This is a question about props, a generalization of operads (used to model operations with several inputs and several outputs).

By forgetting the composition structure of an operad one obtains a so called symmetric module and for any symmetric module one can construct its free operad. These processes define a functorial adjunction and there is a similar adjunction between props and so called symmetric bimodules.

One says that an operad $\mathcal{O}$ is sigma-free if $\Sigma_n$ acts freely on $\mathcal{O}(n)$ for each $n$.

Observation 1: The free operad generated by a symmetric module is a sigma-free operad.

Observation 2: The notion of sigma-free is appropriate in the context of operads to study the homotopy theory of their representations (a.k.a. algebras).

Observation 3: Let $\mathcal{FB}$ be the free prop generated by a sigma bimodule $\mathcal{B}$. Then, if either $\mathcal{B}(n,0)$ or $\mathcal{B}(0,n)$ is nontrivial for some $n$, then $\Sigma_{kn}$ does not act freely on $\mathcal{FB}(kn,0)$ or $\mathcal{FB}(0,kn)$.

Question: Is there a generalization of "sigma-free" to props, which allows for at least the $(n,0)$ parts to be nontrivial and leads to the correct homotopy theory of representations?

Guess: (for categories with a terminal object) the desired class should include props $\mathcal{P}$ with $\mathcal{P}(0,0)=1$, which after modding by the prop ideal generated by $\mathcal{P}(0,n)$ and $\mathcal{P}(n,0)$ for $n>0$, are (bi)sigma-free, i.e. the action of $\Sigma_m\times\Sigma_n$ on $\overline{\mathcal{P}}(m,n)$ is free for $m,n>0$.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy