# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Go back to the page of Subsection 1.1.2.

Comment #35 by Daniel on

"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 $\Lambda^0_0 = \emptyset = \partial \Delta^0$.

Comment #36 by Daniel on

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

Comment #45 by Kerodon on

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

Comment #285 by Stephen L on

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....

Comment #286 by Stephen L on

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...)

Comment #402 by Nicholas Mertes on

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 $\delta^0, \dots, \delta^{i-1}, \delta^{i+1}, \dots, \delta^n$.

There are also:

• 11 comment(s) on Chapter 1: The Language of $\infty$-Categories
• 7 comment(s) on Section 1.1: Simplicial Sets

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).