 What is a Category ($C$)?
 Structured Sets as Categories
 Categories of Structured Sets
 Categories of Types and Terms in Type Theory
 Categories of Categories
 Size of Categories
 Representable Functor
We can think of category theory as a generalized set theory, where in set theory we have sets and $\in$, but in category theory we have objects and arrows, where arrows are just any kinds of mappings.
So we have a kind of composition structure, where ther order of composition doesn’t matter, but the configuration matters.
And rather than reasoning structurally like PL does, it reasons “behaviorally”.
What is a Category ($C$)?
 Data
 Object collections $C_0$,
 Morphisms: Arrow collection or $hom(C)$,
 We write to say $f$ is a morphism from $a$ to $b$.
 We write $hom_C(a,b)$ to denote all morphisms from $a$ to $b$, which is also called the homclass of all morphisms from $a$ to $b$.
 Note that $hom(C)$ is a collection of all $homs$ expanded.
 Boundary maps. domain: , codomain: ,
 Identity morphism: , which is also called loop endomorphism.
 Composition:
 Composition laws
 Unit laws:
 Associativity law: . From $A$ to $B$ to $C$ to $D$ we call it a path.
 If , we say the $f$ and $g$ arrows are parallel.
 Diagram: If we treat objects as vertices and arrows as directed edges, we have a directed graph
or diagram.
 A comutative diagram is one such that all directed paths with the same starting and end points lead to the same result.
 Whiskering: If we have a diagram that commutes, and we add one more arrow into it, we still have commuting diagram.
 Pasting: If we two diagrams, both commute, and they have a common path, then we can “stick” those two diagrams along that path, and the resulting diagram still commutes.
Structured Sets as Categories
We can construct a category from a set.
 Empty Category: $0$, which has no obejcts at all.
 Singleton Category: $1$, which has only one object, and one morphism, which is the $id$ for the object itself.
 Discrete Category: For a set $S’$, we construct a category $S$, where the objects are just the elements of the set, $S_0:=S’$, and the mophisms are only the $id$ for each object, .

Preorder Category: For a preordered set ($P’,\leq$) (a set with a reflexive and transitive binary relation on it), a category $P$ with

objects: $P_0:=P’$

arrows:
 identities:
 composition:

So the simplest category satisfying above requirements is call Interval Category ($I$), where there are only two objects, two $id$ rules, and one arrow from one to the other.

 Monoid Category: For a monoid ()
(a set $M’$, an associative binary operation $*$,
and a unit for the operation $\varepsilon$), the category $M$ with
 objects:
 arrows:
 identity:
 composition:
 An example monoid is ()
Categories of Structured Sets
Some math background: According to Russell’s Paradox, we cannot have a set of all sets, but we can have category of all sets. A class is a collection of sets (or other mathematical objects) that can be unambiguously defined by a property that all its members share. A class that is not a set is called a proper class, and a class that is a set is sometimes called a small class.
 Category of sets $SET$, where objects are just sets and arrows are functions
 Category of preordered sets $PREORD$, where objects are the preordered sets, and arrows are monotone maps (functions that preserve the order).
 Category of monoids $MON$
 objects: monoids
 arrows: monoid homomorphisms (structure preserving maps of monoids)
Categories of Types and Terms in Type Theory
 objects: interpretations of types or typing context, or
 arrows:
 identity:
 compositionn:
 “baby tyep theory” simple example:
Categories of Categories
 What is the morphism of categories? We define functor: For categories $C,D$,
functor $F$ from $C$ to $D$ is:
 a map
 a map such that it
 respects boundaries:
 preserve identity morphisms:
 preserve composition morphisms:
 identity functors and functor composition are just as expected
Size of Categories
Some definitions:
 a collection is either a $proper\ class$, which is $large$, or a $set$, which is $small$.
 $C\ a\ small\ category\ if\ C_1\ is\ small$, meaning $C_1\ is\ a\ set$ (which implies that $C_0\ is\ small$ because of the identity rule: $C_1$ is at least the same size of $C_0$)
 $CAT$: Category of small categories, note: $CAT\notin CAT_0$
 A category is $locally\ small$ means all homs are $small$:
Representable Functor
A representable functor is a functor of a special form that map a locally small category into the category of sets, namely $SET$.
For a category $C$, if we fix an object in category $C$, $X:C$, we can define a functor denoted $F$ or $hom(X\rightarrow )$:
$X$ is known as the representitive of the representable functor $F$.
To proof $F$ is a functor, we need to proof:
Proofs are skipped ;)