1 What is a Category?

A category is just a bunch of dots with arrows going between them. The tricky part is interpreting what those dots and arrows mean.

In general, there are three conditions that these dots and arrows need to satisfy. Every dot needs a special arrow that goes from the dot to itself. We call this special arrow the identity.
Also, if we have an arrow from dot $A$ to dot $B$ and an arrow from dot $B$ to dot $C$, then we have to be able to combine these arrows to get an arrow from dot $A$ to dot $C$.
Finally, combining an arrow with the identity arrow shouldn't change it.

And that's all a category is. Just a collection of dots and arrows satisfying these three rules.

2 What Things Are Secretly Categories?

One simple category is Set, the category of sets. The dots in Set are, of course, sets. The arrows are functions between sets. Every set has an identity map to itself, functions between sets can be composed, and composing a function with the identity map does nothing. So the arrows in Set follow our rules.

A lot of categories follow this model, where the dots are some sort of collection and the arrows are functions between them. For example, the dots in the category Grp are groups and the arrows are group homomorphisms. Every group has an identity homomorphism to itself, the composition of two homomorphisms is a homomorphism, and composing a homomorphism with the identity does nothing. So Grp is also a category.

Along the same lines, we have Ring, the category of rings and homomorphism between them, and Top, the category of topological spaces and continuous maps between them. The list goes on.

But not every category is of this form. As a more exotic example, you can look at a group as a one-dot category. Each element of the group corresponds to an arrow in the category. The identity element of the group is the identity arrow. And combining two arrows corresponds to the group operation. This object is still a category, but in this case it doesn't make sense to view the dot as a set and the arrows as structure-preserving transformations.

3 Why Is This A Useful Notion?

To give some motivation, let's just consider Grp for the moment. The categorical viewpoint is that instead of studying how the individual elements of a group fit together, we should study structure-preserving transformations between groups. Often, when we look at a group, we want to know about its subgroups or quotient groups. But since every subgroup is the image of a homomorphism into the group, we can just study subgroups by studying homomorphisms into the group. And by the first isomorphism theorem, every quotient of a group is the image of a homomorphism out of the group. So we can really study subgroups and quotient groups just by studying group homomorphisms.

The idea of studying structured objects by studying maps that preserve that structure pops up all over in mathematics. Category theory just takes this idea to the extreme by studying only these maps and forgetting everything else about the objects.

The first image is from here

We see ice melting and water boiling every day. But why does it happen? If you think about it, it's kind of weird that the properties of water can change so suddenly. Today, I'm going to talk about why this happens. But first, we have to come up with some equations that we can use to describe the properties of gases.

The Van der Waals Equation

At some point in chemistry class, you might have seen the ideal gas law, $PV = nRT$. An ideal gas is a theoretical gas whose particles take up no volume and don't interact with each other. These properties make ideal gases easy to do math about, and give us nice equations like the ideal gas law. In fact, if we change our units, we can get an even nicer ideal gas law. If we let $N$ be the number of particles and $\tau$ be the temperature in joules, then instead of the normal chemistry ideal gas law, we just have $PV = N\tau$. It's nice and simple.

Unfortunately, if we want to study phase transitions, this ideal gas model is a little bit too simple. Many phase transitions happen because of interactions between molecules, which cannot happen in an ideal gas. If we want to understand why water boils, we'll need a more sophisticated model. We can add in some correction terms to make two of our assumptions a bit more realistic. First of all, instead of assuming our particles are point masses, we can instead give them each a little volume. This decreases the amount of empty space in the container of gas. Furthermore, how much the empty space decreases should depend linearly on the number of gas particles. So we can replace $V$ in our equation by $V - Nb$ for some positive constant $b$. Next, we add in an attractive force between the particles. For particles in the middle of the gas, this doesn't do much. They have particles surrounding them on all sides, so the attractive forces in all directions cancel each other out. But it does affect the particles near the edges of the container. They are pulled back towards the middle of the container since almost all of the other particles are in that direction. The magnitude of this force depends on the number density ($\frac N V$) of particles in the container. The number of particles at the edge of the container is also proportional to the number density of particles. This means that the pressure of our non-ideal gas is decreased by $a \frac {N^2}{V^2}$ for some positive constant $a$. These two modifications give us the Van der Waals equation \[ \left(p + \frac {N^2}{V^2} a\right) (V - Nb) = N \tau \] The Van der Waals equation is still a pretty crude approximation and still only works for dilute gases, but it will allow us to understand phase transitions qualitatively.

Phase Transitions

So, now we've got a fancy new equation to model non-ideal gases. Let's see what it tells us. We'll begin by looking at how pressure varies with volume. I picked some arbitrary $a, b$ and $N$ values and plotted $V$ vs $p$ for various temperatures. The plots look like this.

For $\tau$ below some critical temperature, we see that pressure first dips down, then goes up a bit, and then goes back down again as volume decreases. This is weird. Intuitively, if you squish a substance, the pressure should go up. But there's a region in the plot where decreasing the volume decreases the pressure. As you squish the material more, it resists you less. This seems pretty unrealistic. What's going on there?

To analyze this weird behavior, we need to think about the energy stored by squishing the gas. At some point in physics class, you might have seen that $W = \int F\;dx$. Work (energy) is a force applied over a distance. If we multiply through by $\frac{area}{area}$, we get another form of the equation that is often more useful with gases. \[ W = \int F \cdot \frac{area}{area}\;dx = \int \frac{F}{area} \; d(x \cdot area) = \int p\;dV \] So $\int p \; dV$ can be seen as the energy stored in the system. In fact, if our system is at a constant temperature, then $\int p \; dV$ is the Helmholtz free energy of the system. We denote the Helmholtz free energy by $A$.

Now, we'll try to use the Helmholtz free energy to understand what goes on in the weird region of the graph we identified above. We'll assume that our temperature is constant, so we have that the Helmholtz free energy when the system has volume $v$ is $A(v) = -\int^\infty_v p\;dV$ That weird dip in our $p-V$ graph makes the Helmholtz free energy flatten out a little bit in that area. This means that in that region, the Helmholtz free energy is not a convex function of $V$. In the following plot, I exaggerate the effect a bit, but it makes it a lot easier to see what happens next.

Systems don't like to have high energy, and gases are no exception. A gas tries to minimize its Helmholtz free energy however it can. This is what makes Helmholtz free energy a useful topic to study. And by using a neat trick, a gas can 'cheat' to get a lower free energy than our plot above would predict by taking advantage of all the weirdness that was confusing us earlier. Our plot describes the Helmholtz free energy of a homogenous substance. But a gas doesn't have to be homogenous. Some of it could be in one state, and some could be in another. Normally, this is not a helpful thing to do, so gases don't do it. But because of our weird graph, a gas can use this trick to lower its free energy.

If part of the gas is in the red state and part of it is in the blue state, then the Helmholtz free energy of the gas is a weighted average of the red energy and the blue energy. This means that by splitting itself into two states, the gas can follow the black line on the $A-V$ graph instead of the purple one, thus lowering its energy! This is only possible because of the unusual concavity of the graph, which in turn was caused by the weird dip in the $p-V$ graph. But what does this mean? When does a gas randomly split into two states? When it condenses into a liquid!

As a gas condenses, the liquid and gas form can coexist. This is precisely the gas becoming inhomogeneous as a means to lower its free energy.

So there you have it. Phase transitions occur because a material can decrease its energy by splitting into an inhomogeneous combination of states rather than smoothly changing its properties. The sudden, mysterious change from a gas to a liquid is a just trick that gases play to take advantage of little bumps in their energy curves!

The image of condensation came from here.