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monadic-compl
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monadic-compl
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Date: Fri, 6 Aug 1999 11:41:17 +0100 (BST)
From: Paul Taylor <pt@dcs.qmw.ac.uk>
Subject: categories: monadic completion of adjunctions
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