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.
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$\begingroup$ "Self-dual adjunction" is probably what I'd call it myself. Another famous example is Pontryagin duality. $\endgroup$– Todd TrimbleCommented Jul 14, 2011 at 19:17
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$\begingroup$ If I am parsing your notation correctly, another example of such an adjunction is given by the dual functor on the category of Banach spaces and linear contractions. That it is monadic (the induced monad is given on objects by the bidual) is due to Linton. $\endgroup$– G. RodriguesCommented Jul 14, 2011 at 20:49
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(I'm a bit confused by your notation (what is $I$?), but if you mean what I think you mean...)
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 & Moerdijk, Sheaves in Geometry and Logic, chapter IV, section 5.
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.)
Hayo Thielecke has studied self-adjunctions as a way to understand the notion of 'continuations' in computer science. See his Edinburgh Ph.D. thesis.
answered Jul 14, 2011 at 19:24
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3$\begingroup$ I thought it was Pare who made the observation about monadicity of the power set/object functor (?). $\endgroup$ Commented Jul 14, 2011 at 19:27
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$\begingroup$ Yes, now that I check I see that Johnstone attributes it to Paré in Topos Theory. I must have been thinking of something else. $\endgroup$ Commented Jul 14, 2011 at 20:26
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$\begingroup$ @Finn Lawler: $I$ is a morphism between categories, i.e. a functor. $\endgroup$– beroalCommented Jul 16, 2011 at 10:37
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$\begingroup$ @Finn Lawler: Thanks, that is exactly what I was looking for. The power-object functor is a particular case of contravariant exponential functors. Contravariant exponential functors lead to the continuation monad. I will probably stick to MacLane's term. $\endgroup$– beroalCommented Jul 16, 2011 at 10:43