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Category Theory

For

The Mathematically Inclined Person

A Book for Those who like math and categories but don’t know much about them

Written By

Kasra Mossayebi

2021

# Introduction

In a subject as abstract and unique as category theory, it is often hard to become acclimated to the common techniques and tools in category theory, even though they be extremely common in what has already been studied due to the level of abstraction. To help cure, or at least minimize the massive amounts of confusion that occur that encountering a learning curve like abstraction bring, it’s best to learn the basic intuition behind concepts before generalizing and formalizing them, for abstracting without a general idea of what you’re abstracting leads only to thinking about definitions instead of thinking about how these definitions relate to what they generalize and how they can be used.

In category theory, the most important idea is obviously a *category*. A category consists of three parts:

Objects - Objects are essentially the "elements" of a category, analogous to the elements of a set

Morphisms - Morphisms are structure preserving "maps" or "functions" between objects in a category

Composition of Morphisms - Composing morphisms, like composing functions, can be expressed as \(g\circ f\) = g(f(x)),

but there are also other ways to compose morphisms, so it is important to a category to specify composition

Firstly, it’s good to think of what the word "category" means. A category is defined as a division of people or things that share a common characteristic. In category theory, that definition still rings true, but these categories of mathematical objects all share the same mathematical "structure". For example, a monoid is a set with the added structure of an associative binary operation(one taking two inputs and giving one output, for example multiplication, *(a,b) = a*b = c), and an identity. This means that there can exist a category with objects being monoids. However, there are two parts of the category that we have still not provided: morphisms and composition of morphisms. Well, the first is actually easy because there already exists a name for structure preserving morphisms between monoids. These are called (monoid) *homomorphisms*. The structure preserved by these homomorphisms is, as some may have already guessed, the binary operation of the monoid, along with the condition that the identity is preserved, in other words the identity is mapped to the identity. Now, we can define the composition of morphisms the same we define normal function composition, the composition of two morphisms f:A \(\rightarrow\) B, g:B \(\rightarrow\) C, g \(\circ\) f:A \(\rightarrow\) C with g \(\circ\) f = g(f). The way we just described a category’s properties was woefully inefficient, and category theory helps with this. A profound realization made by category theory is the fact that many of the properties of mathematical systems(such as binary operations satisfying certain properties) can be described in the far more simple and efficient terms of diagrams using arrows. For us, these arrows are morphisms, and these diagrams are called *commutative diagrams*. However, we want to change some of our normal notation for functions, firstly, when we have a function f:A\(\rightarrow\) B, we also have a rule x\(\mapsto\) f(x).Now, we want to remove all unnecessary parenthesis so we will instead write \(x \mapsto f(x)\) as \(x\mapsto fx\), so this would also mean that composition \(\circ(g,f) = g \circ f \mapsto gf\). A typical diagram would look like

First thing you might notice are the arrows going from A to B to C and from A to B. Here, A,B, and C are all objects in a category, and the arrows are morphisms. Here, we say that this diagram commutes if \(g\circ f\) = h. We can visualize this:

Commutes if

Though such a diagram might seem slightly trivial in terms of usefulness, however thanks to the fact that we have already stated that many properties of mathematical systems can be simplified in terms of diagrams using arrows. For example, we have already defined a monoid as a set with a binary operation and an identity satisfying some certain properties. However, there is a far more efficient way to define a monoid in terms of a commutative diagram. Firstly, we recognize that commutative diagrams need not be triangles. Realistically, they can be almost any shape, e.g. squares, pentagons. Note however, they also do not need to be a shape in the sense of a polygon. As a simple example, a commutative diagram in the shape of a square:

Commutes if

Hopefully, from those two examples anyone can try to understand what would make a diagram of a non-polygonal shape commute. Either way, we will only need square diagrams to describe monoids. However before we start, we must describe a few functions,

\(*_M\):\(M \times M \rightarrow M\), \(\mu:1 \rightarrow M\),

Author’s Note: as of current(10-17-21) I haven’t worked on the book for a month thanks to school, but I’ll try to do some more during my free time. I am now studying some AG from Vakil