Motivation

It is likely you were unwittingly exposed to mathematical categories long before you first heard the words "category theory." Real vector spaces and their linear transformations? That's a category. Groups and their group homomorphisms? A category. Rings, fields, or topological spaces (and the "appropriate" maps between them)? All categories.

At the most intuitive level, a mathematical category simply consists of "stuff" (usually mathematical objects with prescribed algebraic structures) and the "maps" between them (usually, but not always, set maps that respect those algebraic structures). Category theory, then, can be thought of as a mathematical language that is broadly applicable across algebra, topology, set theory, logic, and beyond. This is part of what gives category theory its power, namely its ability to universally describe constructions and ideas across different mathematical disciplines. It brings under a single umbrella the study of sets (with their set maps), vector spaces (with their linear transformations), groups (with their homomorphisms), and topological spaces (with their continuous maps), just to name a few.

Category theory studies objects (e.g., groups) and the arrows between them (e.g., homomorphisms). Every general result of category theory is a result that can be interpreted and used in your favorite category. There are maps between categories, called functors, which allow us to connect categories to each other; and there are maps between functors, called natural transformations, which can provide deep insights into fundamental mathematical constructions (e.g., free groups) and operations (e.g., tensor products).

There is a second, less obvious benefit to studying category theory that I personally feel is even more profound. Thinking categorically can push us to embrace new, abstract ideas that we might initially find unintuitive, but which eventually provide incredible new insights. These insights and lessons are sprinkled throughout these notes. Keep your eyes peeled for them!

Formal definitions

Any formal definition of category is admittedly a bit clunky, so remember the general idea: you have objects, and you have arrows between those objects.

Visualization

When we want to visualize categories it's useful to think of the objects as dots and the arrows as ... arrows. For example, we might visualize two objects $a$ and $b$ with some arrows between them as follows:

There are a few things to note. First, if this image is meant to represent an entire category, then it is implicitly assumed that all of the properties required to be a category hold. For example, there must exist an arrow corresponding to the composition $h\circ f$ (and similarly one for $h\circ g$), and since there is evidently only one arrow from $a$ to $a$ in the image above, namely $1_a$, then we must have $h\circ f=1_a$ (and also $h\circ g=1_a$).

That being said, it is much more common to draw a picture like the above to represent a small "part" of a category, in which case we are not meant to assume that the only arrows between $a$ and $b$ are the ones shown. In that (common) case, we are simply visualizing some of the arrows in the category, leaving open the possibility of others. In that case, we would not assume that $h\circ f$ must equal $1_a$.

In all cases, it is common convention (mainly for our own sanity) to omit any arrows that must necessarily be in the category, per the definition of category. So we usually don't draw the identity arrows, nor do we draw the compositions of all composable arrows. We simply assume that those are all present. In our currently example, we might then simply sketch the following diagram:

For one final simplification, it is common to simplify the visual presentation of the objects, either by dropping their labels, or removing the dots, such as below:

More examples can be found here.

Conventions

The language and notation of category theory is not completely standardized, but here are some common conventions.

Abstract categories are sometimes denoted with a single capital script letter, such as $\mathcal{C}$. In this case, the set of objects of the category is often denoted $\mathrm{Ob}(\mathcal{C})$. However, it is also common to simply use $\mathcal{C}$ to refer also to the set of objects, writing $a\in \mathcal{C}$ for an object $a$ in the category $\mathcal{C}$.

If there is no cause for confusion, it's reasonable (and common) to drop all pretension with script lettering and simply use capital letters to denote categories; e.g., the category $C$.

It is common to use the word morphism in place of "arrow." I will personally use "morphism" when working with known algebraic objects (such as groups or modules), as it harkens back to the word "homomorphism" that was (and regrettably still is) used in those contexts. However, for an abstract category I will stick to "arrow."

Given two objects $a$ and $b$ in a category $C$, the collection of arrows between them is usually denoted with some variation of $\mathrm{Hom}(a,b)$ or $\mathrm{Mor}(a,b)$. This derives from the alternative names of "homomorphisms" or "morphisms", noted above. The word "hom-sets" refers to such collections of arrows^[2]. If there are multiple categories under consideration, we might more carefully denote hom-sets as ${\mathrm{Hom}}_C(a,b)$ or ${\mathrm{Mor}}_C(a,b)$. (I have seen some texts add an axiom to the definition requiring all hom-sets to be disjoint, but this doesn't seem to be universal and I don't understand the reason for making the assumption.)

Most (but not all) categories are named after their objects. For example, the category with objects all groups and with arrows all group morphisms is called "the category of groups" and is usually denoted with some variation of $\mathbf{Group}$ or $\mathbf{Grp}$. More examples can be found here.

Set-theoretic issues

When attempting to study all objects of a certain type, it is easy to run into set-theoretic issues along the lines of Russell's paradox. A common convention is to assume there is a big enough set $U$, called a universe, such that all sets one needs to discuss are members of the set $U$. A set $X$ is then called small it is a member of the universe. A set that is not in the universe is called large. (Some sources instead use the language of sets and classes.)

Since each object is uniquely associated with an identity arrow, in a small category the collection of objects is also a small set. Unfortunately, many of the common categories we encounter are not small, i.e., are large.

Will we worry about any of this? We will not. Instead, we will embrace the following quote:

Examples

Examples are abundant! We begin with some really basic categories before discussing the (more complicated) categories you've likely encountered before.

A common convention with very "simple" categories is to simply sketch a visual representation of the category, with dots used to represent objects and arrows to represent ... arrows. It is also common convention not to draw the identity arrows, nor any arrows that are necessarily there by the composition assumption.

Some basic categories

The smallest possible category is the empty category, which has no objects or arrows. This category is usually denoted $\mathbf{0}$. It is the initial object in the category of categories.

The next smallest categories are the categories that have a single object. It is common to let $\mathbf{1}$ denote the category with exactly one object and one arrow (the identity arrow on that object).

Since there is a unique object and unique arrow (the identity arrow on that object), there's no point to even label them. However, if you were to label the object you might^[3] label it as below:

With our convention of omitting identity arrows, we would visualize this category simply as

Continuing the pattern above, the category denoted $\mathbf{2}$ is the category with two objects and exactly one non-identity arrow. Since identity arrows are not usually included when sketching visual representations of categories, a picture of the category $\mathbf{2}$ is

As one last basic example in this specific sequence of categories, the category $\mathbf{3}$ has three objects and the following non-identity arrows:

Note that arrow $1\to 3$ is equal to the composition of the arrows $1\to 2$ and $2\to 3$.

Preorders

A preorder is a category $P$ in which there is at most a single arrow between any two objects. We can then define a relation $\le$ on the objects of $P$ by saying $p\le p^{\prime }$ if and only if there is an arrow $p\to p^{\prime }$ in $P$. This relation is reflexive (identity arrows!) and transitive (composition of arrows). Each of the basic categories above (i.e., $\mathbf{0},\mathbf{1},\mathbf{2},\mathbf{3},\dots$) is a preorder.

Sets as categories

A category is discrete when every arrow is an identity arrow. In other words, it's basically just a set (of objects). For example, a discrete category with six objects might be visualized as below. As usual, the identity arrows are not shown.

On the other hand, for a given set $X$ there is also an associated category (sometimes denote $\mathbf{X}$) whose objects are the subsets of $X$, and where for each pair of subsets $U$ and $V$ of $X$ (i.e., objects of $\mathbf{X}$) there is a unique arrow $U\to V$ exactly when $U\subseteq V$. This category can be useful for understanding "internal constructions" in the set $X$, such as intersections and unions of subsets, in the language of category theory.

Note that both the discrete categories above, and categories such as $\mathbf{X}$, are different from the (large) category $\mathbf{\text{Set}}$, which has as objects all sets and as arrows all set maps. The association to each set $X$ the corresponding category $\mathbf{X}$ is the object map of a functor from the category $\mathbf{\text{Set}}$ to the category $\mathbf{\text{Cat}}$ of categories.

Groups as categories

For a given group $G$ one can associate a category $\mathbf{G}$, which has a single object $\star$ and exactly one arrow $g:\star \to \star$ for each element $g\in G$. Moreover, the composition of arrows in the category $\mathbf{\text{G}}$ corresponds to the group operation in $G$.

For example, if $G$ is a group with four elements (say $G={\mathbf{Z}}_4$), then a picture of $\mathbf{G}$ might look like

Note here that I have chosen to include the identity arrow, since it corresponds to the identity element in $G$ (and it feels weird leaving it out).^[4]

The association to each group $G$ the corresponding category $\mathbf{G}$ is the object map of a functor from the category $\mathbf{Group}$ to the category $\mathbf{Cat}$.

Matrices over a fixed commutative ring

For each commutative ring $R$, the set of all matrices with entries in $R$ is the arrow set of a category ${\mathbf{Matr}}_R$. The objects of this category are the positive integers, and each $m\times n$ matrix $A$ corresponds to an arrow $A:n\to m$. Composition of arrows corresponds to matrix product.

For example, in the category ${\mathbf{Matr}}_{\mathbf{R}}$ of real matrices, the arrows $2\to 4$ are in bijection with the $4\times 2$ matrices with real entries, while the arrows (loops) $2\to 2$ are in bijection with the (square) $2\times 2$ matrices with real entries. The identity arrow on an object $n$ corresponds to the $n\times n$ identity matrix. An illustration of how arrow composition matches with matrix product might be the following:

Note that composition is written algebraically right-to-left ("inside out"), so the composition of the two arrows above corresponds to the arrow labeled by the product of those matrices in the opposite (visual) order.

Opposite categories

For each category $C$, its opposite is the category $C^{\text{op}}$ with the same objects as $C$ and with all arrows "reversed." In other words, for each arrow $f:c\to d$ in $C$ there is a corresponding arrow $f^{\text{op}}:d\to c$ in $C^{\text{op}}$ (and conversely).

Why consider such a category? We'll see.

Large categories

Most of the objects we encounter in math are the objects of some (large) categories. Below is a quick roundup of some with which you might already be familiar:

Suggested next note

Functors