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Stream: learning: questions

Topic: n-equivalence

Leopold Schlicht (Dec 27 2021 at 17:45):

In the intuitive conception of an "$\infty$-category", when should an $n$-morphism be an equivalence ($n$-equivalence)? What is confusing is that to talk about $n$-equivalences the notion of $\infty$-category should already be defined (then one can define "equivalence" coinductively). On the other hand, it seems one needs to have a preexisting notion of "equivalence" already while defining the notion of an $\infty$-category: for instance, when formulating the axiom that $h\circ (g\circ f)$ is equivalent to $(h\circ g)\circ f$ for 1-morphisms $f$, $g$, and $h$ (if one wants to define $\infty$-categories "by hand", i.e., not using simplicial sets).

Leopold Schlicht (Jan 17 2022 at 16:56):

This still confuses me.

Mike Shulman (Jan 17 2022 at 17:22):

Well, the coinductive definition of equivalence requires only the operation of composition to be defined (not its associativity or coherence), which doesn't require knowing what an equivalence is yet.

Mike Shulman (Jan 17 2022 at 17:24):

However, usually what one does is specify as part of the data just the associativity morphism, its inverse, the witnesses that would make them inverses, and all the higher coherence, etc., as primitive parts of the definition, without referring to a general notion of "equivalence". Then after the whole definition is given, one can define equivalences and observe after the fact that the associativity morphisms are equivalences.

Leopold Schlicht (Jan 17 2022 at 19:03):

Very enlightening, thanks!

Leopold Schlicht (Jan 17 2022 at 19:04):

Can the notion of an equivalence in an $\omega$-category be defined without coinduction? (I find coinduction hard to understand intuitively.)

Leopold Schlicht (Jan 17 2022 at 19:07):

Which model for $\omega$-categories do you have in mind in that blog post? (I only know models for $(\infty,1)$-categories, but in these I think one would define equivalences without coinduction. For instance, a 1-morphism in a quasicategory is called an equivalence if its homotopy class is an isomorphism in the homotopy category.)

Mike Shulman (Jan 17 2022 at 23:07):

Any model! It doesn't matter.

Mike Shulman (Jan 17 2022 at 23:08):

You can always unwind the coinductive definition of equivalence to something involving an infinite sequence of things, as for instance in this paper, but it's a lot more complicated to work with.

Leopold Schlicht (Jan 19 2022 at 12:16):

Thanks!