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

[Anders Kock <kock@imf.au.dk>]

Mark Weber says rightly about David Spivak's "monad"
(and it applies to its natural extension to the "cocompletion under k-small coproducts-monad" on Cat as well):

"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."

Yes, you do run into problems; however, they can be solved, as I showed in my Chicago thesis 1967. Namely, take for Set_k the (small) full subcategory of Sets whose objects are the  ORDINAL numbers of cardinality less than the regular cardinal k. Ordinal sum formation then allows you to get the multiplication of the monad to be strictly associative.

Similarly for the "similar monad" mentioned by David
(based on the Grothendieck-construction of categories) - this monad is also in my thesis, and Lawvere reports on it in his "Ordinal sums and equational doctrines", (Seminar on Triples, SLN 80 (1969), see p.152-153. ).

However, these cunning tricks to get strict associativity were in the 1960s forced on us, for historical reasons:  at that time we did not have the notion of 2-dimensional category well enough established to see these cocompletion "monads" in their true 2-dimensional nature.

The "similar monads", based on a suitable Cat_k, are also reported on in loc.cit.; and Cat_k could be replaced by any small category Cat_0 of categories which is stable under the Grothendieck construction, like the category of k-small posets, or of k-small directed categories. (I called these monads  "prelimit monads"; Lawvere calls them Dir_Cat_0. They also appear in Guitart's 1974-article, as referenced in my previous posting.)

Anders Kock




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