I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.

  1. In Proposition, Lemma is used, but I don't see how $\mathcal{D}^{/F(x)}$ matches the hypotheses thereof, in that it is not at all obvious to me that it be $\tau$-filtered and, moreover, that it admit $\tau$-small $\kappa$-filtered colimits, both of which are required in Lemma
  2. Lemma uses Lemma, but this in turn requires that the simplicial set in the middle be weakly contractible, which is not in the case at issue. Is there a way to obviate this, or an alternative proof altogether to Lemma
  3. In Lemma (and dually later on, in Corollary, the square

$\require{AMScd}$ \begin{CD} \mathcal{C}^{p/} @>>> \text{Fun}(K \times \Delta^1, \mathcal{C})\\ @V V V @VV V\\ \ast @>p>> \text{Fun}(K \times \{0 \}, \mathcal{C}) \end{CD}

is declared to be a pullback, which puzzles me, in that while it expresses the condition that the diagram restricted to $K \times \{ 0 \}$ is $p$ it says nothing about the restriction to $K \times \{ 1 \}$, which should be constant.

Answers to some or all these problems would be really appreciated.

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  • 2
    $\begingroup$ My experience is that there are a huge number of mistakes in HTT (but remarkably, none of them seem to be essential), and proofs should be read as suggestions. Here are some comments about your specific questions: $\endgroup$ – dhy Jan 10 at 18:23
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    $\begingroup$ I believe 3.) is actually easily fixed. Simply replace the bottom arrow with $\mathcal{C}\rightarrow\operatorname{Fun}(K\times\{0\}\amalg K\times\{1\},\mathcal{C}).$ On the other hand, I got seriously stuck for a few days on your issue #2. I agree with you that the proof as written does not work, but the result still holds with a different + substantially harder proof (I think you may also need to assume some condition on the cardinalities) - let me try to remember it. I don't immediately recall any thoughts on issue 1.) $\endgroup$ – dhy Jan 10 at 18:29
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    $\begingroup$ Regarding (1), the categorical thinking is as follows: $D_{/d}$ has $\tau$-small $\kappa$-filtered colimits for any $d \in D$ because $D$ does, and in a slice category, colimits are computed levelwise. It is $\tau$-filtered because $C$ is $\tau$-accessible, so every object $d$ is a $\tau$-filtered colimit of objects of $D$, and a cofinality argument then shows that the canonical colimit, indexed by $D_{/d}$, is $\tau$-filtered -- the 1-categorical version of this argument is Prop 1.22 in Adamek and Rosicky, Locally Presentable and Accessible Categories. $\endgroup$ – Tim Campion Jan 12 at 4:33
  • $\begingroup$ $\mathcal{D}$ is actually the essential image of the Yoneda embedding $\mathcal{D}' \to \text{Ind}_{\kappa}(\mathcal{D}')$, where $\mathcal{D}'$ is a not further specified small $\infty$-category. It doesn't admit colimits in general. $\endgroup$ – Giulio Lo Monaco yesterday

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