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Created algebraic theories in functional analysis. I've recently learnt about this connection and would like to learn more so I've created this page as a place to record my (and anyone else's) findings on this. I probably won't get round to doing much before the new year, though.
What's your definition of ‘algebraic’? Johnstone defines an algebraic category to be one that is monadic over Set. The Joy of Cats proves that every monadic category is algebraic (but not conversely). Yet you have a category that is monadic but not algebraic. Maybe you mean what The Joy of Cats calls bounded monadic or bounded algebraic: given by a small set of operations and equations (or possibly other laws).
See! It's proving useful already and I haven't even written anything substantial yet.
More seriously, the distinctions are stuff that I don't know about so are things that I'll need to learn as I work through this stuff. But that's the point of doing it! And by doing it publicly, I get the benefit of people like you shouting from the sidelines "What do you mean by algebraic?" when I didn't even realise that I didn't understand.
(But please note that I won't be able to think about this properly until the new year so please don't get frustrated with me if I don't immediately change stuff.)
Note: I just rewrote comment #2 using better terminology. (I also just rewrote equationally presentable category that way, too, in case you already looked at it closely.)
Actually, I think that you reported Yemon Choi's answer incorrectly; Ban is not monadic. The Joy of Cats agrees; it is not even algebraic, which for them is a weaker condition. But it is essentially algebraic, a yet weaker condition that I found too confusing to summarise at algebraic category. Possibly this means that it can be described using partial operations, which is one meaning of ‘essentially algebraic’, although possibly not Joy's meaning.
I've finally gotten round to following the reference that Todd kindly provided in answer to my question on MO about this (to a section in Toposes, Triples, and Theories). I've started expanding the Banach space example from TTT into this page - not done yet, of course, and not really anything said that isn't in TTT.
Of course, I've probably gotten all my language wrong since I don't yet fully understand all the bits about algebraic versus monadic versus whatever, but I think that I at least now get a picture of the problem with Banach spaces: there aren't enough of them. So it's not that the functor from Ban to Set is bad, it's just that the category of Banach spaces misses out some algebras for the corresponding triple.
I have a sneaking suspicion that (in TTT language) the functor Ban to Set is of descent type, but I've yet to prove it (and TTT doesn't say one way or the other).
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