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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.

FWE
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  • 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. – Jam Sep 07 '18 at 12:17
  • Look at work by Robert Rosen. – Hans Engler Sep 07 '18 at 13:15
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    @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? – FWE Sep 07 '18 at 14:22

5 Answers5

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In evolution.

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

Ittay Weiss
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  • 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. – Did Sep 08 '18 at 23:44
  • I’m not aware of concrete applications. OP was asking for connections between genetics and categories. Hence the link. – Ittay Weiss Sep 09 '18 at 19:17
  • 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). – Did Sep 09 '18 at 19:43
  • The article describes a particular operas relevant to genetics. That is the connection. – Ittay Weiss Sep 09 '18 at 21:01
  • Again, this is the crux of the matter: you state confidently "relevant to genetics", where all I am reading in the paper is mathematics that are surely honorable in themselves and of which some mathematicians say that they could, maybe, in the future, we do not know exactly how, be useful to genetics... Usually, when people decree that theory T is useful to applications A, they describe actual examples of applications in the field A obtained using elements of T and which seem impossible to obtain otherwise. Pity that you do not expand on why this analysis I just described, would be wrong. – Did Sep 09 '18 at 21:10
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    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. – Ittay Weiss Sep 09 '18 at 21:19
  • OK, you now retreat to a rather minimalist position, after having used stronger statements such as "relevant to genetics". If I understand English correctly, that X is relevant to G means that experts in G used at least once (and preferably, regularly) X to get results in G, or that experts in X used X to produce at least one new result (and preferably several new results) in G, not only that experts in X decreed that X is "relevant" to G without ever consulting experts in G for their advice (nor explaining precisely what they mean by "relevant"). – Did Sep 09 '18 at 21:30
  • Your comments are now at the point of being abusive. You keep making the same point again and again and are becoming argumentative. – Ittay Weiss Sep 09 '18 at 21:32
  • Sorry if you feel that way, I was under the impression that I was making the point more precise by reacting to your reactions (which, unfortunately, still do not include specific actual examples, in the common sense of the term that I tried to make clear). So be it. – Did Sep 09 '18 at 21:35
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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

Remy
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  • @ 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? – FWE Oct 19 '18 at 16:30
  • 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. – FWE Oct 19 '18 at 17:38
  • 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.

     – Remy Oct 22 '18 at 20:56
  • 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'.

     – Remy Oct 22 '18 at 21:05
  • 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). – Remy Oct 22 '18 at 21:24
  • I mark this answer as accepted since it gives references to papers making use of CT (e.g. Lawvere theories, sketches, simplicial category, ...) in a biological context (i.e. genetics). Btw since this question is intended a [big list] this should not prevent further answers and references. – FWE Nov 15 '18 at 18:04
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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.

Hans Sieburg
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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.

Steenis
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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.

Larry B.
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