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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 4th 2018

added statement of and references to the weak equivalence with the Fulton-MacPherson operad (here)

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 3rd 2021

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeNov 18th 2021
• (edited Nov 18th 2021)

The statement that $Rect(\square^k \times S, \square^k )$ can be identified with an open subset of $(\mathbb{R}^{2k})^S$ is wrong, isn’t it?

For example, $Rect(\square^1, \square^1)$ can be identified with the set of pairs $(a,b)$ satisfying $0 \lt a \leq 1$, $|b| \leq 1-a$, which is a triangle that includes part of its boundary.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 19th 2021
• (edited Nov 19th 2021)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 19th 2021

Regarding the point you raise: Scanning through Higher Algebra, it looks like the identity map is meant to count as a rectilinear embedding (explicit in Ex. 5.1.0.6, p. 759), while would-be open-ness of the space of rectilinear embedding seems not to be used. That makes me think that the “open subset” on p. 758 is a glitch and that the line was just meant to say that $Rect(-,-)$ is equipped with the subspace topology.

• CommentRowNumber6.
• CommentAuthorHurkyl
• CommentTimeNov 19th 2021
• (edited Nov 19th 2021)

Oh good; I’m not missing something and it’s harmless. Since I’ve just started trying to digest the little cubes operad and still find it somewhat opaque I really didn’t feel confident just removing that line.