# Category Theory

January 4, 2012 21 Comments

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:

A **category** consists of a collection of objects, and for each , a collection of maps satisfying,

- Associativity: Given , , and , .
- Identity: For each , there is an identity morphism such that if , .

Sometimes we write to mean that . Also, when the category is evident, We’ll often drop the subscript in favor of .

Here’s how we like to picture categories:

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

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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.