## March 3, 2013

### Spivak on Category Theory

#### Posted by Simon Willerton

Guest post by Bruce Bartlett

We know about Category Theory for Mathematicians, we’ve all read Category Theory for Physicists, and we also know about Category Theory for Computer Scientists, and we’ve even seen the videos.

But how about Category Theory for Scientists? I spotted this on the arXiv listings.

David Spivak, Category Theory for Scientists.

Abstract: There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity.

I’m afraid this little post is just a shout-out as I’ve only hurriedly browsed through the pages.

Towards the end of the book he gets to sheaves; he is certainly an expert on these as his PhD thesis was on derived smooth manifolds). His motivating example is stitching together pictures of the night sky, which I thought was really cool:

Paging through, I see the Yoneda lemma only gets a small paragraph, with a reference to Mac Lane. I’m kind of sad about that, since I do regard it as the fundamental theorem of category theory. Too bad.

Posted at March 3, 2013 10:29 PM UTC

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### Re: Spivak on Category Theory

Thanks for the write-up!

As for Yoneda, your peer pressure has worked! I’ll add more about it in the next version.

Posted by: David Spivak on March 4, 2013 9:00 PM | Permalink | Reply to this

### Re: Spivak on Category Theory

Do you want to explain why you might refer to it as the Ubuntu-Yoneda Lemma, Bruce? (As opposed to Lawvere referring to it as the Cayley-Dedekind-Grothendieck-Yoneda Lemma.)

Posted by: Simon Willerton on March 4, 2013 9:45 PM | Permalink | Reply to this

### Re: Spivak on Category Theory

Okay, here goes. IsiXhosa has a proverb, umntu ngumntu ngabantu. In this sentence, um = “a”, ntu=”person”, ngu=”is”, nga=”through”, abantu=”people”. So it means literally a person is a person through other people. (Disclaimer: novice at large).

Academics translate the meaning of this proverb as “attaining the totality of being a fully adjusted member of society only through the support, counselling, love, assistance, shelter, example, etc. of one’s fellow human beings”.

I see it as a more elegant version of the Yoneda lemma. A thing is a thing only in the way that it relates to other things. Knowing $\mathrm{Hom}\left(X,A\right)$ for all $A$ is equivalent to knowing $X$. No man is an island. You exist only through, and you are completely determined by, your connections with others. You are nothing more than the sum of your relationships. That kind of vibe.

By the way, the word ubuntu, of Linux fame, is closely related. Ubu=abstract noun prefix, so it literally means “person-ness”, or common human decency. People write essays about it, as in the Wikipedia link, since it is a central theme of African culture.

Posted by: Bruce Bartlett on March 4, 2013 10:20 PM | Permalink | Reply to this

### Re: Spivak on Category Theory

This is fantastic.

Posted by: Emily Riehl on March 6, 2013 4:31 AM | Permalink | Reply to this

### Re: Spivak on Category Theory

Ok, great. Well done on the book.

Posted by: Bruce Bartlett on March 4, 2013 10:24 PM | Permalink | Reply to this

### Re: Spivak on Category Theory

I’m sure a lot of people at the Cafe would agree that the “categorical stance” has the potential to transcend academic disciplines. But actually identifying how to realize this potential is another story. It’s really exciting to see the dialogue advancing on this front!

Spivak’s “olog”s (as in “ontology log”) reminded me of the categorically-motivated work of Reyes et. al. on grammar. For instance, this article (here’s the direct link) analyzes the relationship between count nouns (a man, an amino acid,…) and mass nouns (water, arginine,…) in terms of an adjunction between a category $\mathrm{CN}$ of count nouns and a category $\mathrm{MN}$ of mass nouns:

$•$ The morphisms in these categories are relationships like “a man is a human,” resp. “water is liquid.”

$•$ The left adjoint is pluralization: if “a dog” is a count noun, then “dogs” is a mass noun.

$•$ The right adjoint is less familiar grammatically, but for example, if “water” is a mass noun, then “a body of water” is a count noun.

Like Reyes’s categories $\mathrm{CN}$ and $\mathrm{MN}$, Spivak’s “olog”s form a category whose objects are essentially “real-world types”. The morphisms are also related: the “is a” morphisms of Reyes’s categories $\mathrm{CN}$ and $\mathrm{MN}$ are important examples of Spivak’s “aspects” of ologs (although the latter are more general).

I wonder if Reyes’ analysis might shed light on Spivak’s convention of using e.g. “a man” to denote the set of all men, and his “rules of good practice” for olog notation more generally?

Even Reyes’s diagrammatic notation is very similar to Spivak’s: the objects are denoted by English words in text boxes, and the morphisms are arrows between these.

Posted by: Tim Campion on March 5, 2013 6:49 AM | Permalink | Reply to this

### Re: Spivak on Category Theory

This is very nice! I’ve added it in the “Textbooks” section of the nlab page on category theory.

One minor quibble: I wouldn’t say that the alternative definition of a category in 4.1.1.17 is any “more formal” than the preceeding one. It’s just different; both are equally formal.

Posted by: Mike Shulman on March 5, 2013 2:47 PM | Permalink | Reply to this

### Re: Spivak on Category Theory

By the way, there’s a google doc up on the web if anyone here finds typos or has other comments or suggestions regarding the book. For example, Mike Shulman’s comment above has been implemented in the latest version. Obviously I don’t promise to implement every suggestion, but I do promise to think about it.

Thanks!

Posted by: David Spivak on April 24, 2013 2:26 PM | Permalink | Reply to this

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