Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].

To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.