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:

- Objects and the relationships between them.
- The use of functors to build bridges between different categories.
- Breaking an object up into simple pieces; understanding how to build complicated structure from these pieces. Limits and colimits.
- Localization: shifting view so that two objects you previously viewed as different are now viewed as the same.
- Replacing an object by one which is easier to work with but has the same fundamental properties you are trying to study.
- 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.

Let's start with examples of (1), i.e. of categories themselves:

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 $t$.

States of the economy as time goes on.

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

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

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."

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).

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.

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...

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.

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.

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.

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.

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.

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

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

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).

Geoscience breaks materials down into their simplest constituent pieces.

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

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

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

Political science and the action of individuals.

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.

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 $Cat$.

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.

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.

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).

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).

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

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...

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.

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

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

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

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.

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).

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

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.

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.

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

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).

9more comments