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}

Category Theory II 2: Limits, Higher order functors

1.2 Introduction to Limit 

Analogy : Product to Cones (Limit)

2.1 Five categories used to define Limit:

  1. Index category (I)
  2. Category C: Functors (constant ,  D)
  3. Cones (Lim D)
  4. Functor Category [I, C ]:objects are (constant, D ), morphisms are natural transormations
  5. Set category of Hom-Set Cones [I,C] to Hom-Set C (c , Lim D )

2.2 Naturality

3.1 Examples: Equalizer

CoLimit = duality of Limit (inverted cone = co-Cone)

Functor Continuity = preserve Limit

BM Category Theory 10: Monad & Monoid

10.1 Monad
Analogy

Function : compose “.“, Id

Monad: compose “>>=“, return

Imperative (with side effects eg. state, I/O, exception ) to Pure function by hiding or embellishment in Pure function but return “embellished” result.

10. 2 Monoid

Monoid in category of endo functors = Monad

Ref Book : 

What is the significance of monoids in category theory? by Bartosz Milewski 

Category Theory 9: Natural Transformations, BiCategories

In essence, in all kinds of Math, we do 3 things: 

1) Find Pattern among objects (numbers, shapes, …), 
2) Operate inside the objects (+ – × / …), 
3) Swap the object without modifying it (rotate, flip, move around, exchange…).

Category  consists of :
1) Find pattern thru Universal Construction in Objects (Set, Group, Ring, Vector Space, anything )
2) Functor which operates on 1).
3) Natural Transformation as in 3).

\boxed {\text {Natural Transformation}}
\Updownarrow

\boxed {\text {Morphism of Functors}}

Analogy:

Functors (F, G) := operation inside a container 
\boxed { F :: X \to F_{X}, \:  F :: Y \to F_{Y}}

\boxed {G :: X \to G_{X}, \: G :: Y \to G_{Y}}

Natural Transformation  ({\eta_{X}, \eta_{Y}}) := swap the content ( F_{X} \text { with } G_{X},  F_{Y} \text { with } G_{Y} ) in the container without modifying it.
\boxed{\eta_{X} :: F_{X} \to G_{X} , \: \eta_{Y} :: F_{Y} \to G_{Y}}

9.2 Bicategories 

“Diagram Chasing”:

2- Category

Cat = Category of categories (C, D)

The functors {F, G} instead of being a Set (“Hom-Set”) – like functions  form a function object “Exponentialfunctors also form a category, noted : \boxed {[C,D] = D^{C} }

BiCategory (different from 2-Category): the Associativity and Identity are not equal (as in 2-Category),  but only up to Isomorphism.
Note : when n is infinity,  n-Category & Groupoid (HOTT: Homotopy Type Theory)

Reading Book: chap 10

Curry-Howard-Lambek Isomorphism

Curry-Howard-Lambek Isomorphism

\boxed {\text {Category} \iff \text {Algebra} \iff \text {Logic} \iff \lambda \text {-Calculus}}

Below the lecturer said every aspect of Math can be folded out from Category Theory, then why not start teaching Category Theory in schools.

That was the idea proposed by Alexander Grothendieck to the Bourbakian Mathematicians who rewrote all Math textbooks after WW2, instead of in Set Theory, should switch to Category Theory. His idea was turned down by André Weil.


\boxed { a^{b + c} = a^{b} \times a^{c} }

\boxed {\text {Left-side: Either b c } \to a}

\boxed {\text {Right-side: } (b \to a , c\to a) }

\boxed { (a^{b})^{c} = a^{b \times c}}

\boxed { c \to (b \to a) \sim (b ,c) \to a}

\boxed { (a \times b)^{c} = a^{c} \times b^{c}}

\boxed {c \to (a,b) \sim (c \to a , c \to b)}