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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeDec 6th 2019

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 2nd 2021
• (edited Oct 2nd 2021)

Fixed a small typo in the sequences of equvalences after “Using this…” (here), replacing “$X$” by “$\infty Grpd_{/X}$”.

(Of course one might declare to use the former as shorthand for the latter, but not at this point inside a proof.)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 2nd 2021

I have added statement and proof (here) that for $\mathbf{H}$ cohesive and $X \in \mathbf{H}$, the shape of the slice $\mathbf{H}_{/X}$ is the cohesive shape of $X$.

(This seems to be the kind of statement we would/should have considered in the first wave of edits to this entry, but I don’t see or recall that we did.)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 2nd 2021
• (edited Oct 2nd 2021)

after the old example that cohesive $\infty$-toposes have trivial shape, I have added a remark (here) on how this is a reflection on them being “gros”, and leading over to the new prop on shapes of slices of cohesive $\infty$-toposes.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 3rd 2021

I have worked more on the Definition section (here):

• split up the previous Prop about equivalence of definitions into an actual numbered definition followed by a proposition that two Defs are equivalent

• added a third definition: as image of terminal object under the pro-left adjoint to $LConst$

• added the argument how that is equivalent to the other definition.

• added exact references to page and verse in the articles that have stated these definitions (this was at least unclear in the previous version).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 3rd 2021

The idea-section used to essentially just mention the idea of the shape of $Sh_\infty(X)$ for a topological space $X$, in relation to classical shape.

• two sentences to before this, in order to indicate the idea more generally and mentioning the relation to étale homotopy type,

• a sentence after this, mentioning the relation to cohesive shape for slices of cohesive toposes.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 4th 2021
• (edited Oct 4th 2021)

I have merged the first two Examples-subsections (shape of locally $\infty$-connected and of “retracts”) into a single section, since both subsections were small while having considerable overlap.

Also turned the first remark there (from times back, that the shape is given by the left adjoint when it exists) into a numbered Proposition with a proof, and added the remark that this is a special case of the previous proposition on étale homotopy type.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 5th 2021

I have re-worded (here) the section “Shape of a topological space” (previously it just claimed to point to more details on this case, but didn’t), adding a pointer to Marc’s argument that the notions indeed coicide for compact Hausdorff spaces.