For negative examples and discussion, see chapter 23 of http://katmat.math.uni-bremen.de/acc/acc.pdf Am 20.10.25 um 06:58 schrieb Michael Shulman: Unless I'm confused, I don't think this is true, not if you really meant (as you said) *essentially* algebraic theories. If it were, then for any essentially algebraic theory T, you could form the theory T0 with the same sorts and no equations, so that T is an extension of T0 in your sense. But the category of T0-models is just a power of Set, and the category of models of the essentially algebraic theory T is not usually monadic over a power of Set. For instance, if T is the two-sorted essentially algebraic theory of categories, then T0-Mod is Set x Set, and Cat is not monadic over Set x Set. On Sun, Oct 19, 2025 at 2:53 PM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote: I'm hoping to avoid having to prove by hand that a particular category is monadic, by a finitary monad, over another. It seems to me there should be a very general theorem that if one has an extension of essentially (many sorted) algebaric theories. then the category of models of the extension is monadic, by a finitary monad, over the category of models of the original theory. I'd be happy with some restrictions on the extension: the same set of sorts, no new equations imposed on the operations in the original theory, just new operations and equations on these. Does anyone have a citation for such a result? (Or one equivalent to it phrased in terms of sketches?) Such a result with additional hypotheses (which may or may not apply in my circumstance)? Or a result that obviously implies a result of the sort I want, but is stated in different terms? Best Thoughts, D.Y. You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/mail/deeplink/groupActions?source=EscalatedMessage&action=files&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/mail/deeplink/groupActions?source=EscalatedMessage&action=leave&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>