Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition: * an equivalence in a set ($0$-category) is just an identity; * for each $n \in \mathbb N$ an equivalence between two object (or $0$-cell), let say $a$ and $b$, in a $n+1$-category is just a $1$-cell $f \colon a \to b$ such that exist a $1$-cell $g \colon b \to a$ and two $2$-cells $\alpha \colon g \circ f \to 1_a$ and $\beta \colon f \circ g \to 1_b$ which are equivalence into the $n$-categories $\hom(g\circ f, 1_a)$ and $\hom(f \circ g, 1_b)$ respectively. There is a good formal definition of equivalence also for $\infty$-category?

share|improve this question
add comment

2 Answers

up vote 7 down vote accepted

For some precise definitions and results, see this paper by Eugenia Cheng.

share|improve this answer
    
To be honest, I think that paper is seriously misleading. Her "Unsound definition" 5 is actually a perfectly sound definition; you just have to interpret it coinductively. Unfortunately, coinductive definitions don't seem to be as widely known as inductive ones. –  Mike Shulman Jun 8 '11 at 16:25
    
That's interesting. But offhand, "seriously misleading" sounds like a pretty strong way to put it, as if the paper is invalid somehow. (I believe the paper predates Eugenia's involvement with corecursive definitions of $\infty$-categories. Aside from the label "unsound definition", is there anything actually wrong with the paper?) –  Todd Trimble Jun 8 '11 at 17:57
    
I'm sorry if I came on too strong. I didn't mean to say there was anything wrong with the paper; it's a very nice paper overall! I just meant that specifically in the context of the question "how to define equivalences in ∞-categories?", I think it is a misleading reference to give. To my mind, the best definition is actually the coinductive one, which that paper asserts to be unsound. –  Mike Shulman Jun 19 '11 at 22:46
    
Ouch. :-) Could you provide a better reference, then? –  Todd Trimble Jun 19 '11 at 23:12
    
I hope that Eugenia, if she is reading this, won't take offense at my comment. The fact that coinductive definitions are meaningful seems generally to be a secret well-kept by computer scientists and rarely taught to mathematicians. I only fairly recently learned it myself. In fact I learned it by reading a coinductive definition of infinity-equivalence, thinking "that's nonsense", and going to look it up and finding out that it wasn't! –  Mike Shulman Jun 20 '11 at 3:32
show 6 more comments

A concise coinductive definition can be found, for the case of strict ∞-categories, in the paper "A folk model structure on omega-cat", arXiv. This can be unraveled in order to become equivalent to Eugenia's more explicit version.

share|improve this answer
    
This is very interesting! Presumably you can unroll the coinduction to make a definition in terms of the existence of an infinite binary tree of higher morphisms (although I don't know if this is a useful thing to do). –  S. Carnahan Jun 20 '11 at 10:20
    
Yes, as I said, this version can be unraveled into Eugenia's. Perhaps (probably?) for some applications this would be useful. –  Mike Shulman Jun 20 '11 at 22:06
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.