Hydrodynamics as cospans of field theories into the BF theory

Hydrodynamics is based on conservation laws of currents: one starts from the conserved currents of the theory describing the microscopic dynamics, and provides an alternative parameterisation of these currents in terms of hydrodynamic variables (density, pressure, velocity, etc.). This paradigm has recently been extended to incorporate higher-form symmetries. The conservation law of the $p$-form conserved currents can be regarded as the equations of motion of a $BF$ theory that treats the currents as fundamental fields. We argue that the hydrodynamic approximation to a microscopic theory can be regarded as a cospan of differential graded manifolds $X_\mathrm{micro}\to X_{BF}\leftarrow X_\mathrm{hydro}$, where $X_\mathrm{micro}$ and $X_\mathrm{hydro}$ describe the microscopic and hydrodynamic theories, respectively, and $X_{BF}$ describes the $BF$ theory of conserved currents.