I'm going to try to explain what a category is for people with no knowledge of them. So here goes: In a lot of pure math we study specific mathematical objects and we like to compare how similar they are. (1/19)

In the field topology these objects are called spaces and we can think of them as shapes. In algebra there are many but examples include groups, rings, fields etc.. We don't need to know what any of these are for now. (2/19)

When compare these objects we usually do so with some notion of a "map". So in the case of topology we might think of spaces as shapes. For example maybe we think of a circle, call it S, and a plane (like an infinite piece of paper) call it P. (3/19)

Now if I take my circle S and I place it on top of my plane P this shows one way I may "map" my circle onto the plane. In many ways this is a particularly nice map but it's an example nonetheless. (4/19)

So I may want to be able to refer to the map I made that places the circle in the plane. So I'm going to call it "f" and I'm going to denote it f:S --> P as an arrow pointing S to P. (5/19)

Another easy example of a map is a function. You probably remember functions from high school math, things like f(x) = mx+b or whatever. These take in a real number and spit out a different real number. So functions in this case are pointing the real numbers R to itself (6/19)

We can see this in my arrow notation as f:R-->R like this. (7/19)

Ok so this pattern of objects A,B,C or whatever and maps or arrows between them shows up EVERYWHERE in math. A category is a means of generalizing this structure, or abstracting it. So we get to our definition and we will build piece by piece (8/19)

A category is:
(a) A set of objects {A,B,C,...}
(b) A set of arrows between objects

that satisfy certain conditions: (9/19)

The first condition we care about is composition. What composition says is that if I have an arrow:
f:A --> B pointing A to B and another arrow
g:B --> C pointing B to C then there must exist an arrow
g o f : A --> C pointing A to C (10/19)

We think of this composite arrow as the arrow coming from
A--f-->B--g-->C gluing these two other arrows together (11/19)

The second condition is the notion of an identity arrow, this is essentially an arrow that does NOTHING, think mapping a circle onto itself identically. In a category there exists an identity arrow for every object it has:
A-->A an arrow pointing an object to itself (12/19)

The identity arrow may not be the only arrow pointing an object to itself, what makes it unique is that it does nothing when composed with. What I mean by this is that if I have an arrow
f:A-->B and I compose it with the identity arrow of A
A--id-->A--f-->B =A--f-->B (13/19)

That the composite arrow is just "f" again. So this is super kind of abstract. So let's take a step back again. Let's think about my circle and plane example. If I place a circle onto a copy of itself identically and then place it on a plane... (14/19)

then this is the same as just placing it on the plane for all intents and purposes. (15/19)

Our last condition is associativity of composition. You might remember associativity as property of addition or multiplication. What it says is that x+(y+z) = (x+y)+z in the case of addition. (16/19)

So for composition given three arrows
f: A-->B
g: B--> C
h: C--> D
I can take the composite arrow of f and g first or g and h first and get out the same result. We write this like:
h o (g o f) = (h o g) o f
the 'o' means composition (17/?)

Pictorally:
A--(g o f)-->B--h-->C = A--f-->B--(h o g)-->C
(18/19)

So that's all a category is but it has surprising weight to it. Theres a lot we can learn about math from studying this pretty basic structure. YAY! (19/19)