# BM 5&6 : Category Adjunctions 伴随函子

Adjunction is the “weakening of Equality” in Category Theory.

Equivalence of 2 Categories if:

5.2  Adjunction definition: $(L, R, \epsilon, \eta )$ such that the 2 triangle identities below ( red and blue) exist.

6.1 Prove: Let C any category, D a Set.

$\boxed {\text {C(L 1, -)} \simeq \text {R}}$

${\text {Right Adjoint R in Set category is }}$ ${\text {ALWAYS Representable}}$

1 = Singleton in Set D

From an element in the singleton Set always ‘picks’ (via the function) an image element from the other Set, hence :
$\boxed {\text{Set (1, R c) } \simeq \text{Rc }}$

Examples : Product & Exponential are Right Adjoints

Note: Adjoint is a more powerful concept to understand than the universal construction of Product and Exponential.

6.2

$\boxed {\text {Every Adjunction gives a Monad}}$

vice-versa.

$\boxed {\text {Left Adjoint: L } \dashv \text { R }}$

$R \circ \ L = m = \text { Monad}$

$L \circ \ C = \text { Co-Monad}$

With Product (Left Adjoint) and Exponential (Right Adjoint) => $\text {State Monad}$