Steve Lack writes:
John Baez described the ``platonic idea of an equivalence'' or ``walking equivalence''. (This has also been called the ``free-living equivalence''.) He also describes the free-living adjoint equivalence, and the homotopy-theoretic relationship between the two.
Similarly, one can construct the free-living adjunction. This was done in
S. Schanuel and R. Street, The free adjunction, Cah. Top. Geom. Diff. 27:81-83, 1986.
Thanks! In the next issue of This Week's Finds I'm giving a long introduction to the "walking adjunction", followed by a translation of Michael Mueger's paper
From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories, available at math.CT/0111204
into facts about the "walking biadjunction". I knew someone had studied the walking adjunction but didn't know the reference, and had been meaning to ask here.
The free-living pseudo-adjunction was constructed in
Stephen Lack, A coherent approach to pseudomonads, Adv. Math. 152:179-202, 2000.
Right! Thanks for reminding me! I'll mention that too. Also: has anyone here written about the "walking biadjunction"? It's actually very interesting. Best, jb