Summer REU 2025 - Project arrow

Project summary

One of the most important results in category theory is Yoneda's Lemma. This lemma can take a bit to digest, but as a first approximation we can think of it as the foundation of the philosophy "It's all about the arrows." In short, Yoneda's Lemma tells us that an object in a category is completely characterized by the arrows to (or dually, from) that object.

We are going to take that philosophy to the extreme, by attempting to recharacterize "everything" in terms of arrows. While that sounds like a daunting task, we're going to approach it in a casual, relaxed way, working on concepts and ideas however we like. Some things, like encoding a "commutative diagram" in some category into a single arrow (in a different category) will be remarkably easy. Other things, like dealing with ideals of rings, will (likely) be much trickier. No worries! Our goal is simply to get this process started and see how far we can get this summer.

Meeting notes

Task list

Learn the basics of category theory: fategories, Functors and Nnatural transformations

• In Categories for the Working Mathematician:
  • Read Sections 1.2-1.4
  • Try the following exercises:
    • 1.3: 2, 3c, 4
    • 1.4: 1, 3, 6
• In The Joy of Cats, read/skim:
  • Section 1
  • Sections 3.1-3.3, 3.17-3.20
  • Sections 6.1-6.4

Categorifying the notions of injective and surjective

• In Categories for the Working Mathematician:
  • Read Section 1.5 and the notes at the end of Chapter 1
  • Try the following exercises:
    • 1.5: 1-6
• In The Joy of Cats:
  • Read/skim through Chapters 5, 7 and 8

Yoneda's Lemma: The origin of the motto "It's all about the arrows!"

• Reading through the following notes, which slowly build to Yoneda's Lemma:
  • Universal Properties I - Inspiring Examples
  • Universal Properties II - Commutative diagrams, cones and limits
  • Universal Properties III - Yoneda's Lemma

Using Yoneda's Lemma as a guide

• Suppose $F,G:C\to \mathbf{S}\mathbf{e}\mathbf{t}$ are two functors from a given category $C$ to the category of sets, and suppose $\tau :F\Rightarrow G$ is a natural transformation. Propose a "natural" (pun intended) definition for $\tau$ to be called a natural injection.
• Suppose $a,b\in C$ are two objects in a given category $C$, and $f:a\to b$ is an arrow in $C$. The functor $H$ should provide a natural transformation $H_f:H_a\Rightarrow H_b$. Describe this natural transformation, i.e., for every object $c\in C$, describe the set map $H_a(c)\to H_b(c)$. (Hint: Write down what those sets are.)
• Using your proposed definition for natural injection, show that $f:a\to b$ is a monomorphism in $C$ if and only if the corresponding natural transformation $H_f:H_a\Rightarrow H_b$ is a natural injection.

The team

Robert Easton

rweaston@calpoly.edu

Liya Golden

ligolden@calpoly.edu
!100

Lena Nezamzadeh

lnezamza@calpoly.edu
!100

References

Adamek-Herrlich-Strecker - The Joy of Cats
Mac Lane - Categories for the Working Mathematician