Subject: categories: monadic completion of adjunctions

From: Paul Taylor <[email-masked]>
Date: Fri, 6 Aug 1999 11:41:17 +0100 (BST)

MONADIC COMPLETION OF ADJUNCTIONS

A famous theorem of Jon Beck says that an adjunction
			A
		      ^    |
		      |    |
		    F | -| | U
		      |    |
		      |    v
			C

is monadic (equivalent to its category of Eilenberg--Moore algebras)
iff U satisfies a certain condition involving "U-split coequalisers".

Given any adjunction, of course one may "force" it to be monadic by
replacing the category A by the category of algebras for the monad U.F,
and there is a comparison functor.

(1) My question is this: is there an explicit account in the literature
somewhere of the construction obtained by "freely adding" the conditions
of Beck's theorem (the U-split coequalisers) to A?

(2) Further, has anyone considered what happens when one forces an adjunction
to be BOTH monadic and comonadic?

Steve Lack gave me a verbal answer to the second question, although he
hasn't shown me his notes:
(a) if we force U to be monadic, and then (the new) F to be comonadic,
    (the third) U need NOT be monadic, but
(b) if we force U to be monadic again then (the fourth) F IS still comonadic.

In my investigation of this question and similar ones that have arisen
from my abstract Stone duality programme, I have found a construction 
due to Hayo Thielecke very useful,  though it seems appropriate to 
generalise it considerably beyond either his or my interests:

(3) Given an adjunction, or just the functor U, above, let H (for Hayo) be
the category with
   objects:     those of A
   morphisms:   H(a, b) = C(Ua, Ub)
Does anyone have an account of this category, maybe in the even more abstract
framework of a 2-category?

(4) The usefulness of the category H is that it provides a framework in
which to lift an endofunctor  S:A->A  to  the category of algebras, so
long as T has a "de Morgan dual"   P:C->C such that  U.S=P.U.   For example
if S is (- x a) for some object a in A,  or (which is what interests both
Hayo and me) if U is  Sigma^-: C^op -> C^op   and S is  (- x c), or
the same situation with a symmetric monoidal (closed) structure.

Making use of the construction of H, my sketch solution to question (2)
is as follows:

(i) As wide or lluf subcategories (all objects, some morphisms) of H we find
the categories that freely make U faithful (f), or faithful and reflect
invertibility (fri - this is the usage of "faithful" in Freyd & Scedrov);
call these A_f and A_fri.

(ii) F-|U extends to left adjoints
        F_f -| U_f : A_f -> C    and    F_fri -| U_fri : A_fri -> C
and if F was faithful or fri then so is F_f or F_fri.

(iii) Therefore the processes of making U and F faithful (or fri) commute,
so we need only do each of them once.

(iv) The auxillary category H is also convenient for representing freely
adjoined U-split coequalisers, in a similar way to the "Karoubi completion"
of a category to one with splittings of idempotents, but it's too messy
to describe in this note.

(v) Freely adding U-split coequalisers
    - commutes with freely forcing U (and F) to be faithful, and
    - commutes with freely adding F-split equalisers, and
    - preserves the properties that U or F are faithful or fri, but
    - DOES NOT COMMUTE with freely forcing U to reflect isomorphisms, and
    - DOES NOT COMMUTE with freely forcing F to reflect isomorphisms.

(vi) This justifies Steve Lack's claims. (I have counterexamples in (v),
though not actually to support (2a).)   Resolving the construction of
the monadic reflection of U (and the comonadic reflection of F) into
forcing them to be fri and then to have split co/equalisers, we find that
there is just one obstacle to the commutation of the four operations,
but this is overcome by a single repetition as in (2b).

Paul