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Let $\mathcal{X}$ be the $\infty$-category of $1$-excisive functors from (pointed) spaces to (unpointed) spaces: equivalently, the $\infty$-category of pairs $(X, E)$ where $X$ is a space and $E$ is a …

Marc's examples are good ones, but let me add two more (which are closely related to each other): 1) Let $\mathcal{C}$ be an accessible $\infty$-category which admits small filtered colimits, and let …

If $G$ is a profinite group, then the topos of sets with a continuous $G$-action is Boolean, but the associated $\infty$-topos is usually not hypercomplete.

If by ``effective monomorphism'' you mean the categorical dual of the condition of being an effective epimorphism, then it is not (or at least not obviously) equivalent to the statement that A can be …

For any space $X$, there's an $\infty$-topos of spaces fibered over $X$. The underlying ordinary topos is the category of representations of the fundamental groupoid of $X$. So if $X$ is simply connec …

Let $\mathcal{S}$ denote the $\infty$-category of spaces. For any $\infty$-topos $\mathcal{X}$, there is an essentially unique geometric morphism $\pi^{\ast}: \mathcal{S} \rightarrow \mathcal{X}$. The …

Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the …

I'm going to go out on a limb and suggest that the book HTT never uses anything stronger than replacement for $\Sigma_{15}$-formulas of set theory. (Here $15$ is a randomly chosen large number, and HT …

The projection map $p: \mathbf{R} \rightarrow \ast$ induces a fully faithful embedding of sheaf categories $p^{\ast}: Shv(\ast) \rightarrow Shv( \mathbf{R} )$. This is equally true for sheaves of sets …

Reflecting on Gabe's comment on my original answer, I now think what I wrote is misleading because it conflates two separate (but related) assertions: The existence of strongly inaccessible cardinals …