  Many simplicial complexes arising in practice have an associated metric space
structure on the vertex set but not on the complex, e.g. the Vietoris-Rips
complex in applied topology. We formalize a remedy by introducing a category of
simplicial metric thickenings whose objects have a natural realization as
metric spaces. The properties of this category allow us to prove that, for a
large class of thickenings including Vietoris-Rips and Cech thickenings, the
product of metric thickenings is homotopy equivalent to the metric thickenings
of product spaces, and similarly for wedge sums.
