Category Theory (Apr.2021. updating continuously)

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Definition of a category

  • Objects and arrows exist.
  • There should be identity arrows for all objects.
  • Arrows should be composited.
  • Associativity holds.

Terminology: domain and codomain

$\mathrm{dom}(f)\xrightarrow{f} \mathrm{cod}(f)$

Directed graphs and categories

If $f:A\rightarrow B$ and $g:B\rightarrow C$ exist but $h := g\cdot f$ doesn’t, it violates the rule of composition. Hence the graphs are not categories.

Final object

When every objects in a category has an arrow to the object $F$, the object is called a final object.


  • A Functor is a mapping between categories.
  • It’s like an arrow from category to category.
  • It preseves compositions $F(g\cdot f) = F(g)\cdot F(f)$.
  • Identitiy arrows are mapped to identity arrows.

Contravariant functor

A contravariant functor $F$ from a category $C$ to a category $D$ is simply a functor from the opposite category $\mathrm{C_{op}}$ to $D$, s.t., $$ \mathrm{F}(a) = a, \
f(a) = b, \
\mathrm{F}(f)(b) = a. $$

One sometimes says covariant functor when referring to non-contravariant functors, for emphasis.

Small category and large category

Small category: Small set: Universe (Wiki): Universe (nlab): Grothendieck universe: Red herring:


In a category $\mathrm{C}$, two objects $A$ and $B$ are called isomorphic If therea is arrows $f$ and $g$ such that $f:A\rightarrow B,\ g:B\rightarrow A$ and $f\cdot g =1_A,\ g\cdot f=1_B$. Such $f$ and $g$ are called isomorphism.


A category which has only one object is called monoid.

Product category

Imagine a simple tuple which can be defined from two categories.

Projection functors

Just extract a single category from a product category. $$ \mathrm{pr}_1(C,D) = C \
\mathrm{pr}_1(f,g) = f $$

Dual category

dual category = opposite category = arrows reversed. denoted as A over asterisc, like $A^*$


  • Free cateogry
  • arrow category
  • slice category


$$ \mathrm{Sets}_{\mathrm{fin}} $$

is the category of all finite set sand functions between them. <- very large!

Poset and Preordered set

  • Poset = partially ordered set = reflexivity+transitivity+antisymmetry.
  • Preorder = reflexivity+transitivity


$$ \mathrm{Rel} $$ objects = sets and arrows binary relations $aRb$ is true.

$$ Pos $$ is large also.


$$ Cat $$ the category of all categories and functors (linke self-refrence)


1.5 isomorphism (category)

$$ A \cong B $$

f:A -> B is isomorphism if g:B->A exist s.t. gf = 1A fg=1B

this is one of mophism.

Final object

Mrphism ist arrow

Multi Terminal Objekt are All isomorphic

In Terms of se Theorie, Singleton ist Onkyo Terminal Objekt

preorder set is similar with category without parallel arrows

category product is the most good candidate. most good means unique arrow

we can ask is there a product in this category?

initial morphism

initial property dual -> Universal morphism


  • Universal properties: basical
  • Natural transformation: map from a functor to another functor
  • Corn: continuously drive a structure
  • Limit: No more natural transformation in the corn.Generalized category sproduct?