[[!redirects dagger categories]] #Contents# * table of contents {:toc} ## Definition ## A **dagger category** is a [[dagger precategory]] such that for all $a,b:A$, the function $idtouiso_{a,b}$, as defined in the [[dagger precategory]] article, is an equivalence. The inverse of $idtouiso$ is denoted $uisotoid$. ## Examples ## * Every [[groupoid]] is a dagger category with $f^\dagger = f^{-1}$. * The dagger category $Rel$ of [[set]]s and relations is a dagger category with $f^\dagger$ representing the opposite relation of a relation $f$. * The dagger category [[HilbR]] of Hilbert spaces over the real numbers and linear maps is a dagger category with $f^\dagger$ representing the adjoint linear map of $f$. ## See also ## * [[Category theory]] ## References ## * [[HoTT book]]