Groups of dualities

Abstract:

For arbitrary categories $\mathcal {A}$ and $\mathcal {B}$ , the "set" of isomorphism-classes of dualities between $\mathcal {A}$ and $\mathcal {B}$ carries a natural group structure. In case $\mathcal {A}$ and $\mathcal {B}$ admit faithful representable functors to Set, this structure can often be described quite concretely in terms of "schizophrenic objects" (in the sense of Johnstone’s book on "Stone Spaces"). The general theory provided here allows for a concrete computation of that group in case $\mathcal {A} = \mathcal {B} = \mathcal {C}$ is the category of all compact and all discrete abelian groups: it is the uncountable group of algebraic automorphisms of the circle $\mathbb {R}/\mathbb {Z}$ , modulo its subgroup ${\mathbb {Z}_2}$ of continuous automorphisms.