Re: monad: (k-Set \downarrow -): Set -->Set

[Mark.Weber@pps.jussieu.fr]

Peter Johnstone is right -- the monad David described doesn't exist, thus
neither does the monad morphism I described in my other post (... sorry!).
Perhaps the fibrations monad is still of interest.

One could fix a skeleton of Set_k, and for k = cardinality of natural
numbers this works fine, and the monad on Set you get is the monoid monad.
However for bigger k you're likely to run into problems when trying to do
this sort of thing.

However it isn't true that the monad

>> (... on Cat, where we replace k-Set with k-Cat.)

is the k-coproduct completion monad -- you need to keep k-Set but work in
Cat and take lax slices, ie take the monad on Cat which has underlying
endofunctor

X |-> k-Set // X

(where // means "lax slice") to get the k-coproduct completion monad.

Mark Weber



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