 
[Nov. 12th, 200002:40 pm]
Jason

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. 

