  We consider the problem of magnetic charges in $(2+1)$ dimensions for a torus
geometry in real-space, subjected to an inverted Lorentz force due to an
external electric field applied normal to the surface of the torus. We compute
the Hall conductivity associated with transport of these charges for the case
of negligible gapless excitations and global $\mathrm{U}(1)$ charge
conservation symmetry, and find it is proportional to an integer-valued
topological invariant $\mathcal{Q}$, corresponding to a magnetic quantum Hall
effect (MQHE). We identify a lattice model realizing this physics in the
absence of an external electric field. Based on this, we identify a
generalization of the MQHE to be quantized transport of magnetic skyrmions, the
quantum skyrmion Hall effect (QSkHE), with a $\mathrm{U}(1)$ easy-plane
anisotropy of magnetic skyrmions and effective conservation of charge
associated with magnetic skyrmions yielding incompressibility, provided a
hierarchy of energy scales is respected. As the lattice model may be
characterized both by a total Chern number and the topological invariant
$\mathcal{Q}$, we furthermore outline a possible field theory for electric
charges, magnetic charges, and correlations between magnetic and electric
charges approximated as composite particles, on a two-torus, to handle the
scenario of intermediate-strength correlations between electric and magnetic
charges modeled as composite particles. We map this problem to a generalized
$(4+1)$D theory of the quantum Hall effect for the composite particles.
