Subject: categories: Monads without unit?

From: Jason C Reed <[email-masked]>
Date: Wed, 29 Nov 2000 17:10:21 -0500 (EST)

I had been looking at the standard way the topological closure A |-> CA
can be recovered from the derived set operation A |-> A' (which takes A to
the set of the accumulation points of A) by setting CA := A' u A, and
observed that it can be generalized to categorical language. If you take
thing made up of a coproduct-preserving functor D and a natural
transformation t : D^2 -> D, such that t is `associative' in the usual
sense of asserting that t o t_D = t o Dt, and also 

 p_{X,Y} o t_{X + Y} = (t_X + t_Y) o p_{DX,DY} o Dp_{X,Y}

where p is the canonical natural transformation D(X + Y) -> DX + DY , then
if one defines

TX := DX + X
eta_X := inr_{DX + X}
mu_X := ([1_{DX},1_{DX}] + X) a_{DX,DX,X} (([t_X,1_{DX}] o p_{DX,X}) + TX)

(where [ , ] is copairing and a_{X,Y,Z} is the associativity isomorphism X
+ (Y + Z) -> (X + Y) + Z) then turns out that (T, eta, mu) is a monad.

Has anyone already studied these? I'd be particularly interested in a
type-theoretic or logical (along the lines of the correspondence between 
monads and modal operators) interpretation of such a structure, if any. 

---Jason

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From: F W Lawvere <[email-masked]>
Date: Thu, 30 Nov 2000 10:20:40 -0500 (EST)

Jason Reed's construction, adjoining a unit to obtain a monad,
seems to generalize one of the steps in the proof by Pare', Rosebrugh,
and Wood that any lex idempotent can be split in two steps, one of
the steps involving a left adjoint splitting and the other a right
adjoint splitting.  I believe this result was published about ten years
ago in Australia.

************************************************************
F. William Lawvere			
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284		   
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
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