Re: "prime" monads?

[Vaughan Pratt <pratt@cs.stanford.edu>]

From: Fred Linton
> I was relying on the Axiom of Choice (AC) to know that every vector space
> (scalars from an arbitrary field) has a basis. Paraphrasing: every
> vector space is free. Thinking (loosely) of the Kleisli category as
> the full subcategory of the EM category consisting of the free algebras,
> this means the K and the EM categories "coincide" (to within equivalence
> of categories). The exact nature of the field is irrelevant. But AC (or
> something else strong enough to guarantee every vector space has a basis)
> is crucial.

This seems right for ordered fields.  Is it also true for finite fields
and other non-ordered fields like the complex numbers?

If every Archimedean field k gives rise to a Klem (Kleisli ~ EM) monad
Vct_k, that's uncountably many Klem monads right there, all in the
lattice of submonads of Vct_R.  (The submonads of a monad must form a
complete lattice, more precisely a complete semilattice, if they're
closed under arbitrary intersection; is it always an algebraic
semilattice, as for algebras?)

k doesn't even need to be a field or an additive group, any subrig of R
will do, as Bill was hinting at.  (The coefficients have to form a rig
for T to be functorial.)  Call this Method A for forming a submonad of
Vct_R.

Method B is to limit the operations to those whose weight (sum of
coefficients) is drawn from a submonoid of the monoid R under
multiplication.  (It has to be a submonoid for the multiplication mu and
unit eta to remain defined.)  The smallest such monoid is {1}, which
gave rise to the affine spaces as per my previous post.  R has plenty of
other multiplicative submonoids, such as the set {c^i} for any real c >
0 and i ranging over any submonoid of Z under addition, or for any real
c and i ranging over any submonoid of N under addition.  This can be
combined with Method A (coefficients from a subrig of R) to get even
more submonads.

Questions:

1.  Do all submonads of Vct_R arise as above (coefficients limited to a
subrig of R, weights limited to a multiplicative submonoid of R)?

2.  Among the submonads of Vct_R, are the Klem ones exactly those for
which the set of coefficients is dense in some open interval of R?

>> But not for 1+1+X: the free bipointed sets are in a natural bijection
>> with the non-free ones, the latter having the two points
>> identified
>
> I'm not entirely sure what's "natural" here. Certainly the free bipointed
> sets are in bijective correspondence with the (just plain) sets: that's a
> "consequence," if you like, of the canonical functor from SETS to any
> Kleisli category over SETS being a bijection on objects. So the non-free
> bipointed sets, coinciding as they do with the (singly) pointed sets, are
> likewise in bijective correspondence with the (just plain) sets. But the
> same is true for the free algebras over ANY monad on sets. So? "This"?
>
>> ... --this must be about the simplest nontrivial instance of
>> obtaining an algebra as a quotient of a free algebra, not a bad
>> introductory example when explaining how to get algebras from free
> algebras.

Point taken.  I was trying to say that identifying the two constants of
a free bipointed set was the canonical way of producing any nonfree
bipointed set.  What's a slicker way to say this?

Vaughan