# Most striking applications of category theory?

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:

• Joyal's Combinatorial Species
• Grothendieck's Galois Theory
• Programming (unification as computing a coequalizer, Tatsuya Hagino's categorical construction of functional programming)

I am sure that these only touch on the surface so I would be most grateful to hear of more examples, thank you!

edit: To try and be more precise, "application" in the context of this question means that it makes use of slightly deeper results from category theory in a natural way. So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.

• This question is too board. Category theory play roles in half of mathematics – Shizhuo Zhang Mar 25 '10 at 16:21
• Community wiki? – François G. Dorais Mar 25 '10 at 16:35
• @Martin: My understanding is that Joyal's theory marks the first time the combinatorial interpretation of functional composition was made rigorous, which I think is a great application. Anyway, I'd have to agree that this question is too broad. – Qiaochu Yuan Mar 25 '10 at 16:43
• I'd like to see the question made a lot more focused. Vast swathes of mathematics, from about the Grothendieck era onwards, have used category theory. If you're not careful, this will turn into an unproductive debate about whether you call various things "applications" of category theory, or say that they use categorical tools, or "merely" use categorical language, or display categorical thinking, or whether something that was done with category theory could also be done without, etc. So for the moment I'm going to vote to close -- but if you can sharpen it, I'll cancel that vote. – Tom Leinster Mar 25 '10 at 18:42
• Based on the title, perhaps the interpretation should be "results which essentially depend on a theorem in category theory". – Tom Church Mar 26 '10 at 2:11

For a while, my answer to this question was algebraic K-theory; what little I know of it, I learned from Quillen's paper, and it was a relief to finally see an example of category theory being used in an essential way to do something that was not just linguistic. Quillen defines the higher K-groups of an exact category by forming a quite different category in some combinatorial manner that seems to strip away any vestige of a connection to something non-categorical, and then taking its geometric realization and homotopy groups. The whole process: ring to module category to Q-construction to geometric realization, was the first argument I'd seen that category theory could do more than just rephrase perfectly good theorems confusingly.

(Now my answer would be "perverse sheaves", though.)

One can find a couple of "concrete" striking applications of category theory in algebraic geometry. For example:

Erik Meijer, at the time working for Microsoft, and his group created the dual of an IEnumerable, an IObservable, which led to the Rx Framework. According to Erik, this was an explicit use of category theory -- Erik's new venture is called Applied Duality

The recent developments in homotopical algebra (after 1990) would not be possible without the use of category theory, and more precisely the theory of locally presentable and accessible categories. I am talking about the theory of combinatorial model categories (model categories such that the underlying category is locally presentable).

The most recent book of Nick Katz [see https://web.math.princeton.edu/~nmk/mellin398.pdf ] proves extremely concrete equidistribution theorems for certain families of exponential sums. Categories enter in three essential ways (at least): (1) all the work going to Deligne's Weil II version of the Riemann Hypothesis over finite fields; (2) the theory of perverse sheaves; (3) the Tannakian formalism to recover a group from a category. In this, the new contribution of Katz in this book is (3): essentially, the equidistribution is proved using the Weyl equidistribution criterion, and all analytic estimates follow from (1). But if one doesn't know that there is a group underlying the families of sums (or rather the unitarized Frobenius automorphisms which give rise to these sums), one doesn't know what these estimates are really proving.

For more traditional families of sums, one uses instead Deligne's Equidistribution Theorem, where the group is given concretely as monodromy group of a lisse sheaf, but Katz's family are not parameterized by an algebraic variety, and the Tannakian category arises by looking at a category of perverse sheaves equiped with a suitable form of multiplicative convolution.

This is, I think, completely amazing...

• I agree that this combination of methods is really amazing and a good advertisement for all these abstract concepts. – Martin Brandenburg Mar 22 '15 at 0:52

Noncommutative algebraic geometry(in the sense of Gabriel-Rosenberg, Artin-Zhang,Van den Berg) are based on category(abelian or Grothendieck category). They consider category as category of quasi coherent sheaves on some noncommutative space. This idea was proposed by Grothendieck and then re-quoted by Manin. Without category theory, this subject can not be built. More information is in Noncommutative Algebraic Geometry and Theories of Noncommutative geometry

Another kind of Noncommutative algebraic geometry is based on Functorial POV. It was proposed by Gabriel in the theory of algebraic group and then developed by Grothendieck in commutative algebraic geometry and then Kontsevich-Rosenberg developed noncommutative stack theory via this POV.

Noncommutative derived algebraic geometry is also based on category(triangulated category)theory.

The relevant names(maybe I will miss some of them)are Manin-Beilinson-Drinfeld, Kapranov, Deligne, Bernstein, Bondal-Orlov-Lunts,Kontsevich-Soibelman,Toen,Van den berg, Lurie, Keller,Neeman and others

The finite vector space analog to Ramsey's theorem was proved using categories the paper is available here

• The link in this answer is broken. – David Roberts Mar 28 '11 at 3:25
• I have fixed the link. – Kristal Cantwell Mar 28 '11 at 3:55
• How or why is this any more striking than Ramsey's theorem? – Kevin H. Lin Mar 28 '11 at 5:12
• Additionally, I'm not really seeing how category theory is used. I see the word "category"... – Todd Trimble Aug 19 '12 at 13:45

This question is too vague. Off the top of my head: algebraic topology, homological algebra, etale cohomology (Weil conjectures), homotopical algebra, topological field theory, Mackey functors, Kazhdan-Lusztig theory, ...

• Sorry I have not phrased the questions quite correctly, this is not the sort of answer I was after. – muad Mar 25 '10 at 19:26
• I agree. I think you want to avoid this type of discussion. However I am not clear on what the question should ask. – Bruce Westbury Mar 25 '10 at 19:44
• Bruce, your post is more or less what I would have said. – Charlie Frohman Mar 25 '10 at 22:32
• Muad, Bruce probably does not say only that there are applications of category theory in alg. topology, in homological algebra and so on, but that these whole branches, properly and effectively formulated and practicised are themselves applications of category theory; and deeper you go there, usually the category approach is more indispensable. For example, algebraic topology is study of functors from the category of topological spaces to some algebraic category. – Zoran Skoda Mar 28 '11 at 13:40
• To understand muad's intended question better, consider the question, what applications does set theory have to the rest of mathematics? The knee-jerk answer is that set theory is ubiquitous so the question is too vague. On the other hand, many mathematicians (rightly or wrongly) regard the work of professional set theorists as being totally irrelevant. So I'd interpret muad's question as: If you study category theory for its own sake, will you ever prove a deep theorem with unexpected applications? Or is it, as Miles Reid claims, one of the most sterile of intellectual pursuits? – Timothy Chow Mar 28 '11 at 15:39

First, a comment on studying category theory for its own sake': this slur was very much setting up a straw man. Those accessing the category theory discussion list will know that the discussion there ranges very widely, and actually discusses issues in mathematics, in contrast to other email discussion lists I access.

Second, I have found some elementary facts from category theory very useful; examples are left adjoints preserve colmits, right adjoints preserve limits'. Many years ago, listening to Albrecht Dold on half exact functors made me realise how I could cut down considerably a proof from my thesis by using the basic idea of representable functor: this automatically led to the existence of a homotopy equivalence making a diagram commutative. Again, the theory of ends and coends does make life simpler in discussing geometric realisations.

Third, I have fairly recently realised that the general framework of fibred and cofibred categories is specially useful for discussing pullbacks and pushouts for certain hierarchical structures with which I have dealt. A basic example here is the bifibration (Groupoids) $\to$ (Sets) given by the object functor.

I wish I had a good application in my work of some of the deeper theorems!

Schlessinger's criteria and deformation of Galois representations, see e.g. Mazur's article in Cornell-Silverman-Stevens.

David Spivak has found applications of category theory in many areas outside of pure mathematics, and many are recorded in his book “Category Theory for the Sciences.” He's also done important work regarding the foundations of databases and schema, and it uses non-trivial results from category theory. A whole collection of literature in that direction can be found on his webpage, and probably satisfies the OP's desire that the application use "slightly deeper results" from category theory.

The rest of this answer is a giant list of examples of how you can use category theoretic thinking in pretty much every science. Many of these examples are taken from Spivak's writings. The list here comes from a general-audience talk I gave at my university (lecture notes here) back in 2015, and I figured I may as well post what I came up with somewhere, in case it helps others who need examples of category theory. You should probably take this list with a grain of salt: for many of the items, it would take some work to formalize the relationship to category theory.

The talk tried to highlight the value of category theory in several stages:

1. Objects and the relationships between them.
2. The use of functors to build bridges between different categories.
3. Breaking an object up into simple pieces; understanding how to build complicated structure from these pieces. Limits and colimits.
4. Localization: shifting view so that two objects you previously viewed as different are now viewed as the same.
5. Replacing an object by one which is easier to work with but has the same fundamental properties you are trying to study.
6. Mapping an object to a small bit of information about the object. Showing that two are different because they differ on this bit. Trying to find a complete set of invariants so you know precisely when two are the same.

1. Classical mechanics can be viewed as studying the state of the world around us as time goes on. So it works just like the example above, except an object is the whole state of the universe at time $t$.

2. States of the economy as time goes on.

3. Crystallography: Objects are arrangements of atoms in a molecule, morphism is a symmetry.

4. Databases: an object can be a table, a morphism can be a shared column (called a foreign key).

5. Going a bit more meta, an experiment is like a category. Objects could be observables and a relationship could tell us if they're correlated. Spivak writes: "Reusable methodologies can be formalized, and that doing so is inherently valuable. Category theory also provides a language for experimental design patterns, introducing formality while remaining flexible."

6. Even more meta, the collection of all experiments is a category. Objects are experiments and we say two are related if they got the same conclusion (perhaps just on one question of interest across all experiments).

7. In material science, objects could be materials and we could draw $A\to B$ if A is an ingredient or part of B, so water $\to$ concrete. A different way to view it as a category would be to draw $A\to B$ if $A$ is less electrically conductive than $B$, so concrete $\to$ water.

8. Robert Rosen introduced in the 90s a category of morphogenetic networks to study morphogenetic problems. Objects are elements and their different states, morphisms come from neighborhoods.

Example stolen from Spivak: Category theory can serve as a mathematical model for mathematical modeling. Our minds simultaneously keep several models of the world, often in conflict. The value of a model can therefore be measured by how well it fits with other models. What is true will be present across all models, so we should study the relationship between models.

Now for some examples of functors, (2) in my list of stages of the talk...

1. If A be the set of amino acids and Str(A) the set of all strings formed from A. The process of translation gives a functor turning a list of RNA triplets into a polypeptide.

2. Quantum field theory was categorified by Atiyah in the late 1980s, with much success (at least in producing interesting mathematics). In this domain, an object is a reasonable space, called a manifold, and a morphism is a manifold connecting two manifolds, like a cylinder connects two circles. Such connecting manifolds are called cobordisms. Topological quantum field theory is the study of functors Cob $\to$ Vect that assign a vector space to each manifold and a linear transformation of vector spaces to each cobordism.

3. Suppose you are interested in different algorithms to buy a car. If you fix your preferences then this ordering makes them a category. Consider the price function that tells you the cost of a car, and lands in $\mathbb{R}_{>0}$. In order for this to be a functor it must respect the ordering: is it true that better cars cost more and worse cars cost less? In other words, does the model from category theory match reality? There seems to be debate about this among economists.

4. Suppose you're running an experiment and in all cases so far have observed 4 traits. You've created a mental model for what's going on, but then you observe several cases where only the first 3 traits are true. You shift to a new mental model and that process of shifting your point of view is a functor.

5. An experiment can be thought of as a functor from the category of pairs (Experimenter, Variables) to the category of measurements of the variables under observation. Viewing it this way makes it explicit that the experimenter can affect the outcome, something well-known in psychology and sociology.

Turning to (3), let's think about a natural human tendency: to break things that are hard to understand down into simple pieces, and then try to cobble those pieces together again to understand the original hard thing.

1. Chemistry breaks down to the study of atoms and the molecules they make up.

2. Physics breaks the world down even further, into strings (in the sense of string theory).

3. Molecular Biology studies the cell. Robert Rosen introduced a categorical presentation of (M,R)-systems, which model the activities of a cell. This is a category of automata (sequential machines).

4. Geoscience breaks materials down into their simplest constituent pieces.

5. Neuroscience tries to understand mental processes via the simplest pieces: neurons.

6. Computer Science breaks computation down into 0s and 1s, at the end of the day.

7. Economics and game theory try to isolate a single cause and effect relationship by holding all other variables constant ("decision making on the margin")

8. Political science and the action of individuals.

9. Understanding how materials are built up of their constituent parts. For example, a tendon is made of collagen fibers. Each collagen fiber is made of collagen fibrils (what matters is how these simple pieces are reassembled). A collagen fibril is made up of tropocollagen collagen molecules, i.e. twisted strands of collagen molecules, and you can keep breaking things down this way.

A related example is spider silk, which Spivak has studied.

The process of putting those simple pieces back together again into an understanding of the original problem is an example of a colimit.

1. The current state of any evolutive system is a colimit of previous states. Here, "evolutive system" means a subcategory of time, i.e. for each time $t$ there is a category $K_t$ (the state of the system at time $t$), and for each period $[t,t']$ there is a functor $K_t \to K_{t'}$. So, an evolutive system is itself an example of a functor from time (viewed as a poset) to $Cat$.

2. Emergent Phenomena like the behavior of an ant colony, or of people starting to clap at the end of a performance, or of birds flocking - all examples of colimits.

3. modeling the biological tendency toward homeostasis is again a movement through time, of a collection of individuals following local rules, so it's a colimit.

4. Suppose you have different temperature reading devices measuring a terrain, perhaps with some overlapping areas. You can patch them together to get a maximally accurate reading by taking the colimit. This is simply a categorification of some kind of weighted averaging operation (weighted by knowledge of the devices).

5. Consider outer space. Different astronomers record observations using telescopes. We can patch together different observations of space as a colimit. Objects here using pixels and the set of wavelengths in the visible light spectrum (written in nanometers).

6. The set of laws of the land; are there inconsistencies? Do they assemble properly? This is why we have lawyers.

7. The individuals making up society, and realizing society as the sum of its parts, i.e. at the object built up from all these individuals. When something happens and individuals are effected, the net effects on the colimit can be studied this way.

Turning to (4), localization...

1. Adding more isomorphisms to any of the examples above, e.g. in economics deciding which features of a snapshot in time matter and which don't, and saying two periods are "the same" if they are the same on those features.

2. viewing two different driving routes as the same if they take the same time.

3. viewing two assignments or exercises or exam problems as equivalent if they are the same difficulty and test the same concept.

4. viewing two products as the same if they cost the same and if I don't know/care about any differences in quality.

5. In linguistics, they study phonemes (and morphemes, graphemes, and lexemes, but I won't talk about those), which abstract the types of sounds we hear in speech. The point is to blur away details that cannot serve to differentiate meaning. This is an example of a localization.

6. I could go on, and probably did, but it's not written in my lecture notes from 2015.

Finally, we turn to (5), replacing an object by one which is easier to work with but has the same fundamental properties, and (6) is a special case of (5).

1. Information theory asks: what is the least amount of information required to describe something?

2. Macroeconomics tries to predict behavior at time $t$ based on behavior at time $t'$ just based on the macro environments at those times. It'd be great if you knew which indicators really mattered so you could make predictions like that.

3. Biological classification divides the set of organisms into distinct classes, called taxa. The result is a phylogenetic tree, a partial order on the set of taxa. This is reducing biological information to the information present in the phylogenic tree. Note that the ranking of taxa into kingdom, phylum, etc., can be understood as morphisms of orders. I think I learned this example from Baez.

4. Reducing the information of a human heart to an EKG read-out.

5. In all the examples of categories, I can think of ways to discard extraneous information, giving plenty of examples of localization (4), replacement (5), and compression/invariants (6).

So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.

I believe this is what you seek:

He models financial contracts as objects and then considers the different stages of negotiation as endofunctors on that contract. Of particular importance is the "default", which I gather is like a terminal object.

He takes a foundational view of quantum mechanics and "re-builds" it as a symmetric monoidal category.

Similar to Coecke's program in spirit, they re-build QM and Relativity using Topos.

A great introductory paper to category thoery using an olog as an object (roughly, a concept or semantic) and then the morphisms are how those concepts are combined or modified.

• Hey, can you add a word or two about each of the examples and how they use category theory? – Amir Sagiv Aug 22 '16 at 6:31

Toposes and categorical logic.

• This would be an excellent example if only it were followed up with some detail and references. – Mitch Jul 30 '12 at 15:25