Scares me a little that I get the ‘gist’ of this – i feel like i shouldn’t – i only ever understand the ‘gists’ if i’m lucky
“first through concentration on the particular docrine known as Universal Algebra, making explicit the fibered category whose base consists of abstract generals (called theories) and whose fibers are concrete generals (known as algebraic categories). The term ’Functorial Semantics’ simply refers to the fact that in such a fibered category, any interpretation T 0 → T of theories induces a map in the opposite direction between the two categories of concrete meanings; this is a direct generalization of the previously observed cases of linear algebra, where the abstract generals are rings and the fibers consist of modules, and of group theory
where the dialectic between abstract groups and their actions had long been fundamental in practice.
This kind of fibration is special, because the objects T in the base are themselves categories, as I had
noticed after first rediscovering the notion of clone, but then rejecting the latter on the basis of the principle that, to compare two things, one must first make sure that they are in the same category; when the two are (a) a theory and (b) a background category in which it is to be interpreted, comparisons being models., the category of categories with products serves. “
Found a transcript:
I’d never heard of functoral semantics – my exposure to Category Theory was not at this level but at the much easier description levels found on nLab – my brain’s no good with diagrams that use random variables unless the diagram (sans labels) itself tells a clear story – I’ll give the transcript a read! thank you!