Category Theory Intro

What is a Category

A category $\mathcal{C}$ consists of four things:

• Objects: a class $Obj(\mathcal{C})$ called the class of objects of $\mathcal{C}$
• Morphisms: For all $A,B\in Obj(\mathcal{C})$, a set $Ho{m}_C(A,B)$ of morphisms (called the homset of morphisms from A to B)
• Composition: The morphisms composition function $\circ :Ho{m}_C(A,B)\times Ho{m}_C(B,C)\to Ho{m}_C(A,C)$
• Identity: For all $A\in Obj(\mathcal{C})$, a morphism $1_A\in \mathcal{C}(A,A)$ called the identity on A. They follow the following properties:
  • This composition is associative: if $f\in Ho{m}_C(A,B)$, $g\in Ho{m}_C(B,C)$, $h\in Ho{m}_C(C,D)$ $(hg)f=h(gf)$
  • for all $f\in Ho{m}_C(A,B)$ we have $f{1}_A=f,1_{B}f=f$.

What is a class

A class is a collection of objects. One can think it as a fancy way of calling a collection.

Tip

Every set can be considered as a class

We cannot have a set of all sets (because of Russell paradox) However, we can have a class of all sets, which is called a proper class (a class that is not a set) A proper class cannot be a member of any other set of class.

Difference between Morphism and Function

Aspect	Function	Morphism
Definition	A mapping $f:A\to B$ between two sets	An arrow $f:A\to B$ in an arbitrary category
Domain & Codomain	Both must be sets	Can be any objects (sets, groups, spaces, etc.)
Structure Requirement	No extra requirements beyond “each input has a unique output”	Must preserve the structure of its category (e.g. group law, topology)
Identity	Identity function ${\mathrm{id}}_A$ on set A	Identity morphism $1_A$ on object AA
Examples	Any rule mapping elements of one set to another (e.g. $x\mapsto x^2$)	In Set: functions; in Grp: group homomorphisms; in Top: continuous maps
See Morphisms

Examples 1

Category	Objects	Morphisms $A\to B$	Composition	Identity on A
$\mathbf{Set}$	all sets	all functions $A\to B$	usual composition of functions	identity function ${\mathrm{id}}_A$
$\mathbf{Rel}$	all sets	all binary relations $R\subseteq A\times B$	relational composition: two Relations $R\subseteq A\times B,S\subseteq B\times C$: $S\circ R=\{(a,c)\in A\times C\mid \exists b\in B$ so that $(aRb\wedge bSc)\}$	the equality relation $\{(a,a)\}$
Similar Morphisms can also be multivalued map or partial functions…

Examples 2

Consider an abstract Example Let C be a category, and let A be an object of C We define a category $C_A$ with:

• $Obj(C_A)$= all morphisms from any objects of C to A (e.g. $Z\stackrel{f}{\to }A$)
• Morphisms:
  • 
• Composition:
  • 
  • Because C is a category, we can remove the central arrow according to the composition law,
  • 
  • which gives us the composition in $C_A$ (it commutes)

Examples 3

Consider another coslice category

Let category C=Set and A = a fixed singleton $\{\ast \}$ (any set that contains only one single element)

Define category Set* as the category obtained by considering morphisms in C from the fixed object A to all objects in C.

• An object in Set* is a morphism $f:\{\ast \}\to S$
  • This object should can be described by $(S,s)$ where $s\in S$ is any element of $S$. （a pointed set）
  • A function $f$ from the singleton set $\{\ast \}$ to any set $S$ is uniquely determined by the image of that single point—namely, once the element $s=f(\ast )$ is specified, the entire function $f$ is completely characterized. Thus we don’t need to write out how $f$ looks like.
• A morphism between two such objects is $(S,s)\to (T,t)$ with set function $\sigma (s)=t$
• s and t are here basepoints.

• Category Example.excalidraw

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  Object of Set* are called pointed sets

Example 4

Consider $C_{A,B}$ $Obj(C_{A,B})=$ Morphisms: