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Stream: learning: questions

Topic: infinity naturality

Daniel Plácido (Mar 16 2021 at 14:31):

In the context of quasicategories, a natural transformation between functors $F,G:C\to D$ is a map $\eta:C\times \Delta^1\to D$ such that $\eta(X,0) = FX$ and $\eta(X,1) = GX$ for any $X\in C$. This makes sense. I'm trying to get the meaning of naturality in this context.

Denote the unique 1-simplex in $\Delta^1$ by $\varphi$. Given an object $X\in C$, I believe the right definition for the natural morphism $\eta_X :FX\to GX$ is

$$\eta_X := \eta(s_0(X),\varphi),$$

because we can calculate that $d_1(\eta_X) = FX$ and $d_0(\eta_X) = GX$.

Now I really expected that, at least when $X$ is an object of $C$, we'd have a square
image.png

That is, for any morphism $f:X\to Y$ in $C$ there are 2-simplices $\sigma$ and $\tau$, witnessing the compositions $\eta_Y\circ Ff$ and $Gf\circ \eta_X$, such that $d_2(\sigma) = d_2(\tau)$. However I really stuck here.

Reid Barton (Mar 16 2021 at 14:37):

This is really a question about the universal case, that is, what are the 2-simplices $\sigma$ and $\tau$ in the square $\Delta^1 \times \Delta^1$? (Here the first $\Delta^1$ corresponds to the morphism $f : \Delta^1 \to C$.) You have them drawn already in your diagram.

Reid Barton (Mar 16 2021 at 14:37):

I think it should be $d_1(\sigma) = d_1(\tau)$, by the way.

Reid Barton (Mar 16 2021 at 14:40):

You can write down the maps $\sigma, \tau : \Delta^2 \to \Delta^1 \times \Delta^1$ in terms of degeneracy maps and then check everything algebraically using the simplicial relations, though I think it won't be very enlightening (except to verify that the formalism works as intended).

Daniel Plácido (Mar 16 2021 at 14:58):

Reid, I understand you're saying that the natural squares come from applying $\eta$ to the square $\Delta^1\times \Delta^1\xrightarrow{(f,\text{id})} C\times \Delta^1$?

Reid Barton (Mar 16 2021 at 15:00):

Right, and then composing with $\eta$

Reid Barton (Mar 16 2021 at 15:01):

$\Delta^1 \times \Delta^1$ has various nondegenerate simplices: the four vertices, the four "edges" of the square plus the diagonal, and two 2-simplices

Reid Barton (Mar 16 2021 at 15:02):

The images of these simplices in $D$ will be the various components of your diagram ($FX$, $GX$, $\eta_X$, etc.)

Daniel Plácido (Mar 16 2021 at 15:02):

great, thanks! checking that the boundary is correct should be easy from here