alex hoffnung asked me today about why i think that for purposes of algebraic geometry we should be developing a concept of "doctrine" based on groupoids instead of (as jon beck apparently originally envisioned) on categories, so i want to try to answer that to some extent here.
my first reaction is that one of the big reasons for this is that one of the key examples of a doctrine, namely the doctrine of "dimensional theories", probably needs to be groupoid-based rather than category-based because the objects of a dimensional theory, being "line objects", are invertible under tensor product, and the "inverse object" functor is contravariant, thus not fitting with the most straightforward version of the idea of category-based doctrines.
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