# Category Theory

The idea with category theory is that lots of ideas appear over and over again in every branch of mathematics: We start by defining the object of interest. Then we realize that the interesting stuff comes from the structure preserving maps between objects. When they make sense, we look at images and kernels and quotients (oh my!). Category theory is just a way to point out the similarities in an incredibly generalized setting. So what is a category?

A category is a collection of objects, and a collection of maps between them. We’ll sometimes call these arrows, or maps, or morphisms, but it’s all the same. They’re essentially functions (though to be perfectly precise, they may be maps between proper classes and not sets). They have a few more properties, so let’s spell it out precisely.

#### Definition:

category $\mathcal C$ consists of a collection of objects, $Ob(\mathcal C)$ and for each $X,Y\in Ob(\mathcal C)$, a collection of maps $\hom_{\mathcal C}(X,Y)$ satisfying,

• Associativity: Given $f:W\to X$, $g:X\to Y$, and $h:Y\to Z$, $(h\circ g)\circ f=h\circ(g\circ f)$.
• Identity: For each $X\in Ob(\mathcal C)$, there is an identity morphism $1_X$ such that if $f\in\hom_{\mathcal C}(X,Y)$, $f\circ 1_X=f=1_Y\circ f$.

Sometimes we write $f:X\to Y$ to mean that $f\in\hom_{\mathcal C}(X,Y)$. Also, when the category is evident, We’ll often drop the subscript $\mathcal C$ in favor of $\hom(X,Y)$.

Here’s how we like to picture categories:

We often won’t draw the identity arrows to avoid having the pictures getting to messy.

### 21 Responses to Category Theory

1. Pingback: Examples of Categories « Andy Soffer

2. Pingback: Initial Objects « Andy Soffer

3. Pingback: Terminal objects « Andy Soffer

4. Pingback: Duality « Andy Soffer

5. Pingback: Monos and Epis « Andy Soffer

6. Pingback: Pedantry « Andy Soffer

7. Pingback: Functors « Andy Soffer

8. Pingback: Products (part 2) « Andy Soffer

9. Pingback: Product (part 3) « Andy Soffer

10. Pingback: Back to functors « Andy Soffer

11. Pingback: Naturality (part 1) « Andy Soffer

12. Pingback: Naturality (part 2) « Andy Soffer

13. identities are unique: if we have 1_x and 1’_x, both identites on x, then 1_x = 1_x o 1’_x = 1’_x.