Re: partial adjoints

[Joost Vercruysse <jvercruy@ulb.ac.be>]

Dear Peter,

A few years ago, I have been looking to a generalization of adjoint functors, where there is not necessarily a unit or counit. 
I am not sure if this is relevant for what you want to do, but maybe the technique can be useful. 

Here is a short description:

If F:C\to D and G:D\to C are two functors, and suppose that everything is k-linear (where k is a commutative ring, but this linearity is not essential),
then you can construct a Morita context as follows
(Nat(F,F), Nat(G,G)^op, Nat(1,FG), Nat(GF,1), f , g)
Here Nat(F,F) and Nat(G,G)^op are monoids (k-algebras) for the usual composition of natural tranformations, and Nat(1,FG) and Nat(GF,1) are bimodules between these, e.g. for a\in Nat(1,FG), b\in Nat(F,F) and c\in Nat(G,G)^op, we have b.a.c= Fc\circ bG\circ a=bG\circ Fc\circ a.
The Morita maps are given by f(a\otimes b) = Fb\circ aF\in Nat(F,F) with a\in Nat(1,FG) and b\in Nat(GF,1), and a similar formula for g.

Now you can observe that F and G are adjoint if and only if the above Morita context contains invertible elements that make it a strict context; explictly this means that there exist elements u\in Nat(1,FG) and e\in Nat(GF,1) such that f(u\otimes e)=1_F and g(e\otimes u)=1_G. Of course, u and e are exactly the unit and counit of the adjunction.

The generalized notions of adjoint functors I mentioned above, were obtained by considering arbitrary surjective Morita maps, which means that you have not a single unit and counit, but finite sets of “units” and “counits” which satisfy still an appropriate condition (for this to work you need indeed to work in the k-linear, or at least an enriched, setting). It is temptative to call “quasi-adjoint” functors since it is related to quasi-Frobenius algebras.

Another variation is to consider the case where f(u\otimes e) is not the identity natural transformation on F, but it is a natural transformation that is the identity map on FX for only a (finite) number of objects X in C.
This is related to “coFrobenius coalgebras”.

For details, you can have a look at 
M. Iovanov and J. Vercruysse, Cofrobenius Corings and adjoint Functors, Journal of Pure and Applied Algebra, 212 (9), 2027-2058 (2008).

Best wishes,
Joost.


> On 12 nov. 2016, at 00:26, Peter Bubenik <peter.bubenik@gmail.com> wrote:
> 
> Hello all,
> Sender: categories@mta.ca
> Precedence: bulk
> Reply-To: Peter Bubenik <peter.bubenik@gmail.com>
> 
> Has anyone studied pairs of functors for which there exists a unit,
> but not a counit? Is there a name for such things?
> 
> In the case of interest, the functors are between Cat and Met, the
> category of (extended pseudo-) metric spaces and 1-Lipschitz maps. The
> unit provides a characterization of certain coherent maps of metric
> spaces. Further details may be found in
> https://arxiv.org/abs/1603.07406.
> 
> Thanks,
> Peter
> 



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