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

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

As Peter Johnstone also emphasized in his reply, the construction which  
David Spivak describes,
namely "T(S)=(k-Set \downarrow {S})",  is really a part of a well known 
"monad" on the category of categories: if S is any category, T(S) is the 
free cocompletion of S under k-small coproducts. It is only a monad up 
to canonical isomorphisms, because coproducts are not in general 
strictly associative. This cocompletion "monad"  under coproducts has 
been widely studied under the name "Fam" (because T(S) is the category 
of k-small Families of objects in S). It is an example of a KZ monad.

However,  replacing k-Set by k-Cat provides a monad on Cat which is not 
KZ; David observes:

"(There is a similar monad on Cat, where we replace k-Set with k-Cat.)"

and Peter's reply to this:

"This is correct, and it's well-known: it is the monad which freely adjoins
k-small coproducts to a category. "

does not apply here (it slipped into the wrong place of his reply): 
rather, David's "similar monad" is trying to provide free cocompletion 
under colimits indexed by k-small categories, but does not, until you 
make a category-of-fractions construction on its values. My University 
of Chicago thesis (1967) described this way of making free cocompletions.

This "similar monad" (before doing the fractions-part) has  been studied 
by Guitart, he calls it this monad DIAG. Reference: Guitart, René, 
Remarques sur les machines et les structures. Cahiers de Topologie et 
Géométrie Différentielle Catégoriques, 15 no. 2 (1974), p. 113-144 
(available electronically in NUMDAM).

Anders Kock






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