Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added statement of and references to the weak equivalence with the Fulton-MacPherson operad (here)
added the same statement also at Fulton-MacPherson operad
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.
I suppose this is about this Definition (which, somewhat awkwardly, somebody had copied verbatim from HA p. 758; I have now at least added some hyperlinks and touched the formatting and wording).
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.
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.
Added:
A comprehensive reference for many known models is
1 to 7 of 7