Kerodon

"Example 1.1.2.13. The horn $\Lambda^0_0$ coincides with the $0$-simplex $\Delta^0$."

Should $\Lambda^n_i$ really be defined for $n = 0$? Also, strictly following construction 1.1.2.9, we would have $\Lambda^0_0 = \emptyset = \partial \Delta^0$.

[Typo:] In Exercise 1.1.2.14: "$(f : \mathbf{\Lambda^n_i} \to S_{\cdot}) \mapsto ...$"

Oops; that's right, the horn of the 0-simplex should be empty. Thanks!

So I guess the idea is that though normally (coming from traditional geometry) a simplex is identified only with all faces/edges/points/etc. contained in it, here a standard n-simplex consists of all things that can be embedded in it, included ways of collapsing higher-dimensional simplices onto it. That's a lot of extra parts! Interesting to see where this goes as I read further....

Wait, I got it backwards. $\Delta^n$ is $hom(-,[n])$ can be thought of as "ways of embedding an n simplex in other simplices". And so the yoneda statement is something like

"a way of going from 'generic ways of embedding an n-simplex' to 'a concrete realization of various simplices' is the same as specifying a particular concrete n-simplex . "

(I'm trying to keep my geometric intuition alive as I work through...)

In Construction 1.1.2.9 (The Horn $\Lambda_i^n$): A remark should be added explaining that $\Lambda_i^n$ is the subfunctor of $\Delta^n$ generated by the maps $\delta^0, \dots, \delta^{i-1}, \delta^{i+1},\dots, \delta^n: [n-1]\to [n]$. Then it would be more clear why a morphism of simplicial sets $\alpha: \Lambda_i^n\to S_\bullet$ can be determined by saying where $\alpha$ sends the generators $\delta^0, \dots, \delta^{i-1}, \delta^{i+1}, \dots, \delta^n$.

The set inclusion $[n]\nsubseteq\alpha([m])\cup\{i\}$ in the construction of horns (Construction 1.1.1.9, or Tag 000U) should be replaced by $\alpha([m])\cup\{i\}\subsetneq[n]$.

What is the definition of vertex in Example 1.1.2.5?

I apologize for the previous question: I had overseen the definition of vertices in Definition 1.1.1.12.