Mixed Anomalies of Magnetic Symmetries

We resolve the existence of mixed 't Hooft anomalies between the electric and magnetic (solitonic) symmetries in $\sigma$-models and gauge theories. We identify the anomaly as naturally originating from a higher group in the Whitehead tower of the target space. In particular, the magnetic charges (topological solitons) can be interpreted as higher connections on the stages of the Whitehead tower, when pulled back to spacetime. This allows us to derive the form of the $(d+1)$-dim topological theory which classifies the anomaly by inflow. We also find novel features of these anomalies resulting from the mapping class group of the target space. We give many explicit examples, including a discussion of the cases where the higher magnetic symmetries become non-invertible.