Generalizing the product of sets to a universal property

The concept of product and co-product can be generalized to a universal property. However, “how” to do so is given in a rather unclear manner in most books and internet links. An example would be http://jeremykun.com/2013/05/24/universal-properties/

I have somehow patched together a coherent explanation. I hope this helps anyone starting out in Category Theory.

Let $C$ be a category, and let $A$ and $B$ be two objects in it. How should we define $A\times B$? We define a new category. Here, we select only those objects which have morphisms to both $A$ and $B$. More technically speaking, we create the category $C_{A,B}$. I repeat that not all objects from $C$ are selected. Only those which have morphisms to both $A$ and $B$ are selected in this new category.

Now what are the objects in this new category $C_{A,B}$? These are the newly selected objects along with their morphisms to $A$ and $B$. The morphisms between objects are similar to those in other $C_{A,B}$ categories. I wouldn’t like to go deeper into describing this category, as the details are pretty well-known.

If this category has a final object, then that final object is $A\times B$. Hence, $A\times B$ is universal in the category $C_{A,B}$.

Now an example: let us take the category $(Z,\leq)$. How is $3\times 4$ defined? We know that integers are the objects in this category, and a morphism between objects $a$ and $b$ exists only if $a\leq b$: this morphism is $(a,b)$. In this category, the final object will be $3$, as $3=\min\{3,4\}$. Why $\min\{a,b\}=a\times b$ in the category $(Z,\leq)$ is something that is a good exercise to find out. Just follow the procedure outlined above.

Note that although I have only generalized the product of two sets, the product of any (finite) number of sets can be generalized to a universal property by taking the category $C_{A_1,A_2,\dots,A_n}$

I must admit that my writing is getting more and more incoherent and muddled, and perhaps not as helpful as initially planned. I hope to rectify this from the next post on. I also need to learn how to draw commutative diagrams.