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Stream: theory: algebraic topology

Topic: 2-morphisms

Leopold Schlicht (Nov 15 2021 at 19:39):

Let $S_\bullet$ be an $\infty$-category. What is the definition of a 2-morphism in $S_\bullet$? I'm sure it's not the same as a 2-simplex in $S_\bullet$, since then the 2-morphisms in (the nerve of) an ordinary category would be sequences of morphisms $A\to B\to C$, whereas an ordinary category should have no 2-morphisms at all.

Mike Shulman (Nov 15 2021 at 20:24):

An ordinary category should have only identity 2-morphisms. A sequence of morphisms $A\xrightarrow{f} B\xrightarrow{g} C$ is interpreted as the identity 2-cell from $g\circ f$ to itself.

Daniel Teixeira (Nov 15 2021 at 21:12):

it's a homotopy image.png

Daniel Teixeira (Nov 15 2021 at 21:13):

(in quasicategories)

Daniel Teixeira (Nov 15 2021 at 21:13):

also see Remark 1.16.ii in Groth https://arxiv.org/pdf/1007.2925.pdf

Daniel Teixeira (Nov 15 2021 at 21:15):

the nerve only has trivial 2-morphisms, since such a 2-simplex corresponds to the data $1_y f = g$ implying $f=g$

Leopold Schlicht (Nov 18 2021 at 20:08):

Mike Shulman said:

An ordinary category should have only identity 2-morphisms. A sequence of morphisms $A\xrightarrow{f} B\xrightarrow{g} C$ is interpreted as the identity 2-cell from $g\circ f$ to itself.

Thanks! But I think the last sentence is wrong. A 2-morphism from $f$ to $g$ in the nerve of an ordinary category is a sequence $A\xrightarrow{f} B\xrightarrow{\mathrm{id}_B} B$ such that $f=g$. Hence you are right that the only 2-morphisms are the identity 2-morphisms. But an arbitrary sequence of morphisms $A\xrightarrow{f} B\xrightarrow{g} C$ (i.e., 2-simplex) is in general not interpreted as a 2-cell from $g\circ f$ to itself, as far as I can see. It's a 2-cell only in the case $C=B$ and $g=\mathrm{id}_B$. And then it's a 2-cell from $f$ to $f$.

Leopold Schlicht (Nov 18 2021 at 20:08):

Daniel Teixeira said:

also see Remark 1.16.ii in Groth https://arxiv.org/pdf/1007.2925.pdf

Thanks for the link!

Daniel Teixeira (Nov 18 2021 at 20:34):

I'm not sure if this is what you were implying but in the nerve there is a bijection between 2-simplices with boundary (g,gf,f) and those with boundary (id,gf,gf); the latter are 2-morphisms.

Daniel Teixeira (Nov 18 2021 at 20:35):

More generally, in a quasicategory the compositions are defined up to homotopy (aka up to 2-morphisms)

Daniel Teixeira (Nov 18 2021 at 20:36):

This is Proposition 1.3.4.2 at kerodon https://kerodon.net/tag/0041

Mike Shulman (Nov 18 2021 at 20:36):

It depends how you define "2-cell". If you choose to take "2-cell" to refer only to 2-simplices with one degenerate edge, then yes. But it's also reasonable to call any 2-simplex a 2-cell.

Daniel Teixeira (Nov 18 2021 at 20:41):

ok, agreed

Leopold Schlicht (Nov 19 2021 at 19:46):

Mike Shulman said:

It depends how you define "2-cell". If you choose to take "2-cell" to refer only to 2-simplices with one degenerate edge, then yes. But it's also reasonable to call any 2-simplex a 2-cell.

I haven't seen that definition yet. A 2-cell should have one domain 1-cell and one codomain 1-cell. What should be the domain and the codomain of an arbitrary triangle?

Mike Shulman (Nov 19 2021 at 20:24):

A globular 2-cell has one domain 1-cell and one codomain 1-cell. A simplicial 2-cell has two domain 1-cells and one codomain 1-cell (or, if you orient it the other way, one and two respectively).

Jonathan Weinberger (Nov 19 2021 at 20:32):

@Leopold Schlicht You might want to take a look into Eugenia Cheng, Aaron Lauda: Higher-dimensional categories

Leopold Schlicht (Nov 19 2021 at 20:43):

Thanks!

Leopold Schlicht (Nov 20 2021 at 12:06):

Mike Shulman said:

A globular 2-cell has one domain 1-cell and one codomain 1-cell. A simplicial 2-cell has two domain 1-cells and one codomain 1-cell (or, if you orient it the other way, one and two respectively).

What's the motivation for having simplicial 2-cells?

Leopold Schlicht (Nov 20 2021 at 12:27):

How are (globular) $n$-morphisms in $S_\bullet$ defined? How does one compose them? And if $f$ is an $n$-morphisms, what is the identity $n+1$-morphism $\mathrm{id}_f$?
I can't find any of these definitions in Kerodon or Higher Topos Theory, which is weird, because I thought having morphisms in arbitrary dimensions is the point of $\infty$-category theory.

John Baez (Nov 20 2021 at 12:29):

Simplicial sets have a lot of good properties, and homotopy theorists know how to do a lot of things with them! That's why they like concepts of higher category with simplicial cells.

Some other people hate 'em.