comparing cotriples via an adjoint pair
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
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
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others say it for me) that this was not the first place the result appeared: it (and much more) was in the famous unpublished (and largely unwritten) thesis of Bill Butler.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
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
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
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.
participants (6)
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Claudio Hermida -
Gaunce Lewis -
Michael Barr -
Oswald Wyler -
Prof. Peter Johnstone -
Steve Vickers