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

Comments on Section 1.1

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Comment #10 by Tim Holzschuh on

In the definition of there is a small typo: , not .

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


Shouldn't this be the following?

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 -simplices determined by a singular -simplex given by the formula:


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

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 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, represents a directed graph with three vertices and 5 edges.

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

4.In proof of Proposition, there is no vertical left arrow in the commutative diagram.

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

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