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5 votes

Does every exponentiable ($\infty$-)topos have enough points?

For $1$-toposes, the answer seems to be no: There are locally finitely presentable toposes which don't have enough points. One such example is the Malitz-Gregory atomic topos given in Section 5 of ...
user713327's user avatar
9 votes

Is the adjunction between spaces and chain complexes monadic?

This answer assumes that $\def\Ch{{\sf Ch}}\def\Z{{\bf Z}}\Ch_{≥0}(\Z)$ refers to the derived ∞-category of chain complexes, i.e., with quasi-isomorphisms inverted up to a homotopy. Recall that the ...
Dmitri Pavlov's user avatar
2 votes

Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?

I'm not an expert in ($\infty$-)topos theory, so someone would have to check that this really translates to a proof of what you want, but you can prove the following in homotopy type theory, which is ...
Naïm Favier's user avatar
2 votes

Double category of monads and pseudo monad-morphisms

A monad in a strict 2-category $K$ is equivalent to a monoid in some strict monoidal endomorphism category $K(x,x)$, which is equivalent to a strict monoidal functor $\Delta_a \to K(x,x)$ where $\...
Mike Shulman's user avatar
  • 65.8k
7 votes

Notion of $\kappa$-sifted categories?

In the one-dimensional setting, $\kappa$-sifted categories are studied in §3 of Adámek–Koubek–Velebil's A duality between infinitary varieties and algebraic theories. However, it is shown there (...
varkor's user avatar
  • 9,531
2 votes

Localization and space of morphisms

I asked Ayala about this. He told me that the paper was lacking some justifications and shared with me a proof of Proposition 2.19. His argument can now be found in [Ara24, Theorem 2.24]. [Ara24] ...
2 votes

Cocartesian fibration classifying $\mathrm{Fun}(F,G)$

I am not aware of a reference, so I will give a proof. The proof strategy is to treat the case of ordinary categories first, and then localize the result to get the result for arbitrary $\infty$-...
Ken's user avatar
  • 2,164

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