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 $C_0$, \(A:C:=A\in C_0\)
  • Morphisms: Arrow collection \(C_1\) or $hom(C)$, \(f::C:=f\in C_1\)
    • We write \(f:a\rightarrow b\) 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 hom-class of all morphisms from $a$ to $b$.
    • Note that $hom(C)$ is a collection of all $homs$ expanded.
    • Boundary maps. domain: \(\delta^-\), codomain: \(\delta^+\), \(C(A\rightarrow B):=\{f::C|\delta^-(f)=A\wedge\delta^+(f)=B\}\)
    • Identity morphism: \(id(A):C(A\rightarrow A)\), which is also called loop endomorphism.
    • Composition: \(for\ f:A\rightarrow B,g:B\rightarrow C,we\ have\ f\cdot g=g\circ f:A\rightarrow C\)
• Composition laws
  • Unit laws: \(for\ f:A\rightarrow B,we\ have\ id(A)\cdot f=f=f\cdot id(B)\)
  • Associativity law: \(for\ f:A\rightarrow B,g:B\rightarrow C,h:C\rightarrow D,we\ have\ (f\cdot g)\cdot h=f\cdot(g\cdot h)\). From $A$ to $B$ to $C$ to $D$ we call it a path.
  • If \(\delta^-(f)=\delta^-(g)\&\&\delta^+(f)=\delta^+(g)\), 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, \(S_1:=\{id(x)|x\in S'\}\).

• 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: \(P(x\rightarrow y):= \left\{ \begin{array}{rcl} \{"x\leq y"\} & \mbox{if} & x\leq y \\ \emptyset & \mbox{otherwise} \\ \end{array}\right.\)

    • identities: \(id(x):=x\leq x\)
    • composition: \(x\leq y\cdot y\leq z:=x\leq z\)

  • 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 (\(M',*,\varepsilon\)) (a set $M’$, an associative binary operation $*$, and a unit for the operation $\varepsilon$), the category $M$ with
  • objects: \(M_0:=\{\star\}\)
  • arrows: \(M(\star\rightarrow\star):=M'\)
    • identity: \(id(\star):=\varepsilon\)
    • composition: \(x\cdot y:=x*y\)
  • An example monoid is (\(\mathbb{N},+,0\))

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, \([\![A]\!]\) or \([\![\Gamma]\!]\)
• arrows: \([\![\Gamma\vdash M:A]\!]:C([\![\Gamma]\!]\rightarrow[\![A]\!])\)
  • identity: \([\![x:A\vdash x:A]\!]=id([\![A]\!])\)
  • compositionn: \([\![x:A\vdash [y/M]N:C]\!]=[\![x:A\vdash M:B]\!]\cdot[\![y:B\vdash N:C]\!]\)
• “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 \(F_0:C_0\rightarrow D_0\)
  • a map \(F_1:C_1\rightarrow D_1\) such that it
    • respects boundaries: \(for\ f:C(A\rightarrow B),we\ have\ F_1(f):D(F_0(A)\rightarrow F_0(B))\)
    • preserve identity morphisms: \(F(id(x))=id(F(x))\forall x\in C_0\)
    • preserve composition morphisms: \(F(f\cdot g)=F(f)\cdot F(g)\forall f:X\rightarrow Y,g:Y\rightarrow Z\in C_1\)
    • 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 $small$: \(\forall a,b\in C_0,hom(a,b)\ is\ small\)

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 ;)