  In this article, we introduce and develop the notion of parametrised
Poincar\'{e} duality in the formalism of parametrised higher category theory by
Martini-Wolf, in part generalising Cnossen's theory of twisted ambidexterity to
the nonpresentable setting. We prove several basechange results, allowing us to
move between different coefficient categories and ambient topoi. We then
specialise the general framework to yield a good theory of equivariant
Poincar\'{e} duality spaces for compact Lie groups and apply our basechange
results to obtain a suite of isotropy separation methods. Finally, we employ
this theory to perform various categorical Smith-theoretic manoeuvres to prove,
among other things, a generalisation of a theorem of Atiyah-Bott and
Conner-Floyd on group actions with single fixed points.
