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Jacob's answer to my last question actually answers this one too: if the hypercomplete objects are coreflective as well as reflective, and the coreflector is the same functor as the reflector, then of …

Probably not. Claim: Let $C$ be a small $(\infty,1)$-category with finite limits and colimits which admits an embedding $V:C\to E$ into a Grothendieck $(\infty,1)$-topos preserving finite limits and …

If it may be forgiven to resurrect a very old question, it's worth pointing out that these are not the "homotopy groups of spheres" that appear in synthetic homotopy theory / homotopy type theory. Th …

Jacob's answer to your "for example" question is a good one, but let me be so bold as to try to address the general question. I think there are two different issues in play: the fact that the site of …

The answer to question 1 is yes. To see this, I think it's better to consider a "geometric sequent" to be of the form $$ (x:X), \phi(x) \vdash \psi(x) $$ since this avoids all mention of $\forall$ a …

By definition (e.g. Remark 6.5.4.7 of Higher topos theory), an $n$-topos $\mathcal{E}$ has enough points if for every morphism $f:X\to Y$ in $\mathcal{E}$, whenever $p^*(f)$ is an equivalence for all …

I'd like to mention something that I think hasn't been pointed out yet. The original question began with In set-theoretic language, one fixes some strongly inaccessible cardinal $\kappa$... This imp …

More generally, the issue with such interpretation is that substitution in type theory is interpreted by pullback in category theory, and substitution in ordinary type theory preserves all type-theore …

In fact I believe something quite general can be said: if $p:C\to D$ is any functor that preserves finite limits, then it is a Street fibration if and only if it has a fully faithful right adjoint. A …

One possibility along these lines is large eliminations for higher inductive types. For instance, here is a large elimination rule for the higher inductive interval type $\mathsf{I}$ with $0,1:\maths …