ The article describes a particular operas relevant to genetics. That is the connection. $\endgroup$ – Ittay Weiss Sep 9 '18 at 21:01
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    $\begingroup$ OP was asking about connections. You seem to insist on applications. A connection can be on the level of ideas. If you do not think so, you are certainly entitled to your opinion. I wish you all the best. $\endgroup$ – Ittay Weiss Sep 9 '18 at 21:19
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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.

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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.

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