  Shulman's spatial type theory internalizes the modalities of Lawvere's
axiomatic cohesion in a homotopy type theory, enabling many of the
constructions from Schreiber's modal approach to differential cohomology to be
carried out synthetically. In spatial type theory, every type carries a spatial
cohesion among its points and every function is continuous with respect to
this. But in mathematical practice, objects may be spatial in more than one way
at the same time; for example, a simplicial space has both topological and
simplicial structures.
  In this paper, we put forward a type theory with "commuting focuses" which
allows for types to carry multiple kinds of spatial structure. The theory is a
relatively painless extension of spatial type theory, and enables us to give a
synthetic account of simplicial, differential, equivariant, and other cohesions
carried by the same types. We demonstrate the theory by showing that the
homotopy type of any differential stack may be computed from a discrete
simplicial set derived from the \v{C}ech nerve of any good cover. We also give
other examples of commuting cohesions, such as differential equivariant types
and supergeometric types, laying the groundwork for a synthetic account of
Schreiber and Sati's proper orbifold cohomology.
