Kerodon

[Typo:] "... then the evaluation of $C^{\cdot}$ on the morphism $\delta_i$ determines a map ..."

[Typo:] "Notation 1.1.1.7. For every nonnegative integer $n$ and every integer $i$ with $0 \leq i \leq n$, we let $\sigma^i : [n+1] \to [n]$ denote the function given by the formula ..."

[Typo:] "Remark 1.1.1.10. ... we have canonical bijections $(\operatorname{lim}_{C \in \mathcal{C}} S_{\cdot}(C))_n \simeq \operatorname{lim}_{C \in \mathcal{C}} (S_n(C))$."

Yep; thanks for these!

There is an extra parenthesis in the second equation, right after the $n$: $$(\varinjlim _{C \in \operatorname{\mathcal{C}}} S)_{n}(C) \simeq \varinjlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ) \quad \quad (\varprojlim _{C \in \operatorname{\mathcal{C}}} S)_{n})(C) \simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ).$$

Notation 1.1.1.1 seem to exclude $[0]$. Am I missing something?

Whoops. Yes, the scope of the definition should include [0].

It's not clear to me whether Δ should include [0], since it is said to be equivalent to the category of non-empty finite linearly-ordered sets, and [0] is presumably empty.

OK, I like the idea of a new foundation for mathematics -- I'd found set theory to be a bit lacking as a basis -- but now I'm completely lost from the word go. Can someone recommend prerequisite reading on category theory? I'm not even sure I fully understand what a "category"' means generally in mathematics, let alone thse infinity categories, and I certainly don't understand some of the other terms being used.

[0] denotes the set {0}, which has one element.

"Can someone recommend prerequisite reading on category theory?"

There are several books for beginners. The ones I like are https://www.goodreads.com/book/show/2047855.Category_Theory and https://www.goodreads.com/book/show/22108484-basic-category-theory.

Typo: On the RHS of equation (1.10) in the proof of Proposition 1.1.1.17 the upper and lower indices are swapped.

Yep; thanks!

Typo: It seems that equation (1.3) have the upper and lower indices swapped.

Yep. Thanks!

It looks like there are two Section 1.1.1's: this one and Subsection 04ZB. Is this intentional?

Not sure what the issue is there; the replication doesn't appear in the pdf.