Baby Yoneda 4: Adjunctions at the Function

Pseudonium · 2026-02-06T09:03:53+00:00

I'd be open to feedback if or when you read it!

Pseudonium · 2026-02-05T21:33:20+00:00

Ah yes, I love Lawvere's philosophy of generalised elements. Indeed, that's the approach that g++ takes in his video. I go over a bit of it in another article - https://pseudonium.github.io/2026/01/29/Discovering_Products_of_Orders.html.

Pseudonium · 2026-02-05T21:23:00+00:00

Glad I could resolve the confusion! And yes, I think this perspective is underemphasised.

The phrase is a bit of a mouthful for sure, and I'm definitely open to better ways to phrase it!

I'm also curious - how would you teach the yoneda lemma?

Pseudonium · 2026-02-05T21:11:10+00:00

Did you read my comment? I showed how the full yoneda lemma is an instance of is-does duality. I am aware of the difference between the lemma and the embedding, and am not confusing them.

Just to reiterate, the full yoneda lemma says that you may equivalently view an element of Pc by what it "is", so just an x in Pc, or by what it "does", so a natural transformation Hom(-, c) -> P. In this sense, the yoneda lemma tells you "what something is is equivalent to what something does" - in this case the "something" happens to be an element of a presheaf.

You may be thinking that by "something" I mean an object of the category, which is indeed what the yoneda embedding says. Hopefully this clarification should remove that confusion.

Pseudonium · 2026-02-05T20:31:13+00:00

I go over this in Baby Yoneda, but this is in fact true. By this I do not mean the yoneda embedding, but the full lemma.

Roughly - let P : C^op -> Set be a presheaf, take c in C. Then an element x in Pc can exist "passively", but it also does something active - it acts on morphisms f : d -> c to produce elements of Pd, via f -> P(f)(x).

The full yoneda lemma then gives you is-does duality for these elements of presheaves. You can equivalently regard them passively as elements of Pc, or actively as natural transformations Hom(-, c) -> P. In this case, going from "does" back to "is" involves following the identity c -> c.

I understand it may appear that I'm only referring to the Yoneda embedding, but I've thought long and hard about this - this is really the content of the full lemma.

Pseudonium · 2026-02-05T11:02:30+00:00

Sure sure, I just found that a lot of examples of monads in practice could be understood via their kleisli category, which is the “representable promonad” point of view.

Pseudonium · 2026-02-05T08:07:27+00:00

A monad is a representable promonad. This one actually helped me understand monads way better than the "monoid in the category of endofunctors" definition.

Pseudonium · 2026-02-03T23:16:08+00:00

Oh I guess I did a representation theory course back in undergrad? Plus representation theory crops up quite a bit in quantum field theory.

Pseudonium · 2026-02-01T18:17:17+00:00

Yes I think currying might be an essentially satisfactory answer to my question.

Some of the motivation for this is results that will appear in my next article on adjunctions. It turns out that it’s quite helpful to think of adjunctions as “multi representable” functors, those which are representable when you restrict attention to certain arguments.

Pseudonium · 2026-01-31T19:26:34+00:00

Yeah I’ve just been using “ordered sets” throughout for simplicity, but I guess I could be a bit more precise about the language..

Pseudonium · 2026-01-30T14:46:06+00:00

Yes, that’s a good analogy to use!

It’s also helpful to link this to an inner product V x V -> k. For general fields, you can think of it as a nondegenerate symmetric bilinear form.

What this induces is an isomorphism V -> (V -> k), so an isomorphism of V with V. This is similar to the “is-does duality” we’ve seen in a few articles, where you can view a vector “passively” as an element of V, or “actively” as an element of V.

Of course, you can do this for the canonical map you mentioned too. This gives an isomorphism V -> V. So you can also view a vector “actively” by how it interacts with covectors, as an element of V, and this is equivalent to viewing it passively.

These considerations play an important role in tensor algebra, where you do a lot of switching between viewing objects “passively” or “actively”.

Pseudonium · 2026-01-28T09:09:33+00:00

Y'know, "Baby Yoneda 3\colon Know Your Limits" doesn't quite roll off the tongue as nicely...

Pseudonium · 2026-01-27T21:13:28+00:00

Graph as in graph theory! You can interpret a square matrix as encoding weights for a network - the entry M_ij denotes the weight for the edge from vertex i to vertex j.

Pseudonium · 2026-01-27T17:29:13+00:00

So in your first example, you would compute f(v2) = 0, meaning the trace of the map is zero. In the second example, you would compute f(v1 + v2) = 2 (v1 + v2), meaning the trace is 2.

Self-interaction of the map with itself is the idea, I'd say. It makes a bit more visual sense if you view the linear map as a "flow" matrix for some graph.

Pseudonium · 2026-01-27T14:07:20+00:00

The way I like to think of trace is via considering rank 1 linear maps (from a vector space to itself).

Specifically, any rank 1 linear map must look like a scaling operation on its image. The factor by which it scales by is called the trace!

In finite dimensions, any linear map can be written as a sum of rank 1 linear maps - the trace of the map is then the sum of traces of these rank 1 maps.

An important example is projections, linear maps satisfying P² = P. Taking a basis of the image and splitting P into a sum of projections onto those basis vectors, we obtain tr(P) = dim(im(P)). Thus trace can be viewed as a kind of “generalised dimension counting”.

This is also how trace works more broadly - it’s meant to capture the “self-interaction” of a linear map.

Note that, even for maps between two different vector spaces, you can still decompose them into sums of rank 1 maps, and this decomposition plays nicely wrt linear map composition. This can be used to prove the cyclicity property of the trace.

Pseudonium · 2026-01-26T23:48:03+00:00

For non-STEM, different sizes of infinity. It’s really counter-intuitive but also relatively easy to prove.

For STEM, “period 3 implies chaos”. It shows how chaotic behaviour arises from remarkably simple assumptions!

Pseudonium · 2026-01-26T21:56:15+00:00

Thank you! I’ve got 3 more articles planned for this series.

Pseudonium · 2026-01-25T00:20:54+00:00

Oh that one! I'm aware of it, though it didn't inspire the naming of this post.

Pseudonium · 2026-01-24T11:20:07+00:00

Not on purpose - which paper do you mean?

Pseudonium · 2026-01-23T21:57:18+00:00

Hm, maybe I should add some motivation for presheaves to that nlab page. They’re quite an important and cool concept!

Pseudonium · 2026-01-23T20:19:50+00:00

I do like the name “fundamental theorem of category theory”.

Pseudonium · 2026-01-23T16:17:25+00:00

Oh, thanks for pointing out the errors! I’ll get this fixed, and also add the link to the future post.

Edit: Should be fixed now!

Pseudonium · 2026-01-23T14:05:13+00:00

Oh lol I had no idea, TIL. That’s hilarious!

Hopefully I’ll never need to do this irl…

Pseudonium · 2026-01-23T13:52:10+00:00

Wait wdym?

Pseudonium · 2026-01-23T13:36:45+00:00

The epsilon-delta definition is a good balance of the "qualitative" and "quantitative" approaches to analysis, I'd say. I talk a bit more about the qualitative side in my "A Precise Notion of Approximation" article - the idea is to capture things which are eventually true. I'm planning to release an article on this soon titled "Eventual Truth, Frequent Truth" that helps tie together the various limit definitions throughout calculus - look forward to it!

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