From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9033 Path: news.gmane.org!.POSTED!not-for-mail From: Joost Vercruysse Newsgroups: gmane.science.mathematics.categories Subject: Re: partial adjoints Date: Tue, 15 Nov 2016 22:44:42 +0100 Message-ID: References: Reply-To: Joost Vercruysse NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (Mac OS X Mail 9.3 \(3124\)) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1479322087 5543 195.159.176.226 (16 Nov 2016 18:48:07 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 16 Nov 2016 18:48:07 +0000 (UTC) Cc: categories@mta.ca To: Peter Bubenik Original-X-From: majordomo@mlist.mta.ca Wed Nov 16 19:48:02 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.28]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1c75FY-0007YF-8O for gsmc-categories@m.gmane.org; Wed, 16 Nov 2016 19:48:00 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:53075) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c75Eg-00012C-VP; Wed, 16 Nov 2016 14:47:06 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c75Ei-0006sI-0r for categories-list@mlist.mta.ca; Wed, 16 Nov 2016 14:47:08 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9033 Archived-At: 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.=20 I am not sure if this is relevant for what you want to do, but maybe the = technique can be useful.=20 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=3D Fc\circ bG\circ a=3DbG\circ Fc\circ a. The Morita maps are given by f(a\otimes b) =3D 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)=3D1_F and g(e\otimes u)=3D1_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 = =E2=80=9Cunits=E2=80=9D and =E2=80=9Ccounits=E2=80=9D 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 =E2=80=9Cquasi-adjoint=E2=80=9D 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 =E2=80=9CcoFrobenius coalgebras=E2=80=9D. For details, you can have a look at=20 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 = wrote: >=20 > Hello all, > Sender: categories@mta.ca > Precedence: bulk > Reply-To: Peter Bubenik >=20 > Has anyone studied pairs of functors for which there exists a unit, > but not a counit? Is there a name for such things? >=20 > 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. >=20 > Thanks, > Peter >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]