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Results tagged with infinity-topos-theory
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user 11640
11
votes
1
answer
579
views
What is the connection between Lurie's definition of shape and Čech homotopy?
It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [Higher topos theory] defines this one:
Definition 1.
The shap …
- 15.9k
6
votes
0
answers
283
views
What are the Čech-local equivalences of (simplicial pre)sheaves?
Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield …
- 15.9k
15
votes
2
answers
2k
views
Modern versions of Verdier's hypercovering theorem?
Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a simplicia …
- 15.9k
16
votes
2
answers
674
views
How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot …
- 15.9k