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- [Nov. 12th, 2000|02:40 pm]
Aha! I think I've got the right conditions for it.
The natural transformation t:D^2->D needs to look
associative, and t_{A+B} needs to be 'equal' to
t_A + t_B modulo a few instances of the natural
isomorphism D(A+B)->DA+DB to make the types line
up. Together with a lot of algebraic wonkage, this
seems to be sufficient to guarantee that
T = D + 1
eta = inr_T
mu = ([1_D,1_D]+1) a_DD1 ([t,1_D] p_D1 + T)
define a monad. (where a_ABC : A+(B+C) == (A+B)+C
and p_AB : D(A+B) == DA + DB)
This process, 'premonad' |-> monad
seems analagous to the common
semigroup |-> monoid
by adjunction of unit, and is analagous (by motivation)
to the process
accumulation |-> closure
by setting CX = DX u X.
I'm very curious what the logical/type theoretical
interpretation of this is, though.

Did some revision of topo paper. Must finalize tomorrow.