I am a mathematician (less so) and a cognitive scientist (more so) and I have been participating in and arguing with David and Brendan in their applied category theory seminar at MIT for the last couple of month so it will be a little strange to think through their ideas in their absence.

I have two ideas that I have been playing with that I am hoping is of interest to this community as well:

1. Categorical foundations of cognition. In other words, is category theory the right framework for modeling (some, if not all) cognitive phenomena? I have myself used category theory to understand how language and perception might be related to each other. Is it uniquely suited to do so? How is it better than other tools and frameworks?
2. Philosophical category theory. I don't mean category theory for philosophers or the philosophy of category theory, but something more like philosophical logic, with "logic" replaced by "category theory." Just as philosophical logic is the philosophical exploration of logical ideas and principles (and in that guise much older than mathematical logic), we might want to ask if there are philosophical ideas and principles that are closely tied to category theory - identity is an obvious example.

Looking forward to learning more from John and everyone else.