I'm your huckleberry

category theory studies objects by studying maps between them, and studies constructions of objects S by studying Hom(S,X) which is the collection of maps from S to X. This is cool because: -- you get to understand how things interact with each other. that's a large part of the point.

the point of yoneda's lemma and universal property/functorial definition of things is that while you can construct things by looking at what elements they have inside, to actually understand who someone is you don't look at their organs and cells, right?

You observe how they interact with other people. Same with how for an object S, you actually want to understand Hom(S,X) for general X or Hom(X,S) depending, and that tells you what something is by how it talks to other things.

-- understanding how things talk to each other lets you see some statements as more fundamentally true than others in ways that actually have meaning.

For example it is easy to show that for a finite dimensional vector space V over a field k, Hom(V, k) \simeq V, since they have the same dimension. However to construct the isomorphism between them you have to choose a basis or do something else horrible like that.

However, if you consider the fact that for a finite dimensional vector space V, V \simeq Hom(Hom(V,k),k), you actually don't need to make any such choices to construct this isomorphism. It's simply

v \mapsto (\psi \mapsto \psi(v)).

Which means somehow V \simeq Hom(Hom(V,k),k) is "more true", called "canonically" or "naturally" true, even though both statements are true.

And there's real meaning here, not just aesthetic: if you have a line bundle, which is to say a space where you have a line (1 dimensional vector space) sitting over each point, call it L, with the line over a point p called L_p, then you can consider the line bundle coming from taking Hom(L_p, k) for each point p, call this L^*. Then even though L and L^* are isomorphic over each point, these isomorphisms may not glue to an isomorphism of bundles. However L and (L^*)^* are canonically isomorphic as line bundles.

-- seeing constructions categorically often gives you lots of information for "free", which strongly suggests that this is actually the "correct" way to think about them.

To give a very concrete example from linear algebra (baby's first adjoint functors!): Say you want to think about tensoring with a vector space V that is taking a general vector space and tensoring it with V. There are lots of properties of this construction that might be of interest like the exactness properties and whether it preserves certain other constructions

But (if you defined tensor products by universal property) you can very easily see that it satisfies

Hom(W \otimes V, U) \simeq Hom(W, Hom(V,U))

This is called an adjunction property, that is -- \otimes V is the left adjoint of (Hom(V, --)). And the fact that -- \otimes V is a left adjoint is a very fundamental fact about --\otimes V that automatically gives things like that it preserves colimits and cokernels and is thus right exact.

-- riffing more on the above point, when a construction is right exact but isn't actually exact, to someone who likes things to be "correct" this suggests that you're actually looking at the wrong thing, like you're looking at a truncated version of the more "correct" thing.

in this case, the more correct thing would be the derived tensor product

(note: in my previous point i restricted to vector spaces to make it more relatable, but over a field actually all vector spaces are flat so tensoring with V is just exact. but if you're working over modules or sheaves over a general ring or scheme, you still have right exactness (because you still have the adjoint functors!) but it's only right exact so the derived tensor product is the way to go.)

-- the aesthetics of category theory let you see that the derived tensor product is more "correct" than the tensor product, but the fact that the derived version is more correct/closer to the platonic ideal is not just a matter of aesthetic taste:

Consider for example two lines L and L' in a projective plane. We usually think of their intersection as

O_{L} \otimes_{O_{P^2}} O_{L'} = O_{p}

and when the lines meet transversely this is fine, they intersect at a point, it's all good.

but what happens when the lines coincide? O_{L} \otimes_{O_{P^2}} O_{L} is actually just O_L.

which means that by slightly deforming the situation you have violently changed the algebraic structure. and that's not good--violence is never the answer

But if you take the derived tensor product instead, then for transverse L and L' you still have

O_{L} \otimes_{O_{P^2}}^L O_{L'} = O_{p}

But for the same line with itself you get

O_{L} \otimes_{O_{P^2}}^L O_{L} = [O_L(-1) \to^0 O_L]

whose class in K theory, [O_L] - [O_L(-1)] actually just gives you [O_p], which gives you back invariance of small deformations (for the class in K theory)

(The extra O_L(-1) term in the tensor product records the failure of transversality)

which really means this is the "more correct" version of intersections.

(in the case of this professor for the first half of algebraic geometry i, he kept making mistakes until at some point after a disaster of a lecture he was just like

"screw it, no more plain tensor products, no more QCoh, for the rest of this course and all of next semester we are working in the derived category only and all tensor products will be derived tensor products",

and then he magically stopped making mistakes.)

-- And one point from more recent times: the Geometric Langlands Correspondence!

so back when I was in college, the full categorical geometric langlands correspondence was still conjectural but it was being worked on by a group of very category theory minded people and recently in 2024 they made a huge breakthrough! Which is really cool. (Specifically they showed the global unramified categorical geometric langlands correspondence in characteristic 0)

but basically the geometric langlands correspondence is like a categorified non-abelian version of the Fourier transform: ordinary fourier transforms look at convolutions by decomposing functions into characters. In the geometric langlands picture the analogue of the convolutions are functors acting on sheaves on Bun_G and the \check{G} local systems on X label the spectral side, playing the role of characters.

(When G = G_m this recovers a Fourier-Mukai transform between D modules on line bundles on X and sheaves on the moduli stack of rank 1 local systems.)

and to make this make sense you basically have to build the categories to make sense of the two sides: line bundles on a curve on the automorphic side (Bun_G), rank one local systems on the spectral side, none of it works out of the box; even formulating the conjecture requires thinking really deeply about the right categorical picture.

but over a couple decades, they actually made it make sense! And united representation theory with topology with algebraic geometry with physics.

THE worm, the rich worm, has a noble domain

In the field that is stored with its millions of slain;

The charnel-grounds widen, to me they belong,

With the vaults of the sepulchre, sculptured and strong.

The tower of ages in fragments is laid,

Moss grows on the stones, and I lurk in its shade;

And the hand of the giant and heart of the brave

Must turn weak and submit to the worm and the grave.

Daughters of earth, if I happen to meet

Your bloom-plucking fingers and sod-treading feet--

Oh! turn not away with the shriek of disgust

From the thing you must mate with in darkness and dust.

Your eyes may be flashing in pleasure and pride,

'Neath the crown of a Queen or the wreath of a bride;

Your lips may be fresh and your cheeks may be fair--

Let a few years pass over, and I shall be there.

Cities of splendour, where palace and gate,

Where the marble of strength and the purple of state;

Where the mart and arena, the olive and vine,

Once flourished in glory; oh! are ye not mine?

Go look for famed Carthage, and I shall be found

In the desolate ruin and weed-covered mound;

And the slime of my trailing discovers my home,

'Mid the pillars of Tyre and the temples of Rome.

I am sacredly sheltered and daintily fed

Where the velvet bedecks, and the white lawn is spread;

I may feast undisturbed, I may dwell and carouse

On the sweetest of lips and the smoothest of brows.

The voice of the sexton, the chink of the spade,

Sound merrily under the willow's dank shade.

They are carnival notes, and I travel with glee

To learn what the churchyard has given to me.

Oh! the worm, the rich worm, has a noble domain,

For where monarchs are voiceless I revel and reign;

I delve at my ease and regale where I may;

None dispute with the worm in his will or his way.

The high and the bright for my feasting must fall--

Youth, Beauty, and Manhood, I prey on ye all:

The Prince and the peasant, the despot and slave;