You are dual to everything: The Yoneda Lemma, explained.

2-cell or not 2-cell, that is the question.

What if the universe isn’t actually made of points, waves, fields, particles, or even Lagrangian submanifolds, but instead… natural transformations?

The Yoneda Lemma, a theorem that identifies elements u ∈ F(A) with transformations Φ_ᵤ: h_ₐ → F, makes this view mathematically concrete.

The proof of the Yoneda Lemma takes perhaps three lines (hence why it’s a “lemma” despite its weight).

Its implications, however, reach algebraic geometry, representation theory, and even parts of theoretical physics.

If you truly want to actually understand Tannaka duality, Isbell conjugation, or Grothendieck’s schemes, then you don’t get terribly far without Yoneda.

But how can chasing id_ₐ be this impactful?

The “Yoneda perspective,” that objects are their relations, is forced on you by the math.

The full faithfulness of the Yoneda embedding Y: 𝒞 → [𝒞^ᵒᵖ, Set] is that perspective, formalized. Faithfulness sounds godly, but it’s a technical term meaning injectivity-on-morphisms, which just means different maps stay visibly different, which just means no confusion allowed.

People argue about nonduality vs. duality, but in category theory, duality isn’t optional. Why?

Because it’s not exactly about “things” (per se), nor is it about “relationships” (per se). It’s actually about the parallels between things and their relationships.

Curt’s aside: “Things” in category theory are called objects. “Relationships” are called morphisms. There’s something which is “higher” than this, and that’s a “natural transformation.”

In some sense, you can think of a natural transformation as a “relationship” between relationships, but that would be circular, so you need another name. Categorists call it a 2-cell.

Points, Parts, and Perspective

The Yoneda Lemma sounds like abstract category theory torture. Despite this, it’s actually the closest thing math has to a theory of identity, showing objects are their web of relationships (and importantly, vice versa!). Let’s decode:

\(\text{Nat}(\text{Hom}(-, A), F) \cong F(A) \)

Hom(−, A), often written h^ᴬ, is the contravariant Hom-functor represented by A.

This just means it takes an object X and gives you the set of all “maps” to that object (X → A).

Curt’s aside: I have to put “maps” in quotes because technically these are “morphisms,” and even experts mistake morphisms for maps (in fact, Juan Maldecena did just that in this recent lecture by Jacob Lurie on topological quantum field theory). It’s an easy error because the categorists introduce morphism with every example being maps, and composition looks just like functional composition.

You can think of Hom(−, A) as all the ways other objects can “map into” A.

This just means to step in the shoes of every other object and see what A looks like from that POV.

F is any assignment of sets to objects X and functions F(f): F(Y) → F(X) to maps f: X → Y, in a manner that respects composition and identities.

Now, think of F as some kind of “generalized probe” or “variable set” indexed by 𝒞^ᵒᵖ and think of h^ᴬ as the canonical probe associated specifically with A, capturing its “input signature” (meaning, how objects interact with A).

These are the fun kinds of probes… ;)

You learn more about generalized probes and what that has to do with pulling something out of nothing in physics, in this podcast with Urs Schreiber above.

Nat(h^ᴬ, F) is the set of all “natural transformations” from h^ᴬ to F. The precise definition is tedious, but you can think of it as meaning something like that for each input, the outputs match predictably.

A natural transformation Φ: h^ᴬ → F is a mickle of functions Φₓ: Hom(X, A) → F(X), one for each object X, that play nicely with the maps in 𝒞.

Specifically, for any map f: Y → X, the square below must commute:

\(\begin{align} \text{Hom}(X, A) & \xrightarrow{\Phi_X} F(X) \\ f= -\circ f &\downarrow \ \ \ \ \downarrow{F(f)} \\ \text{Hom}(Y, A) & \xrightarrow{\Phi_Y} F(Y) \end{align}\)

This is known as a coherence condition.

Curt’s aside: Any time you see a complicated diagram like the above, it’s not meant to be something you immediately recognize as true. In fact, it’s not something you’re immediately meant to know the meaning of! You are meant to pause, and write out each arrow on your own, compose, and think “what does this mean?” This post on took a few weeks to put together so feel free to go through it slowly. Take your time. Just get wet.

The Yoneda Lemma states this entire, potentially brobdingnagian, set of coherent families {Φₓ}_{X ∈ 𝒞} is in one-to-one correspondence with the single, potentially simple, set F(A).

This is super interesting.

Why is this the case? And what does it mean?

Given an element u ∈ F(A), you construct a natural transformation Φ_ᵤ: h^ᴬ → F.

This means, if someone hands you an element of the object F(A), someone else may hand you something else. That “something else” happens to be not just anything, but specifically this guy Φ_ᵤ: h^ᴬ → F.

For the next section I’m going to go about doing a proof, which you can skip if you like and move onto the section directly afterward...

Nasty Symbol Chasing Proof

For any object X, define the component (Φ_ᵤ)ₓ: Hom(X, A) → F(X) by sending a morphism g: X → A to the element F(g)(u) ∈ F(X).

You have to check this is natural (it is, essentially because F is a functor).

Conversely, given a natural transformation Φ: h^ᴬ → F, how do we get an element of F(A)?

We use the component of Φ at A itself, Φ_ₐ: Hom(A, A) → F(A), and feed it the most natural element of Hom(A, A) there is: the identity morphism id_ₐ.

So, the element is Φ_ₐ(id_ₐ) ∈ F(A).

The two constructions are inverse to each other. Let’s prove it.

Start with u ∈ F(A). Construct Φ_ᵤ. Now apply the second construction to Φ_ᵤ. We get (Φ_ᵤ)_ₐ(id_ₐ). By definition of Φ_ᵤ, this is F(id_ₐ)(u). Since F is a functor, F(id_ₐ) is the identity on F(A). So F(id_ₐ)(u) = (the identity on F(A))(u) = u.

We got u, bro.

Start with Φ: h^ᴬ → F. Construct u = Φ_ₐ(id_ₐ). Now construct Φ_ᵤ from this u. We need to show Φ_ᵤ = Φ. This means showing their components are equal for every object X. So, pick X and pick any g ∈ Hom(X, A). We need to show (Φ_ᵤ)ₓ(g) = Φₓ(g).

By definition, (Φ_ᵤ)ₓ(g) = F(g)(u). Substituting u = Φ_ₐ(id_ₐ), we get (Φ_ᵤ)ₓ(g) = F(g)(Φ_ₐ(id_ₐ)).

Look at the naturality square for Φ with the map g: X → A:

\(\begin{align} \text{Hom}(A, A) &\xrightarrow{\Phi_A} F(A) \\ g = -\circ g &\downarrow \quad \downarrow F(g) \\ \text{Hom}(X, A) &\xrightarrow{\Phi_X} F(X) \end{align}\)

Naturality means F(g) ∘ Φ_ₐ = _Φ_ₓ ∘ h^ᴬ(g). Applying both sides to id_ₐ ∈ Hom(A, A):

\(F(g)(\Phi_A(\text{id}_A)) = \Phi_X(h^A(g)(\text{id}_A))\)

\(F(g)(\Phi_A(\text{id}_A)) = \Phi_X(\text{id}_A \circ g)\)

\(F(g)(\Phi_A(\text{id}_A)) = \Phi_X(g)\)

The left side is exactly (Φ_ᵤ)ₓ(g). So, (Φ_ᵤ)ₓ(g) = Φₓ(g). Done.

The proof is just chasing the identity morphism id_ₐ through that naturality square.

The square forces a connection between what Φ does at A (specifically to id_ₐ) and what it does at any other X (to any map g: X → A).

Because this connection holds universally for any natural Φ, the value Φ_ₐ(id_ₐ) perfectly captures Φ, and conversely, any potential value u ∈ F(A) can consistently define such a Φ.

That diagram encodes the entire equivalence.

Within the Web

Okay, let’s ditch the dense formalism and capture the conceptual shift.

What earthquake lies hidden in Nat(h^ᴬ, F) ≅ F(A)?

Well, it shatters the concrete notion of elements as just static ‘things’.

This is because it dictates that the set F(A) (the specific way some structure or property F manifests when probed at the object A) is identical to the set of all natural transformations mapping from A’s canonical probe h^ᴬ (recall, representing all ways into A) into F.

What this means is that knowing a single element u ∈ F(A) is nothing less than knowing an entire, category-wide dynamical process, a natural transformation Φ_ᵤ.

This process Φ_ᵤ provides a globally consistent rule translating every possible map into A (an element of h^ᴬ(X) for some X) into a corresponding structure within F (an element F(g)(u) in F(X)).

That seemingly localized element u is the global transformation Φ_ᵤ.

The point contains the entire map; the local is the global process.

That’s the profound equivalence.

Concrete Nonsense

You may think “Curt… Come on, man. You’re good looking and all but this still sounds like high-altitude category theory, disconnected from the grit of physics and reality.” Well, thank you for the totally non-contrived flattering but let’s think about it: How do you define something like the “space of all possible field configurations” on spacetime? Or a superspace with anticommuting coordinates for fermions? Or an orbifold spacetime with singularities?

As Urs Schreiber emphasizes, physicists often work implicitly as if these spaces exist, manipulating things locally on charts. Topos theory, built on the Yoneda principle, provides the rigorous framework.

A generalized space X is defined by how it’s probed by simpler charts C (it’s literally the functor X: C ↦ Plt(C, X)).

Now, bring in the lemma: F(A) ≅ Nat(h^ᴬ, F).

Let F be some physical structure defined over all probes (perhaps the configuration of a background gauge field, or some observable property). F(A) is the specific value or state of that structure when measured or observed using probe A.

The lemma says this local value is equivalent to knowing Φ_ᵤ, the complete, consistent set of rules governing how any map into A interacts with the global structure F. It connects local physics data (the state u at probe A) to global consistency rules (the natural transformation Φ_ᵤ).

Physics needs laws that hold universally, not just locally.

By analyzing the natural transformations and dualities (adjunctions) that exist between these functorial spaces within the “topos of physics,” structures such as the superpoint, and even the constraints of 11D supergravity, emerge logically.

The Yoneda framework, where objects are defined by their probe interactions (A ≅ h^ᴬ), is what allows these structures to be identified and their properties derived from fundamental principles, rather than being put in by hand.

So, even the messages you write to your friend, to your lover, or to your parent who passed away… the ones you wrote but never sent… they’re there in the imprint of you. They’re inscribed in how you interface with the world.

You don’t need the explicit. Paradoxically, the implicit carries the explicit.

This isn’t to say you’re defined by everything but that you’re dual to everything else. That’s different. And everything else also has that same contingency.

All is defined with respect to one another, up to “isomorphism”, in a sense.

And this is birthed from Yoneda’s lemma.

I want to hear from you in the Substack comment section below. I read each and every response.

—Curt Jaimungal

PS: This isn’t the whole story. There are limitations. I’ll explain that in an upcoming post, so feel free to subscribe to see the limitations of these (supposed) conclusions.

PPS: Please do consider becoming a paying member on this Substack. This is how I earn a living, as I’m directly reader-supported. Moreover, you’ll get a slew of exclusive content. If you like the “free” content, you’ll love the members-only content. Anyhow, thank you dearly either way.

PPPS: There are more cells than just 2-cells. In fact, you can even have “infinity categories.” Here’s a lecture explaining this concept with Emily Riehl.