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Added to references
Axel Osmond, On Diers theory of Spectrum I: Stable functors and right multi-adjoints, (arXiv:2012.00853)
Axel Osmond, On Diers theory of Spectrum II: Geometries and dualities, (arXiv:2012.00853)
There should be some Isbell duality connection with Osmond’s ideas, no?
added pointer to
In Multimonads and multimonadic categories, Diers writes
We use the notion of multiadjunction developed in [4]
[4] is
Where Diers then refers us back to
Added a reference to Diers spectrum.
Added a section on multi-monads
Any functor $U: A\to B$ which has a left multi-adjoint generates a multi-monad on $B$. Categories $A$ which can be reconstructed from this multi-monad are called multi-monadic (Diers 80).
Multi-monadic categories on $Set$ can be characterized in the following way: they are regular, with connected limits, with coequalizers of coequalizable pairs, their equivalence relations are effective, their forgetful functors preserve coequalizers of equivalence relations and reflect isomorphisms. Unlike monadic categories they need not have products. Examples include local rings, fields, inner spaces, locally compact spaces, locally compact groups, and complete ordered sets. (Diers 80, p.153)
Seems like important ideas, these ’multi-’ constructs. Why do we hear so little about them?
How about some right multi-adjoints and multi-comonads - any nice cases?
Link to multimonad.
It looks like the “abstract polynomial functors” of Street’s “Variation on a comprehensive theme” are exactly the functors with left multiadjoints.
Moved the lengthy definition/discussion of $Fam(D)$ (here) out of the following theorem environment. Added links to (dual of) free coproduct completion, where this is all discussed further.
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