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Applied Category Theory #258

Open ajschumacher opened 3 years ago

ajschumacher commented 3 years ago

thought this was kind of funny, but maybe not really true: "Category theory is kind of emo; you ask “what’s a category?” and it says “you wouldn’t understand”."

ajschumacher commented 3 years ago

Some quotes on Category Theory in Halmos are kind of interesting...

ajschumacher commented 3 years ago

Preface

"The basic idea of category theory—which threads through every chapter—is that if one pays careful attention to structures and coherence, the resulting systems will be extremely reliable and interoperable." (page x)

"Readers become students when they work through the exercises; until then they are more tourists, riding on a bus and listening off and on to the tour guide." (page xi)

Might mean "mere" instead of "more"...

Author David Spivak has done some work funded by the Air Force Office of Scientific Research that relates to the book:

The book was developed, it seems, in part through a class involving forums hosted by the Azimuth Project:

"The Azimuth Project is a group effort to study the mathematical sciences for “saving the planet.” Our goal is to make clearly presented topic literature, and to help people work together in study and research efforts."

ajschumacher commented 3 years ago

Chapter 1. Generative effects: Orders and Galois Connections

"observing a complex system is rarely enough to predict its behavior because the observation is lossy." (page 1)

"For example, think of X as the subject of an experiment and Y as a meter connected to X, which allows us to extract certain features of X by looking at the reaction of Y." (page 1)

Hasse diagrams are named after Helmut Hasse.

Equation 1.3 (page 5) is a diagram with no letters or numbers. Fun!

"In mathematical logic, false implies true but true does not imply false. That is “P implies Q” means, “if P is true, then Q is true too, but if P is not true, I’m making no claims.”" (footnote 2, page 6)

I got slightly twisted up because I hadn’t thought about this in a while...

Recall that “implies” does not mean “causes” or “proves.” The statement false implies true is true, but is essentially “making no claims,” as they say.

This is different from having a contradiction P ^ ¬P, which can prove anything. Following Rhoades:

The equivalence between equivalence relations and partitions and surjections is pretty neat.

They introduce on page 13 a fun empty semicolon (I’m not sure how to reproduce it) which behaves like a pipe operator: g(f()) as f ; g.

poset: partially ordered set, a skeletal preorder (no distinct but equivalent members)

Definition 1.31 of a graph is fun: G = (V, A, s, t) but Vertices and Arrows are just elements, with s and t functions that map each Arrow to its Source and Target.

"The general Yoneda lemma is a powerful tool in category theory, and a fascinating philosophical idea besides." (page 20)

Hmm! What's this, then?

It's "arguably the most important result in category theory" according to WikiPedia, eh? Hmm...

Daggers and skeletons, oh my!

"Show that a skeletal dagger preorder is just a discrete preorder, and hence can be identified with a set." (page 21 )

"Perhaps this justifies the terminology: the joining of two sets is their union, the meeting of two sets is their intersection." (page 25)

"The inequality implies that we see something when we observe the combined system that we could not expect by merely combining our observations of the pieces. That is, there are generative effects from the interconnection of the two systems." (page 26)

"It is one of the pleasures of category theory that adjoints so often turn out to have interesting semantic interpretations." (page 33)

ajschumacher commented 2 years ago

now reading Category Theory for the Sciences...