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 be a category, and let and be two objects in it. How should we define ? We define a new category. Here, we select only those objects which have morphisms to both and . More technically speaking, we create the category . I repeat that not all objects from are selected. Only those which have morphisms to both and are selected in this new category.
Now what are the objects in this new category ? These are the newly selected objects along with their morphisms to and . The morphisms between objects are similar to those in other 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 . Hence, is universal in the category .
Now an example: let us take the category . How is defined? We know that integers are the objects in this category, and a morphism between objects and exists only if : this morphism is . In this category, the final object will be , as . Why in the category 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
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.