a “self-dual” adjunction - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:22:51Z http://mathoverflow.net/feeds/question/70361 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70361/a-self-dual-adjunction a “self-dual” adjunction beroal 2011-07-14T18:13:37Z 2011-07-14T19:24:54Z <p>Is there a name for $(U,\eta)$ such that $(\eta, \eta^{op}):U^{op}\dashv U$ (is an adjunction). To clarify — $C:category$, $(I,I^{op})$ is the contravariant isomorphism with $I:C^{op}\to C$, $U:C^{op}\to C,\ U^{op}:=I^{op}\circ U\circ I^{op},\ U^{op}:C\to C^{op}$, $\eta:id(C)\to U\circ U^{op},\ \eta^{op}:= I^{op}\eta I,\ \eta^{op}:U^{op}\circ U\to id(C^{op})$. E.g. in CCC, contravariant exponential functor and $\eta(a):=\lambda(x:a) f.f\ x$ is such an adjunction.</p> http://mathoverflow.net/questions/70361/a-self-dual-adjunction/70366#70366 Answer by Finn Lawler for a “self-dual” adjunction Finn Lawler 2011-07-14T19:24:54Z 2011-07-14T19:24:54Z <p>(I'm a bit confused by your notation (what is $I$?), but if you mean what I think you mean...)</p> <p>I don't think there is an 'official' name for these things, but I've seen the term 'self-adjoint' used, sometimes qualified by 'on the left' or 'on the right' according to whether $U \dashv U^{op}$ or $U^{op} \dashv U$. See e.g. Mac Lane &amp; Moerdijk, <em>Sheaves in Geometry and Logic</em>, chapter IV, section 5.</p> <p>I believe it was Manes who observed that the power-object functor $P \colon E^{op} \to E$ of an elementary topos $E$ is not only self-adjoint on the right but monadic, with the corollary that toposes have finite colimits. (See loc. cit.)</p> <p>Hayo Thielecke has studied self-adjunctions as a way to understand the notion of 'continuations' in computer science. See his Edinburgh Ph.D. thesis.</p>