Open 2D TFTs admit initial open-closed extensions

We show that any open 2-dimensional topological field theory valued in a symmetric monoidal $\mathrm{\infty }$-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of $E_1$-Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over $\mathbb{Q}$ and Wahl-Westerland and Wahl over $\mathbb{Z}$, and serves as a crucial ingredient to Lurie’s “non-compact cobordism hypothesis” in dimension 2. As part of the proof we also give a description of slice categories of the d-dimensional bordism category with boundary, which may be of independent interest.