  General relativity may seem very different from quantum theory, but work on
quantum gravity has revealed a deep analogy between the two. General relativity
makes heavy use of the category nCob, whose objects are (n-1)-dimensional
manifolds representing "space" and whose morphisms are n-dimensional cobordisms
representing "spacetime". Quantum theory makes heavy use of the category Hilb,
whose objects are Hilbert spaces used to describe "states", and whose morphisms
are bounded linear operators used to describe "processes". Moreover, the
categories nCob and Hilb resemble each other far more than either resembles
Set, the category whose objects are sets and whose morphisms are functions. In
particular, both Hilb and nCob but not Set are *-categories with a noncartesian
monoidal structure. We show how this accounts for many of the famously puzzling
features of quantum theory: the failure of local realism, the impossibility of
duplicating quantum information, and so on. We argue that these features only
seem puzzling when we try to treat Hilb as analogous to Set rather than nCob,
so that quantum theory will make more sense when regarded as part of a theory
of spacetime.
