# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 23rd 2019

I forget if I ever knew the following:

What is there to the assumption that a given cohesive $\infty$-topos admits an $\infty$-site of definition all whose objects have (under Yoneda embedding) contractible shape?

Is this automatic? Is it a weak extra assumption? A strong extra assumption?

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJun 23rd 2019

Isn’t that essentially the “locally $\infty$-connected” condition at infinity-connected (infinity,1)-site?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 23rd 2019

Yes, in that terminology I am asking: How strong is the condition that a cohesive $\infty$-topos admits any locally $\infty$-connected $\infty$-site?

1. My intuition such as it is is that it is quite strong. For example in algebraic geometry, affines will typically not be contractible. Even in the pro-étale topos of Scholze and Bhatt I expect that the affine line is not contractible for instance.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJun 23rd 2019

Re #4: …and for that reason, those toposes are not, I believe, cohesive. (-:

Re #3: C3.6.3 of the Elephant implies that any cohesive 1-topos has a locally 0-connected site, and by Prop. 1.3 of remarks on punctual local connectedness it can be taken to have finite products as well. I don’t have time to look up the proofs right now, but I would expect that they generalize at least partially to the $\infty$-case.

2. Re #5: That sounds right!