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}

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