Generalizing the product of sets to a universal property

by ayushkhaitan3437

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.

Advertisements