1.5Categories

A category $\sf{C}$ consists of:

1. A class $\Obj(\sf{C})$ of objects of the category.

2. For every $A, B \in \Obj(\sf{C})$, a set $\Hom_{\sf{C}}(A, B)$ of morphisms between objects, with the following properties:

   1. Identity. For every $A \in \Obj(\sf{C})$, there exists a morphism $1_A \in \Hom_{\sf{C}}(A, A)$, called the identity on $A$.

   2. Composition. For any $A, B, C \in \Obj(\sf{C})$, there is a composition operation: morphisms $f \in \Hom_{\sf{C}}(A, B)$ and $g \in \Hom_{\sf{C}}(B, C)$ determine a morphism $gf \in \Hom_{\sf{C}}(A, C)$.

   3. Associativity. For any $A, B, C, D \in \Obj(\sf{C})$, morphisms $f \in \Hom_{\sf{C}}(A, B)$, $g \in \Hom_{\sf{C}}(B, C)$, $h \in \Hom_{\sf{C}}(C, D)$ compose associatively. That is, $(hg)f = h(gf)$.

   4. Unitality. For any $A, B \in \Obj(\sf{C})$ and any morphism $f \in \Hom_{\sf{C}}(A, B)$, we have that $f1_A = f$ and $1_Bf = f$.

We can define a category $\Set$, whose objects are all possible sets and whose morphisms are functions mapping one set to another.

Suppose $S$ is a set with an relation $\sim$ that is reflexive and transitive. This determines a category whose objects are the elements of $S$ and whose morphisms are such that whenever $a, b \in S$, $(a, b) \in \Hom(a, b)$ if $a \sim b$ and $\Hom(a, b) = \emptyset$ otherwise. We verify the axioms as follows:

1. Identity. Note, for any $a \in A$, $a \sim a$ and so $1_a = (a, a) \in \Hom(a, a)$.

2. Composition. Suppose $f \in \Hom(a, b)$ and $g \in \Hom(b, c)$ for some $a, b, c \in S$. Then note $a \sim b$ and $b \sim c$, so by transitivity, $a \sim c$, meaning the morphisms $f, g$ determine the morphism $gf \in \Hom(a, c)$.

3. Associativity. This follows from the transitivity of $\sim$.

4. Unitality. This follows trivially from how composition is defined.

Many important categories fall out of this exam. For a trivial one, consider any set $S$ and $=$ as its relation (which is actually an equivalence relation!). As such, all the morphisms are identity morphisms—such a category is called discrete.

Branching off from previous discussion, we can endow $\bZ$ with the $\le$ relation ($a \sim b$ iff. $a \le b$). This induces a category whose objects are integers and whose morphisms are connections $n \to m$ for every $n \le m$.

We can create more abstract—and consequently more powerful—categories: one such example is the slice category. Let $\sf{C}$ be any category with $A \in \Obj(\sf{C})$. We can define a category $\sf{C}/A$ as follows:

1. Objects. An object $f \in \Obj(\sf{C}/A)$ is a morphism $f \in \Hom_{\sf{C}}(B, A)$ for any $B \in \Obj(\sf{C})$. That is, the slice category $\sf{C}/A$ comprises all the morphisms going to $A$ for its objects.

2. Morphisms. Suppose $f_1, f_2 \in \Obj(\sf{C}/A)$ such that $f_1 \in \Hom_{\sf{C}}(C_1, A)$ and $f_2 \in \Hom_{\sf{C}}(C_2, A)$ for $C_1, C_2 \in \Obj(\sf{C})$. Then a morphism $\sigma \in \Hom_{\sf{C}/A}(f_1, f_2)$ is defined such that $f_1 = f_2 \sigma$. Graphically, $\sigma$ is a morphism in $\sf{C}$ making the following triangle commute.

Verification of the category axioms is trivially shown by pulling back to $\sf{C}$, so they are omitted.

In a similar, yet opposite, sentiment, we define the coslice category. Let $\sf{C}$ be any category with $A \in \Obj(\sf{C})$. We can define a category $A/\sf{C}$ as follows:

1. Objects. An object $f \in \Obj(A/\sf{C})$ is a morphism $f \in \Hom_{\sf{C}}(A, B)$ for any $B \in \Obj(\sf{C})$. That is, the coslice category $A/\sf{C}$ comprises all the morphisms that map $A$ to some other object in the category.

2. Morphisms. Suppose $f_1, f_2 \in \Obj(A/\sf{C})$ such that $f_1 \in \Hom_{\sf{C}}(A, C_1)$ and $f_2 \in \Hom_{\sf{C}}(A, C_2)$ for $C_1, C_2 \in \Obj(\sf{C})$. Then a morphism $\sigma \in \Hom_{A/\sf{C}}(f_1, f_2)$ is defined such that $\sigma f_1 = f_2$. Graphically, $\sigma$ is a morphism in $\sf{C}$ making the following triangle commute.

Note the similarities between the coslice category and the slice category.

Consider the set $\bZ$ endowed with the relation $\le$, and let $\sf{C}$ be the category induced by this relation. Fix $3 \in \bZ$, and consider the slice category $\sf{C}/3$. Then the objects $f \in \Obj(\sf{C}/3)$ are morphisms $f \in \Hom_{\sf{C}}(n, 3)$ for any $n \le 3$. The morphisms are maps between two objects, so they take the form $\sigma \in \Hom_{\sf{C}}(n, m)$ for $m \le n \le 3$.

Let $\sf{C} = \Set$ and let $A$ be a fixed singleton $\{*\}$. Then the objects of $A/\sf{C}$ are precisely set-functions of form $f: \{*\} \to S$. Suppose $f_1: \{*\} \to S$ and $f_2: \{*\} \to T$. Then a morphism $\sigma: S \to T$ is determined in the sense that $\sigma(f_1(*)) = f_2(*)$.