Two that I know of, when I worked there:

- Amazon Device Econ modeling platform, for econometrics

- Amazon Tax calculation platform, for things like VAT

But more broadly, that’s how you develop software: you’re theory building.

1. Develop language and rules in that language.

2. Give them semantic meaning in a diagram and set of example cases.

3. Develop a corresponding type theory for that.

4. Implement that synthetic domain theory in a general type theory — eg, a programming language.

You don’t have to know that it’s topos theory to do those steps, but each is better when you’re explicit about them — and knowing how to guide the first step to get good results at the last step is definitely applied topos theory.

Are you implying that the people who worked on those two platforms had no idea what topos theory is (but they were "applying" it nonetheless)? Or that they explicitly modeled their stuff in topos theory, and that improved upon the former way they had been doing it without topos theory? I don't count the former as an application of CT, but I would count the latter.

References to primary sources would be appreciated, or feel free to email me at [email protected] with the name or contact information of someone who directly worked on those systems that I could talk to.