comparing cotriples via an adjoint pair

comparing cotriples via an adjoint pair

[Gaunce Lewis]

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

Re: comparing cotriples via an adjoint pair

[Michael Barr]

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

On Thu, 20 May 2004, Gaunce Lewis wrote:

> 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
>
>
>
>

Re: comparing cotriples via an adjoint pair

[Prof. Peter Johnstone]

On Thu, 20 May 2004, Gaunce Lewis wrote:

> 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
>
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

Re: comparing cotriples via an adjoint pair

[Oswald Wyler]

On Thu, 20 May 2004, Gaunce Lewis wrote:

> 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

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

Re: comparing cotriples via an adjoint pair

[Steve Vickers]

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.

Gaunce Lewis wrote:

> 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

Re: comparing cotriples via an adjoint pair

[Claudio Hermida]

Gaunce Lewis wrote:

> 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


Here's a related reference: the appendix of

C.Hermida and B.Jacobs, Structural Induction and Coinduction in a
fibrational setting, Information and Computation 145(2) 107-152,1998.

stablishes the suitable 2-functoriality of categories of (co)algebras
for endofunctors as inserters (which does not follow straightforwardly
from their weighted limit formulation) and the more or less immediate
corollaries of induced adjoints, without any coequalisers or additional
structure. It is remarkably simple but fairly useful.

The result (and the argument) extends literally to the case of
Eilenberg-Moore algebras for monads: using the (old) notion of morphism
of monads from (a), given a pseudo-morphism of monads (f,\theta):M -> N
(where \theta is iso), if f has a right adjoint g, adjoint transposition
of \theta yields (g,\theta'):N -> M right adjoint to (f,\theta) in Mnd
(b), and 2-functoriality of algebras yields the desired adjunction
between the categories of algebras (commuting with the forgetful
functors, of course). Once again, no structure is required on the
categories involved.
(a) R. Street, The formal theory of monads, JPAA 2 (1972) 149-168

(b) R. Street, Two constructions on lax functors, Cahiers top. et geom.
diff. 13 (1972) 217-264.


Claudio

PS: There is a more liberal notion of 2-cell for Mnd, essentially
arising from internal category theory, but I don't know its impact in
the above adjoint results.