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  • $\begingroup$ Thank for explaining that. Can you get rid of the $\kappa$-tuples? $\endgroup$ Commented Dec 31, 2021 at 15:14
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    $\begingroup$ The tuples never break ranks, so you should have $\kappa=1$ by adding types and terms with equations to make them products, projections and pairings. I'm just being a categorist tidying up the mess of universal algebra. Maybe there is a purely categorical theorem of when there's a dual adjunction between (l)extensive categories. No disrespect to Broodryk, but I smell an idea that needs more than one insight, just as the double pullback formulation of extensivity did. $\endgroup$ Commented Dec 31, 2021 at 20:01
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    $\begingroup$ This argument is for one-sorted algebraic theories. I don't think it would be appropriate to introduce a new sort here. $\endgroup$ Commented Jan 1, 2022 at 1:22
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    $\begingroup$ One-sorted algebraic theories were a gratuitous handicap of the mid 20th century akin to the unit fractions of the ancient Egyptians. $\endgroup$ Commented Jan 1, 2022 at 12:35
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    $\begingroup$ You are welcome to state and prove a theorem for many-sorted algebraic theories if you like. $\endgroup$ Commented Jan 1, 2022 at 12:45