User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:42:33Z http://mathoverflow.net/feeds/user/27086 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109147/how-to-endow-an-n-fold-segal-space-with-a-symmetric-monoidal-structure How to endow an n-fold Segal Space with a symmetric monoidal structure? samh123 2012-10-08T13:06:45Z 2012-10-09T11:04:40Z <p>I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: <a href="http://mathoverflow.net/questions/81425/what-is-a-symmetric-monoidal-infty-n-category" rel="nofollow">http://mathoverflow.net/questions/81425/what-is-a-symmetric-monoidal-infty-n-category</a> but I'm still not sure how to formulate this in terms of complete Segal spaces. If I just try to copy the definition from Lurie's book "Higher Algebra" I guess I get something like this: </p> <p>Let $X:\Delta^{op}\rightarrow Top$ be a complete Segal space ( or $(\infty,1)$-category) then a symmetric monoidal structure on $X$ is a morphism $F: X^{\otimes}\rightarrow N(Fin_{*})$ between two Segal spaces with $X^{\otimes}_{&lt;1>}:=F^{-1}(&lt;1>)=X$ (so $F$ is a natural transformation $F$= { $F_n: X_n^{\otimes} \rightarrow N(Fin_{*})_n$} and I guess $F^{-1}(&lt;1>)$ = { $F_n^{-1}(&lt;1>)$ }$_{n\geq 0}$ ) with the following two properties:</p> <p>1)$F$ is a coCartesian fibration</p> <p>2)For all $n\geq 0$ the morphisms $p^i:&lt; n > \rightarrow &lt;1>;p^i(i)=1;p^i(j\neq i)=*$ induce an equivalence of Segal spaces: $X_{&lt; n >}^{\otimes} \rightarrow X_{&lt;1>}^{\otimes}\times...\times X^{\otimes}_{&lt;1>}$. </p> <p>$Fin_*$ is the category of pointed sets $&lt; n >$ = { * ,1,...,n } and functions with $f(*) = *$ as morphisms and $N(Fin_{*}):\Delta^{op} \rightarrow Set$ is the nerve of $Fin_{*}$ </p> <p>My question is if this is correct or is there a mistake (I'm sure there is) or some additional condition that I'm missing? How do I generalize this to n-fold Segal spaces? Just replace everything with n-fold versions? I'm pretty new at this stuff so I'm quite uncertain how to formlate this. As far as I know there is not much literature on this topic. </p>