  Drazin inverses are a fundamental algebraic structure which have been
extensively deployed in semigroup theory and ring theory. Drazin inverses can
also be defined for endomorphisms in any category. However, beyond a paper by
Puystjens and Robinson from 1987, not much has been done with Drazin inverses
in category theory. As such, here we provide a survey of the theory of Drazin
inverses from a categorical perspective. We introduce Drazin categories, in
which every endomorphism has a Drazin inverse, and provide various examples
including the category of matrices over a field, the category of finite length
modules over a ring, and finite set enriched categories. We also introduce the
notion of expressive rank and prove that a category with expressive rank is
Drazin. Moreover, we study Drazin inverses in mere categories, in additive
categories, and in dagger categories. In an arbitrary category, we show how a
Drazin inverse corresponds to an isomorphism in the idempotent splitting, as
well as explain how Drazin inverses relate to Leinster's notion of eventual
image duality. In additive categories, we explore the consequences of the
core-nilpotent decomposition and the image-kernel decomposition, which we
relate back to Fitting's famous results. We then develop the notion of Drazin
inverses for pairs of opposing maps, generalizing the usual notion of Drazin
inverse for endomorphisms. As an application of this new kind of Drazin
inverse, for dagger categories, we provide a novel characterization of the
Moore-Penrose inverse in terms of being a Drazin inverse of the pair of a map
and its adjoint.
