The n-Category Café

If it weren’t $n$-categories which gripped us, would it make a difference to the way the blog works?

Maybe there is indeed a correlation between

a) the desire to see the “big picture” in the background, to understand the natural nature of the objects of your research

b) the desire to extract the $n$-categorical structure governing the objects of your interest

c) the desire to communicate ideas and share insights

Hi,

I just wanted to mention my appreciation for the existence of blogs such as this one. I meant to speak up last time there was something posted about the blog itself, but I couldn’t work up the nerve.

I’m just a 16 year old with a rather keen interest in mathematics and related areas, and I’m glad to have places like this, especially so since I’m mostly teaching myself until I’m able to get into university. Although a fair bit, perhaps even most of the material posted here goes far over my head, I can usually find some interesting detail that doesn’t, and I always leave with plenty of ideas for new things to look into. As a whole, the discussion here makes for great inspiration to keep working, so that the more complicated stuff won’t be out of reach.

So, thanks!

Since you’re talking about the goal of the blog, a question about category theory (including it’s generalisations) which I’ve been meaning to ask for a while (and which I haven’t seen directly commented on anytime since inauguration). [My first degree was in maths but it’s been a long while so forgive any mis-rememberings of details in example in the following]

In maths there’s two ends of a spectrum: “understanding why and having a framework for things” and “concrete details”. So, for example take differential equations. At one end you’ve got understanding of existance/uniqueness/stability/computability/etc of solutions and how this varies for partial d.e.’s as dimension/form/etc varies, properties of systems of eqs, etc, proofs about other properties, etc. In a sense, this is how you “understand” differential equations. At the other end you’ve got “interesting details”: knowledge of special solutions to particular equations, how far you can push perturbation around a solution of a similar system, numerical methods of solution, etc. In a lot of ways this is just as intellectually challenging as other stuff and requires clarity of thought, but they aren’t giving you a glimpse of the “‘big picture’ in the background”.

So, the question is: is category theory essentially about the framework or is it also useful in working on “the details”? I notice that virtually all the posts seem to be about how you can “view thing X in a category-theory context as…”. One of the things that seemed interesting about (what little I followed about) the categorifying geometry threads was that you might end up taking the categorical framework you’d created inspired by existing geometry and “instantiated” part of that with new concrete objects - which we don’t already have a geometrical viewpoint on - and following through to thus come up with a geometrical viewpoint on something novel, which might be useful. But maybe that was me misunderstanding things.

I suspect there’s a trivial sense in which category theory applies to “the details”, in the sense that you do everything in terms of category theory. What I’m really asking is if category theory is efficacious (in some sense)for working with “the details”.

May I suggest replacing “deflated” by “distilled”, an action that shrinks while increasing value.

Off-topic: I introduced rings yesterday in a graduate algebra class, as one-object additive categories. Not yet sure how people liked that.

Lol - my favourite part in the interview is when Urs says:

Urs: And I remember one day, I was walking home in Hamburg, and I somehow felt dissatisfied. And then, I had this idea. I said, ‘why don’t we have a new blog where we discuss these things’? And I think, before I even went home, I went to an internet café - because I don’t have internet at home -

That was the original n-category cafe! Its just like Brougham bridge - where Hamilton had his moment of inspiration about quaternions, and scrawled it on the stones.

Perhaps in years to come, there will be a similar placque outside this internet cafe in Hamburg,

Here as he walked by on the ? of ?, Urs Schreiber in a flash of genius wrote the email which founded the n-category cafe.

Please take this with a pinch of salt!

But its fun to be at the n-category cafe. (YMCA?)