Dear Marta, Thanks very much for your detailed answer. First, I agree with you that it is wrong to equate category theory with logic and computation. I don't know whose mistake this is, but it certainly is one. However, it seems to me that people may be led to category theory, or more generally any interesting subject matter, for bad reasons --- and that this is sometimes good. (*) Category-land, as Jean B?nabou calls it, may look like a fortress from the outside. So maybe we should be softer towards people trying to make it look attractive and open. What do you think? Best wishes, Tom (*) It was John Baez, `making a fool of himself' as usual, who made me think I could enter the fortress back in 2006. I'll probably never be as fluent as native category theorists, but I nevertheless think my research has improved since then. On 06/08/2013 11:39 AM, Marta Bunge wrote:

Dear Tom,

I will do my best to answer your question writing on mi iPad - my only tool here in Greece for the summer.

The first part of the article by Julie Rehmayer is standard and I find no fault with it. It is in the second part that the author reveals her ignorance of the subject. The assertion "the theory of programming languages and the field of logic can be seen as essentially identical to category theory " is pure nonsense, even more so as it is given an outrageous name - "computational trinitarianism" even if spoken in jest.

Why is this assertion wrong? Because it distorts completely the nature and role of category theory.

Firstly, category theory is a field in itself, its main virtue being, not to compute but to better understand, simplify, unify several mathematical fields and in turn their applications. Secondly, category theory serves as a foundation of mathematics that is closer to mathematical practice than set theory ever was.

Within category theory there are, as you know, several distinct but related areas - fibered categories, algebraic theories monads and their algebraic, topos theory, model structures, 2-categories ( and reasonably beyond without falling into science fiction). These in turn have been succesfully employed in several areas in mathematics, such as algebraic geometry, homotopy theory, differential geometry and topoly, functional analysis, as well as in computer science, logic, model theory, physics, linguistics. Some of those applications have promoted further developments of the theory of categories itself.

Instead of pointing this out, the author of this article jumps to mention solely applications which are sure to impress the naive reader, such as quantum information theory, biological systems, linguistics, even music, as if the fundamental role of category theory in mathematics were not as important or even more so. In short, her report on category theory is more typical of a tabloid than of serious scientific journalism.

The last part is an instance of what I mean by doing more harm than good. Instead of directing the reader to the best texts for introducing the subject of category theory to a wide audience, she implicitly recommends a book by David Spivak, " published" in Arxives. This book. which I perused solely on account of the article by Julie Rehmeyer, promotes category theory mostly as a language for recording data bases in an efficient way. It dismisses the far superior text by F. W. Lawvere and S. Schanuel, Conceptual Mathematics, Cambridge University Press, 1997. The latter not only instructs but motivates. Even for an expert in the subject, to read it is pure pleasure. Another valuable elementary textbook is one by M. Barr and C. Wells, Category Theory for Computing Science, CRM, Third edition, 1999.

I hope that this partially answers your question.

Kind regards, Marta

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