Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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.

pic of cover

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:

pic of stars

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

TrackBack URL for this Entry:

11 Comments & 0 Trackbacks

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 Hom(X,A)Hom(X, A) for all AA is equivalent to knowing XX. 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 CNCN of count nouns and a category MNMN of mass nouns:

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

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

\bullet 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 CNCN and MNMN, 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 CNCN and MNMN 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 a bit after the fact, but I want to say this anyway, because linguistic misunderstanding is rampant enough without it being spread by smart people!)

I haven’t read the paper, but I’m really hoping they’re not actually claiming that “dogs” is a mass noun. It’s count, just like “dog” is. Nouns fall into two major classes by numberability, count and mass, with count nouns being subdivided into singular and plural. The usual semantics given to this is that count nouns can be individuated into discrete smallest units, while mass nouns cannot. It’s a rough approximation, however, because plenty of examples can be trotted out that show that the division is not so clean. More generally, single/plural/mass is best described as a purely syntactic class issue, that often tracks some semantic properties but doesn’t have to. The standard way of figuring out which of the three a noun is is roughly as follows: if it can come after “a” and/or agrees with the verb “be” as “is”, it’s singular; if it cannot appear with “a” before it and agrees with “be” as “are”, it’s plural; if it cannot appear with “a” before it and agrees with “be” as “is”, it’s mass. So:

a dog is shaggy (grammatical) dog is shaggy (ungrammatical) a dog are shaggy (ungrammatical) dog are shaggy (ungrammatical)

so “dog” is singular.

a dogs is shaggy (ungrammatical) dogs is shaggy (ungrammatical) a dogs are shaggy (ungrammatical) dogs are shaggy (grammatical)

so “dogs” is plural.

a coffee is tasty (ungrammatical) coffee is tasty (grammatical) a coffee are tasty (ungrammatical) coffee are tasty (ungrammatical)

so “coffee” is mass.

We have to be careful here, tho, because English allows null derivations in between these, so you can get stuff like “dog is disgusting” with the understanding that you mean something like dog meat not dogs-as-animals, and conversely, “I’ll have a coffee” is understood to mean that you’re ordering a cup of coffee or something like that.

Posted by: Darryl McAdams on June 7, 2013 8:41 PM | Permalink | Reply to this

Re: Spivak on Category Theory

I agree that ‘dogs’ is a plural count noun rather than a mass noun, but I think ‘coffee’ as a count noun (meaning ‘a cup of coffee’, as you said) has pretty firmly entered the English lexicon by now. Likewise with ‘data’ as a mass noun (although I still make an effort to use it as a count noun when speaking of small numbers of data).

Posted by: Mike Shulman on June 8, 2013 2:13 AM | Permalink | Reply to this

Re: Spivak on Category Theory

This reminds me of some other fun examples. ‘Spaghetti’ I think qualifies as a mass noun, and yet originally it was a plural of ‘spaghetto’ (dim. of spago = string, twine). As in “Waiter! There’s a dead fly on this spaghetto!”).

(Looking this up online just now, I see that ‘spaghetto’ is also a colloquialism meaning ‘fright’, and the Urban Dictionary says that it can refer to spaghetti made from ramen noodles and ketchup. Nice.)

Then there are count nouns which in their singular forms were originally plural forms. The word ‘agenda’ is Latin for “things to be done”, and now is used to mean a list of things to be done (hence can be pluralized as ‘agendas’), but few people nowadays use agendum for a single item on the list.

The other day I was at the pizzeria, and as an impulse purchase, I asked for two cannoli (and that’s exactly what I said, “oh, and I’d like two cannoli”). The guy behind the counter, a native Italian speaker too, confirmed “okay, two cannolis”. I don’t think I’ve ever heard anyone in the US ask for a connolo (“little tube”) for dessert; you’d say “I’d like a connoli” instead.

Posted by: Todd Trimble on June 8, 2013 7:36 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 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.


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

Post a New Comment