# Stability of accessible $\infty$-categories under some operations

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 5.4.4.3, Lemma 5.4.4.2 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 5.4.4.2.
2. Lemma 5.4.5.11 uses Lemma 5.4.5.10, 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 5.4.5.11?
3. In Lemma 5.4.5.13 (and dually later on, in Corollary 5.4.6.7), 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.

• 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:
– dhy
Jan 10, 2019 at 18:23
• 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.)
– dhy
Jan 10, 2019 at 18:29
• 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. Jan 12, 2019 at 4:33
• $\mathcal{D}$ 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. Jan 14, 2019 at 13:50
• @GiulioLoMonaco $\mathcal C$ is $\kappa$-accessible. Then $\tau$ is chosen to be strictly bigger than $\kappa$ such that $\mathcal C$ is still $\tau$-accessible. Then $\mathcal D$ is equivalent to the full subcategory of $\mathcal C$ consisting of the $\tau$-accessible objects. $\mathcal C$ has $\kappa$-filtered colimits and the $\tau$-accessible objects are closed under $\tau$-small colimits. So $\mathcal D$ is indeed closed under $\tau$-small $\kappa$-filtered colimits. It's the fact that $\tau$ is strictly bigger than $\kappa$ which ensures that $\mathcal D'$ is not arbitrary. Jan 22, 2019 at 17:22

For (3) it really appears to be a typo and can be fixed as in the comment of dhy. For (1), as explained by Tim in the comments, there is actually no mathematical issue in the sense that $$\mathcal{D}^{/F(x)}$$ is in fact $$\tau$$-filtered and admits $$\tau$$-small $$\kappa$$-filtered colimits. The first property is by the fact that the presheaf $$\mathrm{Map}(-,F(x))$$ on $$\mathcal{D}$$ becomes representable in $$\mathcal{C} = \mathrm{Ind}(\mathcal{D})$$, which is equivalent to its unstraightening being $$\tau$$-filtered. The existence of $$\tau$$-small $$\kappa$$-filtered colimits follows as Tom explains from the fact that $$\mathcal{D}$$ can be taken to be the full subcategory of $$\tau$$-compact objects in $$\mathcal{C}$$ (this is maybe the main missing explanation in the proof as Jacob wrote it). Since $$\mathcal{C}$$ admits $$\kappa$$-filtered colimits by the choice of $$\kappa$$ and $$\mathcal{D}$$ is closed under any $$\tau$$-small colimit that exists in $$\mathcal{C}$$ it follows that $$\mathcal{D}$$ admits $$\tau$$-small $$\kappa$$-filtered colimits.