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Universal property of graph cobordisms

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.