  We study the (generalized) transition probability spaces, in the sense of
Mielnik and Cantoni, for spacetime quantum states in loop quantum gravity.
First, we show that loop quantum gravity admits the structures of transition
probability spaces. This is exemplified by first checking such structures in
covariant quantum mechanics, and then identify the transition probability
spaces in spin foam models via a simplified discrete version of general
boundary formulation. The transition probability space thus defined gives a
simple way to reconstruct the discrete analog of the Hilbert space of the
canonical theory and the relevant quantum logical structures. Second, we show
that the transition probability space and in particular the spin foam model are
2-categories. Then we discuss how to realize in spin foam models two proposals
by Crane about the mathematical structures of quantum gravity, namely the
quantum topos and causal sites. We conclude that transition probability spaces
provide us with an alternative framework to understand various foundational
questions of loop quantum gravity.
