Coherent objects in a hypercomplete $\infty$-topos

Question

In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\mathcal{X}^{coh}$ spanned by the coherent objects is essentially small. I will not go too deeply into the proof of this statement, as it uses a lengthy recognition criterion for coherent objects (Prop. A.6.5.7), but the main point is that the presentability ensures the existence of an essentially small subcategory $\mathcal{X}_0$ of $\mathcal{X}$ such that every object of $\mathcal{X}$ can be covered by objects of it; and local coherence makes it possible to choose $\mathcal{X}_0 \subseteq \mathcal{X}^{coh}$ so that we can further go on to enlarge it as to fill $\mathcal{X}^{coh}$ up completely.

In Warning A.6.6.2, Lurie claims that "using a more refined argument", one could show that the local coherence of $\mathcal{X}$ could be dropped in this statement, namely every hypercomplete $\infty$-topos has an essentially small subcategory of coherent objects; but this hypercompleteness condition can't be dropped: A counterexample would be the Tangent topos of the $\infty$-category of Spaces $\mathcal{X} = \operatorname{Exc}^1(\mathcal{S}_*^{fin}, \mathcal{S})$, whose objects can be identified with pairs $(X,E)$ of $X \in \mathcal{S}$, $E$ a local system of spectra on $X$. The coherent objects here are, as one can show, exactly those pairs $(X,E)$ where $X$ is coherent as a space; and since the $\infty$-category of spectra is not essentially small, the subcategory of those can't be as well.

However concerning the first claim of the Warning, namely the "more refined argument" that Lurie is talking about, I have no idea how this argument would look like. It sounds like the above argument could somehow be saved, but I think the remarks I gave above make clear why this isn't a very straightforward task - does anyone know how this works?

Answer 1

Let $\kappa$ be a regular cardinal. Say that an object $X$ in $\mathcal{X}$ is almost $\kappa$-compact if the truncation $\tau_{\leq k}X$ is a $\kappa$-compact object of $\mathcal{X}_{\leq k}$ for every $k$. I claim that if $\mathcal{X}$ is hypercomplete then the full subcategory of $\mathcal{X}$ on the almost $\kappa$-compact objects is essentially small. This answers the question since coherent objects in an $\infty$-topos are almost compact.

Let $\mathcal{X}^{\kappa}$ be the full subcategory of $\mathcal{X}$ on the $\kappa$-compact objects. Enlarging $\kappa$ if necessary we may assume that $\mathcal{X}^{\kappa}$ is closed under finite limits and generates $\mathcal{X}$ under colimits.

Equip $\mathcal{X}^\kappa$ with the Grothendieck topology where a sieve is covering if and only if it contains a finite collection of morphisms which is jointly effectively epimorphic, and let $\operatorname{Sh}(\mathcal{X}^\kappa)^\text{hyp}$ be the corresponding $\infty$-topos of hypersheaves. The inclusion $\mathcal{X}^\kappa \rightarrow \mathcal{X}$ extends to a left exact localization functor $p: \operatorname{Sh}(\mathcal{X}^\kappa)^\text{hyp} \rightarrow \mathcal{X}$.

Let $q$ be the right adjoint to $p$ and observe that $q$ preserves $\kappa$-filtered colimits. Combining this with the fact that every effective epimorphism in $\mathcal{X}$ is a $\kappa$-filtered colimit of effective epimorphisms between $\kappa$-compact objects we deduce that $q$ preserves effective epimorphisms. Since $q$ is also left exact we have that $q$ commutes with the truncation functors $\tau_k$ for every $k$.

Assume now given an almost $\kappa$-compact object $X$ of $\mathcal{X}$, and write $X$ as a $\kappa$-filtered colimit of $\kappa$-compact objects $Y_\alpha$. Let $k \geq 0$. Since $\tau_{\leq k}X$ is $\kappa$-compact we may find an index $\alpha_0$ such that $\tau_{\leq k}X$ is a retract of $\tau_{\leq k}(Y_{\alpha_0})$. It follows that $\tau_{\leq k}q(X) = q(\tau_{\leq k}X)$ is a retract of $q(\tau_{\leq k}Y_{\alpha_0}) = \tau_{\leq k}q(Y_{\alpha_0})$. Since the topology on $\mathcal{X}^\kappa$ is finitary we deduce that $q(X)$ is a coherent object of $\operatorname{Sh}(\mathcal{X}^\kappa)^\text{hyp}$. Our claim now follows from the fact that $X = p(q(X))$, since the full subcategory of $\operatorname{Sh}(\mathcal{X}^\kappa)^\text{hyp}$ on the coherent objects is essentially small.

(More concretely, one can also deduce smallness by showing that, under the above assumptions on $\kappa$, every almost $\kappa$-compact object admits a hypercover by $\kappa$-compact objects. This is essentially what the above argument is doing anyways, but it can also be proven more concretely by inductively building such a hypercover using the fact that almost $\kappa$-compact objects are closed under finite limits).