In this post, we will discuss categories with universal property. We recall that categories with a universal property are one which has a terminal object. We will also see a connection between mathematical constructs which might seem not related at first, but by the power of category theory we will see that they are indeed related.
Recall, from set theory we have the direct product of two sets. This will be a final object in a certain category. In general, we can define the product of two objects in a general category $\mathcal{C}$; however, it might or might not exist.
For the definition to make sense we have have to introduce the category $\mathcal{C}_{A,B}$. Suppose that $A$ and $B$ are objects in a general category. We will construct $\mathcal{C}_{A,B}$ as follows:
1)$Obj(\mathcal{C}_{A,B})$ = morphisms defined by the following diagram below:
\begin{equation*}
\xymatrix{
& A \\
Z \ar[ur]^{h} \ar[dr]_{g} \\
& B
}
\end{equation*}
One can think of this think of the objects of this category are the 3-tupe $(Z,h,g)$. That is, the objects of this category are actually two morphisms from arbitrarily object Z into fixed A and B respectively.
2)Morphisms are commutative diagram given by the following:
\begin{equation*}
\xymatrix{
& & A \\
Z_1 \ar@/^/[urr]^{h_1} \ar@/_/[drr]_{g_1} \ar[r]^{\sigma} & Z_2 \ar[ur]^{h_2}\ar[dr]_{g_2} \\
& & B
}
\end{equation*}
Now, that we defined the category $C_{A,B}$. We can define the product of two objects $A$ and $B$ in a specific category.
Definition 1:
Suppose that $\mathcal{C}$ is a category such that A and B are arbitrary objects. Then, the product denoted by $A\times B$(if exists) is defined to be the final object if it exists in the category $\mathcal{C}_{A,B}$. That is, given any object $(Z_1,h_1,g_1) \in Obj(\mathcal{C}_{A,B})$, there exists a unique morphism a morphism $\theta : Z_1 \rightarrow A\times B$ such that the diagram below commutes:
\begin{equation*}
\xymatrix{
& & A \\
Z_1 \ar@/^/[urr]^{h_1} \ar@/_/[drr]_{g_1} \ar[r]^{\theta} & A\times B \ar[ur]^{h_2}\ar[dr]_{g_2} \\
& & B
}
\end{equation*}
In the category of Sets. Product do indeed exist. We shall prove this.
Theorem 1:
Suppose that $\mathcal{C} = Set$. The regular product $A\times B$ is final object in the category $\mathcal{C}_{A,B}$.
Proof:
Consider the following diagram:
\begin{equation*}
\xymatrix{
& A \\
A \times B \ar[ur]^{\pi_A} \ar[dr]_{\pi_B} \\
& B
}
\end{equation*}
Here, $\pi_A$ and $\pi_B$ are the regular projection maps from the product into A and B respectively. We would like to show that given any object $(Z,h_1,g_1)$, then there exists unique $\theta$ that makes the diagram below commutes.
\begin{equation*}
\xymatrix{
& & A \\
Z \ar@/^/[urr]^{h_1} \ar@/_/[drr]_{g_1} \ar[r]^{\theta} & A\times B \ar[ur]^{\pi_A}\ar[dr]_{\pi_B} \\
& & B
}
\end{equation*}
Consider $\theta = h_1 \times h_2 : Z \rightarrow A\times B$ defined as $\theta(z) = (h_1(z),h_2(z))$. Then, this makes the diagram above commutes. Also, it is unique as this definition is forced by the commutativity of the diagram above. $\square$
If we consider the category $(\mathbb{Z},\leq)$. Then, we would like to see if this category has a product or not. Suppose that it does indeed exist for now. Suppose m and n are integers. Denote their product as $m \times n$. Then, if the product exist we must have that $m \times n \leq n,m$. Also, if there exists another integer $s$ such that $s \leq m$ and $s \leq n$, then $s \leq m \times n$. Then, $m \times n = min(m,n)$. Thus, category theory says that min and the product are really related to each other.
Now, we will explore the dual aspect of products. In the category of Sets, we can define the notion of being disjoint union. Is it possible to generalize this notion to arbitrarily categories ? The answer to that question is yes.
To do that, we will construct a new category which is really the dual. This category is obtained by reversing the arrows of the category $\mathcal{C}_{A,B}$. However, let us define it formally for completeness sake. Suppose that $\mathcal{C}$ is a category such that $A$ and $B$ are objects in this category. Construct a new category, which we will denote by $\mathcal{C}^{A,B}$.
$Obj(\mathcal{C}^{A,B})$ = morphisms given by the diagram below:
\begin{equation*}
\xymatrix{
& A \ar[dl]_{h} \\
Z\\
& B \ar[ul]^{g} \\
}
\end{equation*}
We can think of the objects as the tuple $(h,g,Z)$. I wrote this differentially than tha category $C_{A,B}$, to differentiate that the arrows are emanating from A and B not from Z.
Morphisms: Are given by the following commutative diagram:
\begin{equation*}
\xymatrix{
& & A \ar[dl]^{h_2} \ar@/_/[dll]_{h_1} \\
Z_1 & Z_2 \ar[l]_{\theta} \\
& & B \ar[ul]_{g_2} \ar@/^/[ull]^{g_1}
}
\end{equation*}
That is we can think of the morphisms between $(h_2,g_2,Z_2)$ and $(h_1,g_1,Z_1)$, as a morphism between $Z_2$ and $Z_1$ in $\mathcal{C}$ such that the commutative diagram above is satisfied.
Recall, in sets we can form the disjoint union between two Sets. We can generalize this notion to arbitrary category.
Definition 2:
Suppose $\mathcal{C}$ is a category such that A and B are arbitrary objects in this category. The coproducts(if exists) is the initial object in the category $A,B$. We shall denote the coproducts as $A \coprod B$. That is given any object $(h,g,Z)$, there exists a unique morphism $\theta$ from $A \coprod B$ to $Z$, such that the diagram below commutes:
\begin{equation*}
\xymatrix{
& & A \ar[dl]^{h} \ar@/_/[dll]_{h_1} \\
{A \coprod B} & Z \ar[l]_{\theta} \\
& & B \ar[ul]_{g} \ar@/^/[ull]^{g_1}
}
\end{equation*}
In is left as exercise to the reader that the product is indeed an initial object in the $\mathcal{C}_{A,B}$ with $\mathcal{C}$ being the category of sets. As, there are many ways to form disjoint union, so this will right away tells us that all those ways of forming disjoint union are isomorphic to each other. This is the power of category theory. In the category $(\mathbb{Z},\leq)$, one can check that the coproduct in this category correspond to taking the maximum.
Now, we are completely done with chapter 1 of allufi summary. In the next few posts, we will solve all the problems in chapter 1. Keep posted.
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