Re: "prime" monads?

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

Fred E.J. Linton wrote:
> There ARE a few monads for which the Kleisli and E-M categories
> "coincide," however, beyond the identity monads. First example
> coming to mind is the FreeVectorSpace monad on SETS. I'm sure
> other Categories-readers will point out more.

Fred, your numbering makes the example preceding your "first example"
(the identity monad) your zeroth example, but I'll number them 1 and 2
anyway.  Example 2 should work for all Archimedean fields (and
presumably all ordered fields).  Does it also work for fields not of
characteristic 0, or for the complex number field?

For more examples, all on Set, how about the following?  Example 3
extends example 1 (i.e. the identity monad is a submonad of example 3),
the rest (examples 4-9) are submonads of example 2.  This should bring
the crop of examples of Kl=EM up to 6 if I'm not mistaken (a big if),
with examples 7-9 not contributing.  All of 1-9 are finitary monads,
unlike for example the covariant power set monad whose signature is a
proper class.

3.  The lift monad 1+X on Set.  Obviously Kl=EM in this case.

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--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.

4.  The FreeAffineSpace monad on Set as the submonad of example 2
consisting of those operations whose coefficients sum to 1, i.e.
barycentric weightings where the individual weights can be any real.  I
noticed this the other day and assumed it must be well known (is it?).
It gives an alternative to France Dacar's equations yesterday showing
that the affine spaces are algebraic over Set.  The Kleisli-EM
identification certainly holds for finite-dimensional affine spaces (any
quotient is a space of lower dimension, but all spaces of a given finite
dimension are dense and therefore isomorphic); is there any reason why
the proof (with Choice) for infinite-dimensional vector spaces would not
go through for the affine case?

By way of insight into this monad, if you take those operations whose
coefficients sum to 3 say (still as a "submonad" of example 2), the
multiplication produces exactly the operations whose coefficients sum to
9, so that would-be monoid isn't closed under multiplication.  If you
think to fix this by taking the operations whose coefficients sum to 0
(remember we're allowing negative coefficients) then that attempt
doesn't have a unit.  That just leaves the barycentric operations as the
only possible submonad of example 2 consisting of operations of a fixed
total weight.

My guess would be that Kl=EM for the next two, by density (cf. examples
7 and 8).  Do statisticians know about these monads?

5.  Conservative statistics.  The submonad of example 4 where the
coefficients must be positive (and hence in (0,1]).  The next example
shows the sense in which this is "conservative."

6.  Lossy statistics.  The submonad of example 2 where both the
individual coefficients *and* their sum (over the coefficients of an
operation) are in (0,1].  Lossy statistics permits its sample spaces to
leak out through a wormhole.

The following two are examples in this vein where EM > Kl.

7.  The lattice monad (JAMS-friendly lattices, not that quaint
order-theoretic stuff that went out with the Bauhaus,
http://www.math.rutgers.edu/~zeilberg/Opinion81.html).  The submonad of
example 2 restricted to integer coefficients.  The free lattice on n
generators is Z^n; for n <= 2 I think all non-free quotients must be a
circle, cylinder, or torus, but for n = 3, (x,y,z) |--> (2x+z,2y+z)
produces the 2D checkerboard as an infinite quotient of Z^3 with no
cycles, pointing up the role of density in 4 above.

8.  Affine lattices.  The intersection of examples 4 and 7 (still
allowing negative coefficients).   This is the appropriate monad for
studying crystalline structure, which has no origin.  The free lattices
are the cubical crystals and all other crystalline structures are
non-free.  Unless I'm missing something the quotient (x,y,z) |-->
(2x+z,2y+z) works here too to show EM > Kl.

9.  The intersection of examples 5 and 7 (same as 6 and 7).  Isn't this
just example 1?

The chances of all my guesses being right are pretty slim, but if they
are we'd now be up to six examples of Kl=EM, namely 1 to 6 above.

What is the structure of the submonads of a monad on Set?  Anything like
algebraic lattices?

Vaughan