Universal property of graph cobordisms

Abstract

We exhibit the symmetric monoidal $\infty$-category $\mathrm{GrCob}$ of graph cobordisms between spaces as a full $\infty$-subcategory of the $\infty$-category $\mathrm{Psh}(\mathrm{Gr})$ of presheaves over $\mathrm{Gr}$, where $\mathrm{Gr}$ is the symmetric monoidal $\infty$-category of graph cobordisms between finite sets. We also describe a universal property for $\mathrm{GrCob}$: it is, in a suitable sense, the free symmetric monoidal extension of $\mathrm{Gr}$ endowed with the factorization homology ${\int }_X\ast$ of the universal ${\mathbf{E}}_{\infty }$-Frobenius algebra for all spaces $X$. As a corollary, we identify the space of universal natural operations on the factorization homologies, in particular the Hochschild homology, of ${\mathbf{E}}_{\infty }$-Frobenius algebras.

14 pages, comments welcome!