Let $\mathcal{C}$ be a small category. The nlab-article on Isbell duality tells us that there is a functor $$\mathcal{O} : [\mathcal{C}^{\mathrm{op}},\mathsf{Set}] \to [\mathcal{C},\mathsf{Set}]^{\mathrm{op}},\, X \mapsto \bigl(c \mapsto \hom(X,\hom(-,c))\bigr)$$ which is left adjoint to the functor $$\mathcal{S} : [\mathcal{C},\mathsf{Set}]^{\mathrm{op}} \to [\mathcal{C}^{\mathrm{op}},\mathsf{Set}],\, A \mapsto \bigl(c \mapsto \hom(A,\hom(c,-))\bigr),$$ and the fixed objects on either category are called Isbell-self-dual. For example, any (co)variant representable is Isbell-self-dual. This is just a reformulation of the Yoneda Lemma.

Question. What are some interesting examples of categories $\mathcal{C}$ for which not just the representables, but many objects are Isbell-self-dual? Where perhaps the classification of Isbell-self-dual objects is a non-trivial matter?

For example, when $\mathcal{C}=G$ is a group, so that we get an adjunction between right $G$-sets and left $G$-sets, then either $G$ is trivial and it is the unique Isbell-self-dual left/right $G$-set, or $G$ is non-trivial and in this case also the initial and the terminal left/right $G$-sets $0,1$ are Isbell-self-dual. These are all. So this is quite boring, the duality doesn't tell us anything interesting here. A similar thing will happen when $\mathcal{C}$ is a groupoid. I wonder for which $\mathcal{C}$ something interesting happens. Clearly many interesting dualities arise from related adjunctions, but here I am specifically interested in the general Isbell duality discussed at the nlab-article.

  • $\begingroup$ The notation for the adjoints $\mathcal{O}\dashv \text{Spec}$ suggests that nontrivial (and in fact quite complicated sometimes) examples arise in algebraic geometry: see this MO and this link $\endgroup$ – Fosco Loregian Oct 9 '16 at 16:03
  • $\begingroup$ @FoscoLoregian: Please consider the last sentence of my question. $\endgroup$ – HeinrichD Oct 9 '16 at 17:13
  • $\begingroup$ Partially answered here mathoverflow.net/questions/260335 $\endgroup$ – HeinrichD Mar 10 '17 at 11:07

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