# Kerodon

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

## Comments on Section 1.1

Go back to the page of Section 1.1.

Comment #10 by Tim Holzschuh on

In the definition of $|\Delta^n|$ there is a small typo: $(t_0,...,t_n) \in [0,1]^{n+1}$, not $(t_0,...,t_n)\in [0,1]$.

Comment #11 by Kerodon on

Yes, that sounds better. Thanks!

Comment #111 by Emily Pillmore on

Currently the definition of the faces of an n-simplex is the finite collection

$(d_i \sigma)(t_0,...,t_{n-1}) = \sigma (t_0,...,t_{i-1},0,t_i,...,t_{n-1})$.

Shouldn't this be the following?

$(d_i \sigma)(t_0,...,t_{n-1}) = \sigma (t_0,...,t_{i-1},0,t_{i+1},...,t_{n-1})$

Sorry for bothering if not! Thanks for all your hard work.

Comment #112 by Emily Pillmore on

Currently the degeneracy maps are defined to be the set of singular $(n+1)$-simplices determined by a singular $n$-simplex $\sigma : |\Delta^n| \rightarrow X$ given by the formula:

$(s_i \sigma) = \sigma (t_0,t_1,...,t_{i-1}, t_i + t_{i + 1}, t_{i + 2},...,t_{n+1})$.

Shouldn't these degeneracy maps be a map defined thusly?

$s_i : Sing_n(X) \rightarrow Sing_{n+1}(X) : \sigma \mapsto \sigma(t_0,...,t_{i-1}, t_i, t_i, t_{i + 1},...,t_{n+1})$

Thanks, Emily

Comment #113 by Kerodon on

I don't think so; those constructions don't look like they will preserve the property that the coordinates will add up to 1. (Another description of the n-simplex is that it's the set of n-tuples (s_1, ..., s_n) satisfying 0 <= s_1 <= ... <= s_n <= 1; this is related to the description on the page by setting s_i = t_0 + t_1 + ... + t_{i-1}. If you use these new coordinates, then the formula for the face map will look like your proposal for the degeneracy map.)

Comment #114 by Emily Pillmore on

Ah, I see my mistake. Thank you!

Comment #218 by Peng DU on

I'm on the road of a sketch of reading this Kerodon website, and found some small issues in §1.1.

1.I have a small issue about the convention in https://kerodon.net/tag/000Y. I think we'd better not make sense of the symbol Λ^0_0. For if we let Λ^0_0=∅, then consider the Kan fibration ∅→Y for any non-empty Y, it can't have lifting for Λ^0_0=∅→Δ^0 (of course, Λ^0_0=∅→Δ^0 isn't a trivial cofibration, my convention is that all the inclusion of horns Λ^n_i↪Δ^n should be trivial cofibrations).

1. The diagram in Remark 1.1.5.3 https://kerodon.net/tag/001G, represents a directed graph with three vertices and 5 edges.

3.Before Notation 1.1.8.7, there is the paragraph:"It is possible to deduce Proposition 1.1.8.4 and Corollary 1.1.8.5 in a completely formal way from Lemma 1.1.8.6, since every simplicial set can be presented as a colimit of simplices (see Proposition 1.1.6.18 below)." But Proposition 1.1.6.18 doesn't exist.

4.In proof of Proposition 1.1.8.22 https://kerodon.net/tag/00H5, there is no vertical left arrow in the commutative diagram.

There are also:

• 12 comment(s) on Chapter 1: The Language of $\infty$-Categories

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