Dear David I'm not sure whether your specific construction has been looked at or named, but it is a "submonad" of something very well known. Let's work with 2 universes of sets, your "Set" being the topos of small sets, and SET being a topos of sets large enough to include {arrows of Set} as an object. For those not so comfortable doing this, fix two regular cardinals far enough part, read "Set" as "the category of sets of cardinality less than the smaller cardinal", and "SET" as "the category of sets of cardinality less than the bigger one", and far enough apart means Set is a category internal to SET. Now the 2-category CAT (of category objects in SET) contains Set as an object so one may form the slice 2-category CAT/Set in the strictest sense (1-cells being triangles commuting on the nose). The big brother of your monad is a 2-monad on this 2-category. The endofunctor part does the following: S:A-->Set |--> Set \downarrow S --> Set This is the underlying monad of what could be called the "fibration" 2-monad. That is applying to a functor produces the free split fibration on what you started with. This construction works at the following generality: replace CAT by any 2-category with comma objects and Set by any object therein, and the first paper to see fibrations as algebras of a monad in this way was R. Street "Fibrations and yoneda's lemma in a 2-category" SLNM 420 1974 The relation between your monad and this one is that there's a canonical inclusion Set --> CAT/Set which regards any Set S as a functor S:1-->Set, and this functor is the 1-cell data for a monad morphism (in the sense of Street: "Formal theory of monads") from your monad to the monad I described. With best regards Mark Weber

Dear Categorists,

Does anyone know a name for the monad described below and/or whether it has been studied?

Let k-Set denote the category of k-small sets (for some small regular cardinal k). For a set S, we denote by

T(S)=(k-Set \downarrow {S})

the set whose elements are pairs (K,f), where K is a k-small set and f:K-->S is a function. This construction is functorial in S. I claim that the endo-functor T: Set -->Set is a monad. The identity transformation S-->T(S) is given by "singleton set" and the multiplication transformation TT(S)-->T(S) is given by Grothendieck construction.

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

Does this monad T have a name? Has it been studied?

Thank you, David

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

In the case where k=omega, T is the well-known finite multiset monad, which associates to each S the free commutative monoid generated by S (whose elements are also known as finite multisets in S).

For other k, I would call this the "monad of multisets of size less than k". I think this works for any infinite small cardinal, not just regular ones.

You really do need the regularity. Otherwise removing brackets from a k-small multiset of k-small multisets might yield something bigger than a k-small multiset, and then one cannot define a multiplication for the monad. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]