Adjoint functors induced by adjoint linear transformations

Self Adjoint Linear Transformations

1 Definition of the Adjoint Let V be a complex vector space with an inner product < , and norm , and suppose that L : V → V is linear. If there is a function L * : V → V for which Lx, y = x, L * y (1.1) holds for every pair of vectors x, y in V , then L * is said to be the adjoint of L. Some of the properties of L * are listed below. Proof. Introduce an orthonomal basis B for V. Then find the m...

Dualities and Equivalences Induced by Adjoint Functors

Adjoint Functors and Heteromorphisms

Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important in mathematics. Our focus here i...

On Adjoint and Brain Functors

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heterom...

Adjoint functors; categories in topology

In this section, we develop the some important categorical definitions and ideas which will be used throughout this paper. For a more complete treatment, the interested reader should consult either [ML-1971], [H-1970] or [M-1967]. Definition 1.1: A metacategory (which we typically denote as C or D) is a pair C = (OC,MC) where OC is considered to be the collection of objects of C and MC is consi...