The n-Category Café

In Weeks 101–150 I focused strongly on topics connected to particle physics, quantum gravity, topological quantum field theory, and $n$-categories. However, I digressed into topics ranging from biology to the fiction of Greg Egan to the game of Go. I also explained some topics in homotopy theory in a series of mini-articles:

• A. Presheaf categories.
• B. The category of simplices, $\Delta$.
• C. Simplicial sets.
• D. Simplicial objects.
• E. Geometric realization.
• F. Singular simplicial set.
• G. Chain complexes.
• H. The chain complex of a simplicial abelian group.
• I. Singular homology.
• J. The nerve of a category.
• K. The classifying space of a category.
• L. $\Delta$ as the free monoidal category on a monoid object.
• M. Simplicial objects from adjunctions.
• N. The loop space of a topological space.
• O. The group completion of a topological monoid.

You can reach all these mini-articles from the introduction.

One annoying thing is that I now move in circles where it feels like all this stuff is considered obvious. When I was first learning it, I didn’t feel that everyone knew this stuff — so it was exciting to learn it and explain it on This Week’s Finds. Now I feel everyone knows it.

So, I have to force myself to remember that even among the mathematicians I know, not all of them know all this stuff… so it’s worth explaining clearly, even for them. And then there’s the larger world out there, which still exists.

I think what happens is that when scientists start discussing technical concepts like ‘group completion’ or ‘heterochromatin’, they scare away people who don’t know these terms — and attract people who do. So, without fully realizing it, they become encased in a social bubble of people who know these concepts. And then they feel ignorant because some of these people know more about these concepts than they do.

This phenomenon reminds me of the hedonic treadmill:

The process of hedonic adaptation is often conceptualized as a treadmill, since no matter how hard one tries to gain an increase in happiness, one will remain in the same place.

I think this phenomenon is especially strong for people like me, who roam from subject to subject rather than becoming an expert in any one thing. These days I feel ignorant about particle physics, homotopy theory, higher categories, algebraic geometry, and a large range of other topics. Whenever I blog about any of these things, some expert shows up and says something more intelligent! It tends to make me scared to talk about these subjects, especially when I know enough that I feel I should know more.

I fight this tendency — and I’m admitting it now to help myself realize how silly it is. But it’s funny to look back to my old writings, where I had the brash self-confidence of youth, and hadn’t yet attracted the attention of so many experts.

It’s also funny to think about how these scary ‘experts’, who I may picture as vultures sitting on nearby trees waiting to swoop down and catch any mistake I make, are actually people eager to be admired for their knowledge, just like me.