Subject: categories: comparing cotriples via an adjoint pair

From: Gaunce Lewis <[email-masked]>
Date: Thu, 20 May 2004 23:28:18 -0400

I have encountered a situation in which I have two categories C, D which
are related by a pair of adjoint functors L from C to D and R from D to
C.  Also, there is a cotriple S on C and a cotriple T on D.  Finally, there
is a natural isomorphism f from RT to SR.  It seems that if a couple of
diagrams relating f to the structure maps of the cotriples commute, then
there is an induced adjoint pair relating the two coalgebra categories.  Is
this, or something similar to it, in the literature in some easily
referenced place?

Thanks,
Gaunce

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From: Michael Barr <[email-masked]>
Date: Fri, 21 May 2004 16:30:36 -0400 (EDT)

Although I cannot be sure, this looks an awful lot like an adjoint triple.
I think Charles and I had a section of TTT on this.  It was certainly not
new with us and may have even been in Harry Appelgate's thesis back around
40 years ago.

Michael

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From: Oswald Wyler <[email-masked]>
Date: Sat, 22 May 2004 11:17:59 -0400 (EDT)

> this, or something similar to it, in the literature in some easily

This situation has been encountered since at least 1970 by various
categorists, including myself.  A relevant paper is:
  D. Pumpl\"un, Eine Bemerkung \"uber Monaden und adjungierte Funktoren,
  Math. Annalen 185, 329-337 (1970).
If Gaunce's two commuting diagrams are the usual ones, then his
conjecture is correct.  Observe that in this situation, we have not just
a pair but a quadruple of dual categories, replacing C and D by their
duals, or inverting the direction of arrows, or both.

This may just be a "folk theorem", but it should have been published by
someone, somewhere, and I would also like to have an easily accessible
reference, or references.

Oswald Wyler

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From: "Prof. Peter Johnstone" <[email-masked]>
Date: Fri, 21 May 2004 21:54:06 +0100 (BST)

> this, or something similar to it, in the literature in some easily
See (for the dual situation) an old paper of mine:

Adjoint lifting theorems for categories of algebras, Bull London Math.
Soc. 7 (1975), 294--297.

I should say (before others say it for me) that this was not the first
place the result appeared: it (and much more) was in the famous
unpublished (and largely unwritten) thesis of Bill Butler. But Gaunce
asked for a published reference.

Peter Johnstone

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From: Steve Vickers <[email-masked]>
Date: Mon, 24 May 2004 10:04:38 +0100

This paper may also be relevant (again in the dual situation, with monads):

   Jean-Pierre Meyer "Induced functors on categories of algebras",
Mathematische Zeitschrift 142 (1975) 1-14.

This relaxes the condition that it should be a natural isomorphism
between RT and SR. Instead it has a monad functor from (D,T) to (C,S)
and a left adjoint monad opfunctor. It constructs an adjoint pair of
functors between the algebra categories. However, it does assume that
one of the algebra categories has coequalizers.

For monad functors and opfunctors see

   Ross Street "The formal theory of monads", Journal of Pure and
Applied Algebra 2 (1972) 149-168.

Steve Vickers.