https://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi ]]>

Recently we had discussion of "subculture"; problems of communication between category theorists and other mathematicians, and between strains of category theorists like those on the interface with homotopy theory and those in pure category theory, or those on the interface with algebraic geometry and so on; and even between different schools and practitioner circles like Canadian/American (MacLane, Lawvere, Joyal...), Australian (Kelly, Street), French ([[Charles Ehresmann]]/Benabou), French (Grothendieck), French (Simpson, Toen et al.), Russian (Beilinson and followers, Kontsevich and followers), John Baez "circle", Georgian school (Janelidze, Jibladze...), *n*lab...

I have moved a query box discussion from [[TAC]] entry to here:

]]>[[Urs Schreiber]]: Lately I have been wondering what will be happening to this unpopularity of category theory among AMS in light of recent developments. Before long and opposing category theory in math will be a bit like opposing the use of complex numbers (there was a time when that was vehemently opposed by some, too) and currently the impetus of this development comes notably from US researches, and there notably from AMS grantees.

[[Zoran Škoda]]: I recall from 1990s that the whole Grothendieck school was at that time very unpopular in US, and recall mean jokes about Grothendieck (like "the French mathematician whose only example of a big theory was a circle") and even recall being scorned by an algebraist because of using schemes in a discussion; the popularity of stacks (and similar notions) in recent mathematical physics changed the balance in the geometric part of the story since then. But I am not optimistic that so soon we could see more general change beyond central parts of pure mathematics and formal mathematical physics. I mean, we do see the huge coming influence of category theory in central parts of pure mathematics (algebra, topology, modern geometry), but not much in most of analysis, including so popular PDEs as well as in stochastics and probability; then influence in one direction of mathematical physics, but I would not say in mainstream theoretical physics (to name few major areas say highTc superconductivity, perturbative analysis of standard model and its extensions, turbulence theory, plasma physics and so on) either. Combinatorics is in between, it is huge area where most often deep knowledge of categories will not help radically, but there are so many examples of wonderful cooperation. We should keep in mind that modern geometry, algebra and topology while so central to us they are not nearly a half of an average department in math, the rest being mathematical biology, financial math, PDEs, ODEs, probability, algorithms, complexity theory, operator spaces, metric geometry, complex systems, game theory and so on...

New entry history of mathematics and a couple of minor changes at philosophy.

]]>Discussion about historical influence of MacLane’s CWM textbook from a query box in *n*lab is transferred here. Toby, Todd or Mike, if you want it back please rollback the entry. I put the link to the original entry.

The reaction is to

Categories Workis the standard reference for category theory, and we may often cite it here. Almost all of its terminology is widely adopted, although strangely its approach to foundations is not.

Discussion:

*Todd asks*: In what sense do you mean “strangely”? Do you mean that it’s strange that more categorists don’t like Mac Lane’s approach because it deserves better, or that it’s strange that Mac Lane missed here when he so often got things right? Or something else?

*Toby replies*: I mean something more neutral: It's strange that people don't follow him here, when they follow him on so much else.

Mike Shulman: I’m curious why you say that people don’t follow him. Mac Lane’s approach to foundations is, I believe, the assumption of one Grothendieck universe. Most non-category-theorists who I talk to, and many category theorists, don’t really make any explicit foundational choice (perhaps partly because of ignorance of the options); thus I don’t think they could be said either to follow *or* to not follow any particular approach. And of the category theory I’ve read which does make reference to foundations, one universe seems to be a fairly common assumption.

*Toby*: Maybe I'm thinking of less sophisticated references than you are, but it seems to me that when people just dash off a definition of category, trying to make sure that it's correct but otherwise not giving much thought to foundations, that they almost invariably adopt the set/class distinction of NGB set theory. And this is equivalent to ZFC, weaker than ZFC plus one uncountable Grothendieck universe. Someone taking more care with foundations might well prefer to assume an uncountable Grothendieck universe (or equivalently an uncountable inaccessible cardinal), which makes things easier to work with, but I don't see it otherwise. (I suppose that now we should hunt down specific references to make our respective points.)

*Mike*: I don’t really think I ought to spend the time hunting up (or down) references on such a not-very-consequential point. Now that I think about it, you’re right that many people do think of large categories as proper classes. But I think there are also enough of people who like to think of at least one universe (and usually need no more than one). Maybe we could leave it at “some do, some don’t”?

*Toby*: I'll agree to that, although I stand by the ’not’ that modifies ’widely adopted’ in this case. If you think that you know how you'd like to phrase it, please go ahead.