The n-Category Café

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.

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.

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.

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!