Mathematics is a *huge* subject.

Category theory is one area of mathematics dedicated to exploring the commonality of structure between different branches of mathematics.

Categorical language allows us to ascend a layer of abstraction, and recognize the obvious underlying principles that guide seemingly unrelated concepts. Generality facilitates connections.

#### What is a category?

A category $C$ consists of:

- a class of objects $Ob(C)$
- For every ordered pair of objects $X$ and $Y$, a set $C(X,Y)$ of morphisms with
*domain*$X$ and*range*$Y$ [$C(X,Y)$ is possibly empty] Note: $f \in C(X,Y)$ $\equiv$ $f : X\rightarrow Y$ $\equiv$ $X \overset{f}{\rightarrow} Y$. - For every object an identity morphism $Id_x \in C(X,X)$.
- A composition law $$C(X,Y) \times C(Y,Z) \rightarrow C(X,Z)$$ $$(g,f) \rightarrow f\cdot g$$

The concept of *composition* follows naturally from the definition of path equivalence in graph theory: *Two paths with the same source as destination are equal.* Additionally, categories must satisfy the laws of *associativity* and* identity*.

#### Category Laws

Categories must obey 2 laws:

- Composition must be associative:
- Every object $a$ in $C$ has a morphism $id_a$ which is equivalent to a loop in graph theory. The identity morphism $id_a$ connects the object $a$ to itself, $$id_a: a \rightarrow a$$.

*Two paths are equal if the source and destination of the paths are equal.* With this in mind, we can represent the category laws of identity and associativity with diagrams.

Follow the arrows, recall that we write function composition backward! If we traverse $f$ then $g$, it is the convention to write $g \circ f$

Examples:

Groups, together with group homomorphisms, form a category (we will discuss these next lecture for those who have not danced with abstract algebra).

Each of the natural numbers is a category:

#### Categories have some nice properties

Any property which can be expressed in terms of (category, objects, morphism, and composition):

**Dual:**$D$ is $C$ with reversed morphisms**Initial:**$Z \in obj(C)$ s.t. $\forall Y \in obj(C)$, #$hom(Z,Y) =1$. In other words: an object is initial if there exists a unique morphism from that object to any other object in $C$.**Terminal:**$T \in obj(C)$ s.t. $T$ is initial in the dual of $C$**Functor:**Structure preserving mapping between categories

#### Homomorphims Between Categories: What the func is a functor?

An example of associating morphisms from $C$ to $D$ with the functor $F(C)$.

#### A Reflection on the Unfication of Familiar Concepts

A few categories you have likely encountered before without recognizing it:

- Set (sets and functions)
- Vec (vectorial spaces and linear transformations)
**Top (topological spaces and continuous maps)**- Grp (groups and homomorphisms) — we will be discussing these next lecture for those who have not danced with abstract algebra.
- Ab (abelian groups and homomorphisms)
- Hask (Haskell types and functions)
- Cat (categories and functors)

A categorical key to highlight some of the relationships between structure preserving maps:

Here are some more advanced examples for those who feel groovy and algebraic:

- $R$-Mod (R-modules and homomorphisms)
- $Gr_R$ ($\mathbb{Z}$-graded $R$-modules and graded $R$-module homomorphisms)