Subject: categories: Re: comparing cotriples via an adjoint pair

From: Claudio Hermida <[email-masked]>
Date: Mon, 31 May 2004 09:52:12 -0400

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.