Properties of Morphisms by Jeremy Kun.
From the post:
This post is mainly mathematical. We left it out of our introduction to categories for brevity, but we should lay these definitions down and some examples before continuing on to universal properties and doing more computation. The reader should feel free to skip this post and return to it later when the words “isomorphism,” “monomorphism,” and “epimorphism” come up again. Perhaps the most important part of this post is the description of an isomorphism.
Isomorphisms, Monomorphisms, and Epimorphisms
Perhaps the most important paradigm shift in category theory is the focus on morphisms as the main object of study. In particular, category theory stipulates that the only knowledge one can gain about an object is in how it relates to other objects. Indeed, this is true in nearly all fields of mathematics: in groups we consider all isomorphic groups to be the same. In topology, homeomorphic spaces are not distinguished. The list goes on. The only way to do determine if two objects are “the same” is by finding a morphism with special properties. Barry Mazur gives a more colorful explanation by considering the meaning of the number 5 in his essay, “When is one thing equal to some other thing?” The point is that categories, more than existing to be a “foundation” for all mathematics as a formal system (though people are working to make such a formal system), exist primarily to “capture the essence” of mathematical discourse, as Mazur puts it. A category defines objects and morphisms, but literally all of the structure of a category lies in its morphisms. And so we study them.
If you are looking for something challenging to read over the weekend, Jeremy’s latest post on morphisms should fit the bill.
The question of “equality” is easy enough to answer as lexical equivalence in UTF-8, but what if you need something more?
Being mindful that “lexical equivalence in UTF-8″ is a highly unreliable form of “equality.”
Jeremy is easing us down the road where such discussions can happen with a great deal of rigor.