Michael Barr and I came across upon the following elementary fact, which does not seem to have been noticed so far. (However, Michael's conviction that something like this should be true was based on his proof of theorem 3 in "The Point of the Empty Set", Cahiers XIII-4(1972), pp.357-368.) Let A and B be categories, F:A->B and G:B->A functors, h:id->GF and e:FG->id natural transformations. Then F is left adjoint to G as soon as the transformations a = eF.Fh and b = Ge.hG are either both split epi or both split mono. PROOF. First observe that the naturality alone implies eF.aGF = eF.FbF FbF ------------------------------- | | | FhGF FGeF V FGF ----------> FGFGF --------> FGF | | | | | | |id |eFGF |eF | | | V aGF V eF V FGF -----------> FGF ----------> F Now if some (natural!) right inverses a' and b' of a resp. b are given, we derive eF.a'GF = eF.Fb'F Fb'F FbF FGF ----------> FGF ----------> FGF | | | | |a'GF |a'GF | | V FbF V FGF ----------> FGF | | | | |aGF |eF | | V eF V FGF -----------> F The unit and the counit of the adjunction F-|G will be eta = h :id->GF epsilon = e.Fb' :FG->FG->id The adjunction identities follow: (eta)G G(epsilon) G ------------> GFG ----------> G | | | | | | |b' |GFb' |id | | | V (eta)G = hG V Ge V G ------------> GFG ----------> G F(eta) (epsilon)F F ------------> FGF ----------> F | | | | | | | | |a' a'GF| |Fb'F |id | | | | V F(eta) = Fh V V eF V F ------------> FGF ----------> F When a and b are split monics, one can dualize the above arguments by switching e and h; or simply consider the opposite categories of A and B. Regards to all, Dusko Pavlovic =================================================================== Subj: AUTOMATED REASONING JOB From: dfs@hilbert.maths.utas.edu.au Junior Research Fellow AUTOMATED REASONING PROJECT Applications are invited for a Junior Research Fellowship at the University of Tasmania for a period of 6 months, starting 15 August 1992, with a possibility of renewal for a similar period. The salary will be in the range: $31980-38,950pa, and a contribution toward the cost of a fare to Tasmania will be made. The successful applicant will join a small group working on an ARC-funded Project entitled: A model and programming language for reasoning incorporating Grobner base methods. Applicants should have an Honours degree in Computer Science or Mathematics, with a major in the other discipline. A strong background in algebra, and practical programming experience are desirable. The principle task of the appointee will be disciplined research programming: to refine, extend and test a programming language for reasoning that currently exists in prototype form, written in Mathematica. The language is based on ideas from algebra, category theory and functional programming. Substantial programming skills will be needed, but experience with Mathematica is not required. The position offers the opportunity to combine creative programming in an exciting project with a taste of life in Tasmania --- if it's not the best life-style in the world, what is? Applicants should forward their Curriculum Vitae to reach The Staff Officer, University of Tasmania, PO Box 252C, Hobart, Tasmania 7001, AUSTRALIA by 25 May 1992. They should also ask three referees to submit letters in support of their application direct to the Staff Officer. All correspondence should quote Ref: 48/92. The University of Tasmania is an equal opportunity employer. Email enquiries about the position, and notice of intention to apply may be addressed to the undersigned. ------------------------ Desmond Fearnley-Sander, Mathematics Department, University of Tasmania, PO Box 252C, Hobart, Tasmania 7001, AUSTRALIA. Phone: 002 202445 Fax: 002 202867 AARnet: dfs@hilbert.maths.utas.edu.au ------------------------ ===================================================================