Abstract Algebra/Category theory

Category theory is the study of categories, which are collections of objects and morphisms (or arrows), from one object to another. It generalizes many common notions in Algebra, such as different kinds of products, the notion of kernel, etc. See Category Theory for additional information.

Definitions & Notations

Definition 1: A (locally small) category $𝒞$ consists of

A collection $\mathrm{O}\mathrm{b}\mathrm{j}(𝒞)$ of objects.

A collection $\mathrm{A}\mathrm{r}\mathrm{r}(𝒞)$ of morphisms.

For any $X,Y\in \mathrm{O}\mathrm{b}\mathrm{j}(𝒞)$, $\mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$ is the subcollection of $\mathrm{A}\mathrm{r}\mathrm{r}(𝒞)$ of morphisms from $X$ to $Y$, where each $\mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$ is required to be a set (hence the term locally small).

These obey the following axioms:

There is a notion of composition. If $X,Y,Z\in \mathrm{O}\mathrm{b}\mathrm{j}(𝒞)$, $f\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$ and $g\in \mathrm{H}\mathrm{o}\mathrm{m}(Y,Z)$, then $f$ and $g$ are called a composable pair. Their composition is a morphism $g\circ f\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Z)$.

Composition is associative. $f\circ (g\circ h)=(f\circ g)\circ h$ whenever the composition is defined.

For any object $X$, there is an identity morphism ${\mathrm{i}\mathrm{d}}_X\in \mathrm{H}\mathrm{o}\mathrm{m}(X,X)$ such that if $Y,Z$ are objects, $f\in \mathrm{H}\mathrm{o}\mathrm{m}(Y,X)$ and $g\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Z)$, then ${\mathrm{i}\mathrm{d}}_X\circ f=f$ and $g\circ {\mathrm{i}\mathrm{d}}_X=g$.

Note that we demand neither $\mathrm{O}\mathrm{b}\mathrm{j}(𝒞)$ nor $\mathrm{A}\mathrm{r}\mathrm{r}(𝒞)$ to be sets; if they are both in fact sets, then we call our category small.

Definition 2: A morphism $f$ has associated with it two functions $\mathrm{d}\mathrm{o}\mathrm{m}$ and $\mathrm{c}\mathrm{o}\mathrm{d}$ called domain and codomain respectively, such that $f\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$ if and only if $\mathrm{d}\mathrm{o}\mathrm{m}f=X$ and $\mathrm{c}\mathrm{o}\mathrm{d}f=Y$. Thus two morphisms $f,g$ are composable if and only if $\mathrm{c}\mathrm{o}\mathrm{d}f=\mathrm{d}\mathrm{o}\mathrm{m}g$.

Remark 3: Unless confusion is possible, we will usually not specify which Hom-set a given morphism belongs to. Also, unless several categories are in play, we will usually not write $X\in \mathrm{O}\mathrm{b}\mathrm{j}(𝒞)$, but just "$X$ is an object". We may write $X\stackrel{f}{⟶}Y$ or $f:X\to Y$ to implicitly indicate the Hom-set $f$ belongs to. We may also omit the composition symbol, writing simply $gf$ for $g\circ f$.

Basic Properties

Lemma 4: Let $X$ be an object of a category. The identity morphism for $X$ is unique.

Proof: Assume $i$ and $j$ are identity morphisms for $X$. Then $i=i\circ j=j$.

Example 5: We present some of the simplest categories:

i) $0$ is the empty category, with no objects and no morphisms.

ii) $1$ is the category containging only a single object and its identity morphism. This is the trivial category.

iii) $2$ is the category with two objects, $X$ and $Y$, their identity morphisms, and a single morphism $f\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$.

iv) We can also have a category like $2$, but where we have two morphisms $f,g\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$ with $f\ne g$. Then $f$ and $g$ are called parallel morphisms.

v) $3$ is the category with three objects $X,Y,Z$. We have $f\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Y)$, $g\in \mathrm{H}\mathrm{o}\mathrm{m}(Y,Z)$ and $h=gf\in \mathrm{H}\mathrm{o}\mathrm{m}(X,Z)$.

Initial and Final Objects

Definition An object $I$ in a category is called initial or cofinal, if for any object $X$ there exists a unique morphism $f:I\to X$

Lemma If $I$ and $J$ are initial objects, then they are isomorphic.

Proof: Let $i:I\to J$ and $j:J\to I$ be the unique morphisms between $I$ and $J$. Given that both $I$ and $J$ have a unique endomorphism because of their initiality, this morphism must be the identity. Therefore $i\circ j$ and $j\circ i$ are the respective identity morphisms, making $I$ and $J$ isomorphic.

Definition An object $F$ in a category is called final or coinitial, if for any object $X$ there exists a unique morphism $f:X\to F$

Lemma If $T$ and $S$ are final objects, then they are isomorphic.

Proof Pass to isomorphicness of initial objects in the cocategory.

Some examples of categories

• $𝐒𝐞𝐭$: the category whose objects are sets and whose morphisms are maps between sets.
• $𝐅𝐢𝐧𝐒𝐞𝐭$: the category whose objects are finite sets and whose morphisms are maps between finite sets.
• The category whose objects are open subsets of ${ℝ}^n$ and whose morphisms are continuous (differentiable, smooth) maps between them.
• The category whose objects are smooth (differentiable, topological) manifolds and whose morphisms are smooth (differentiable, continuous) maps.
• Let $k$ be a field. Then we can define $k-𝐕𝐞𝐜𝐭$: the category whose objects are vector spaces over $k$ and whose morphisms are linear maps between vector spaces over $k$.
• $𝐆𝐫𝐨𝐮𝐩$: the category whose objects are groups and whose morphisms are homomorphisms between groups.

In all the examples given thus far, the objects have been sets with the morphisms given by set maps between them. This is not always the case. There are some categories where this is not possible, and others where the category doesn't naturally appear in this way. For example:

• Let $𝒢$ be any category. Then its opposite category ${𝒢}^{op}$ is a category with the same objects, and all the arrows reversed. More formally, a morphism in ${𝒢}^{op}$ from an object $X$ to $Y$ is a morphism from $Y$ to $X$ in $𝒢$.
• Let $M$ be any monoid. Then we can define a category with a single object, with morphisms from that object to itself given by elements of $M$ with composition given by multiplication in $M$.
• Let $G$ be any group. Then we can define a category with a single object, with morphisms from that object to itself given by elements of $G$ with composition given by multiplication in $G$.
• Let $𝒞$ be any small category, and let $𝒟$ be any category. Then we can define a category ${𝒟}^{𝒞}$ whose objects are functors from $𝒞$ to $𝒟$ and whose morphisms are natural transformations between the functors from $𝒞$ to $𝒟$.
• $𝐂𝐚𝐭$: the category whose objects are small categories and whose morphisms are functors between small categories.