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I am starting higher Segal space (while sitting in a talk by Mikhail Kapranov about them…)
There is a similar “recursive” idea in Kontsevich’s definition of $(\infty,n)$-categories which he used at the end of 1990s. Unfortunately, it has never been published. I heard one exposition of it, but did not keep much notes.
Added the new paper - Higher Segal spaces I.
Eventually we need to add some warning. Apparently the Dycker-Kapranov-style Segal higher spaces are equivalent to $(\infty,1)$-operads, certainly not to $(\infty,2)$-categories or similar.
[edit: I have added a warning to the entry.]
This seems to provide a bridge between them and Lurie:
In analogy to the situation for $(\infty, 1)$-categories, there are various models for the notion of an $(\infty, 2)$-category. To describe the bicategorical structures appearing in this work, we will use Segal fibrations. In fact, we will also use the dual notion of a coSegal fibration. These and other models for $(\infty, 2)$-categories, as well as their relations, are studied in detail in the comprehensive treatment [Lur09b]. (p. 163)
In 9.3 they associate an $(\infty, 2)$-category to a 2-Segal space.
Thanks.
Now I see that the statement that DK 2-Segal spaces are $\infty$-operads is essentially in the article, in 3.6.
(We were supposed to discuss the DK-article in our seminar this semester, but somehow we didn’t get around to. Or not yet maybe.)
The nlab article higher Segal Space is very unclear – there are two completely different, unrelated types of objects that are sometimes referred to as “higher Segal spaces”. One is n-fold Segal Space, a model of $(\infty,n)$-categories. The other is the Dyckerhoff-Kapranov notion of a $d$-Segal space, which models something like an $(\infty,1)$-category, but without uniqueness of composites (for $d\geq 2$), and higher associativity only in dimension $d$ and above (higher associativity conditions are governed by $d$-dimensional polyhedra, related to $d$-dimensional field theories).
The article as it stands refers to both notions. The “Idea” section refers to $n$-fold Segal spaces.
But we already have a separate page for n-fold Segal spaces. So I’m pretty sure the intention was that this page refer to the Dyckrhoof-Kapranov notion, and not the $n$-fold Segal space notion. So I think the article needs a major cleanup.
Does anybody object to making higher Segal spaces discuss only the Dyckerhoff-Kapranov notion (except for adding some discussion of the difference)?
Please feel invited to work on the entry and improve it, as you see the need.
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