[0,1] is a commutative quantale when equipped with its usual multiplication. You can lift the monoidal product structure to sheaves on [0,1] (viewed as a frame) via Day convolution. So we recover a topos where the truth values are probabilities.
People who have attempted to build toposes with probabilities as truth values have also failed to notice this. Take Isham and Doering’s paper, for example, (which I personally am quite averse to because they bullishly follow through on constructing toposes with certain properties which are barely justified). They don’t even think about products of probabilities.
I think the monoidal topos on the unit interval merits some serious investigation.
[0,1] is a commutative quantale when equipped with its usual multiplication. You can lift the monoidal product structure to sheaves on [0,1] (viewed as a frame) via Day convolution. So we recover a topos where the truth values are probabilities.
People who have attempted to build toposes with probabilities as truth values have also failed to notice this. Take Isham and Doering’s paper, for example, (which I personally am quite averse to because they bullishly follow through on constructing toposes with certain properties which are barely justified). They don’t even think about products of probabilities.
I think the monoidal topos on the unit interval merits some serious investigation.