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Madeleine Birchfield (Nov 18 2024 at 12:28):fosco said:
Which is also what takes aback many mathematicians who believe building language and streamlining ideas is grunt work, and we should instead single-handedly solve <big name>'s conjecture.
Why do many mathematicians have this attitude in the first place?
fosco (Nov 18 2024 at 18:48):Madeleine Birchfield said:
fosco said:
Which is also what takes aback many mathematicians who believe building language and streamlining ideas is grunt work, and we should instead single-handedly solve <big name>'s conjecture.
Why do many mathematicians have this attitude in the first place?
Hah, good question. I think it comes, in part, from the image of mathematicians pictured by the media. Related meme:
Mike Shulman (Nov 18 2024 at 21:27):I agree that the image of mathematicians in the media is often misleading, but my own experience has been that most professional mathematicians are well-aware of this, and are also aware that big conjectures are rarely solved entirely by eccentric geniuses working in isolation but instead by putting together small contributions from many people, and moreover that solving big conjectures is not the only or even the primary goal of mathematics. I feel like valuing small contributions is an orthogonal question to valuing language-building.
Mike Shulman (Nov 18 2024 at 21:29):(I do agree that language-building is undervalued in mathematics in general.)
John Baez (Nov 18 2024 at 21:52):I think rather few mathematicians are actually working to solve "big conjectures". But big conjectures do serve as a quick way of answering the question "what are you trying to do and does it really matter?" Not a very good way... but for example:
There's a lot of research connected to elliptic curves, modular curves, automorphic forms, L-functions, and a vast array of technology connected to these things, and people working on these things can point to the $1,000,000-dollar prizes for the Riemann Hypothesis and Birch--Swinnerton-Dyer conjecture as a quick way of noting that someone cares about this stuff - much easier than actually explaining what it is, or why it matters.
John Baez (Nov 18 2024 at 21:53):This is one reason I was happy to help propound the [[cobordism hypothesis]] and more general [[tangle hypothesis]], biggish conjectures that require a pretty good theory of -categories or -categories to prove.
John Baez (Nov 18 2024 at 22:04):I encourage other folks to propound intuitively appealing but (it may turn out) hard-to-prove conjectures in category theory, both to encourage its development and to give researchers goals to point to. Language building is very important, and often you can only find the good conjectures after you've built the language in which they can be stated - but it's hard for most people to understand the virtues of language-building, while the idea of "can you prove this?" seems easier to understand.
In categorical terms "can you prove this?" is about verifying a property, while many deeper problems involve finding an interesting structure, finding interesting stuff, etc.
John Baez (Nov 18 2024 at 22:07):Note also that I called the tangle hypothesis a "hypothesis" because it was not precise enough to be a conjecture. So in fact I was calling for the development of structures, stuff etc. required for precise statement of a conjecture! But like Grothendieck's [[homotopy hypothesis]], there was a "pre-conjecture" very clearly begging to be stated more precisely.
There are definitely dangers in stating "pre-conjectures" that need to be made more precise: it's easy to get lost in vagueness. We don't want category theory to be seen as the home of pre-conjectures. So try to pose some precise, appealing yet difficult conjectures!
Emilio Minichiello (Nov 22 2024 at 13:12):Huh this just made me think, is there a list out there of open problems in category theory? I love the Grothendieck style of language building in areas like category theory and homotopy theory, but I do worry at times that we don’t pose enough conjectures in these fields (with John as a very notable exception) as noted in Barwick’s Future of Homotopy Theory. In fact I would bet much of that article applies to category theory as well.
John Baez (Nov 22 2024 at 18:08):I don't know a list of open problems in category theory. We should create one on the nLab, once we get a few really good ones.
To create good open problems, it helps to look for currently inexplicable patterns in data. But this doesn't require long hand calculations or computer calculations. For the cobordism and tangle hypothesis, the data was simply a small chunk of the periodic table of n-categories, together with some strong hints from homotopy theory. But due to how category theory involves so much "language building", these had to be "hypotheses" rather than "conjectures" - part of proving them was developing the language to state them precisely.
I bet @Mike Shulman could make up some good conjectures, or at least hypotheses. Probably other people here could do it too.
A lot of doing this is giving yourself permission to go out on a limb.
Jean-Baptiste Vienney (Nov 22 2024 at 19:22):I think I have a potentially interesting open question: Is a monoidal category such that every object is a counital comagma in a unique way necessarily a cartesian monoidal category?
The reason why I think it is interesting is that the answer is "yes" if we replace "monoidal" by "symmetric monoidal" but the proof makes essential use of the symmetry. On the other hand, I don't know how to find any counterexample without the symmetry.
Mike Shulman (Nov 22 2024 at 20:47):Does "there should be a definition of X with Y properties" count as a "hypothesis"? I mentioned one like that somewhere around here recently: there should be a construction of a double-Kleisli virtual double category from two VDC monads and a horizontal distributive law, in which monoids are a notion of "generalized polycategory".
Mike Shulman (Nov 22 2024 at 21:00):Here's a conjecture that I really wish someone would solve: the type theory of Agda can be interpreted in any Grothendieck topos. This is imprecise only insofar as "the type theory of Agda" hasn't been written down formally in mathematical language.
Mike Shulman (Nov 22 2024 at 21:22):I could probably go through my old papers and formulate more hypotheses. Here's another one: Steve Lack and I formulated a context of -categories in which we could exactly characterize the 2-dimensional limits of diagrams consisting of strict and lax morphisms for algebras over a 2-monad that can be lifted from the base category. There's a dual version for strict and oplax morphisms. But is there a context of this sort that includes both lax and oplax morphisms? There are limits of that sort that do lift, e.g. the comma object of an oplax morphism over a lax one. But (it seems that) you can't compose a lax morphism with an oplax one to get anything interesting, so it's unclear whether there's some kind of "enhanced 2-category" that includes them both. There is a double category of lax and oplax morphisms, but is there a kind of limit in double categories that would capture this?