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 $\int_X*$ 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.
Categories: math.AT