Hello & Here we go

Hello!  I’m a terrible writer, so I’m going to dive right it. My selfish goal is to gain a thorough understanding of category theory, but that road is not a straight shot.  It requires a great deal of knowledge and experience from all of mathematics to really grok many of the abstract methods employed.  This blog is my way of keeping track of the myriad of examples and topics that motivate these ideas.

Categories:

Setting the setting (prologue?).  If I’m going to talk about category theory, I should probably say what a category “is”.  A category C, intuitively, consists of a collection of “objects” with similar properties and a collection of “arrows” (also often called “morphisms”) between objects.  For some examples, think of:

  • The category Set with objects being sets and arrows being functions between sets.
  • The category Grp with objects being groups and arrows being group homomorphisms.
  • The category Top with objects being topological spaces and arrows being continuous functions between topological spaces.
  • If R is a ring, then the category R-Mod is the category with objects left R-modules and arrows R-module homomorphims. (This includes things like the category of vector spaces over a field)
  • Any partially ordered set (C,\leq) can be turned into a category as well.  We define the objects of this new category are the elements of C, and for any two objects x and y of C, there is one and only one arrow from x to y if and only if x \leq y.

and so on.  You get the idea.   Formally, C consists of a class Ob(C) of objects and a class Hom(C) of arrows, such that

  • Each arrow f has a unique source object a and final object b.  We write this as f: a \to b.
  • For any two objects a and b of C there is a set of arrows from a to b, called \text{Hom}_\textbf{C}(a,b).  If a' and b' are two objects (with a \neq a' and b \neq b'), then \text{Hom}(a,b) and \text{Hom}(a',b') are disjoint.
  • For any three objects a,b and c of C, there is a binary operation \text{Hom}(a,b) \times \text{Hom}(b,c) \to \text{Hom}(a,c) called “composition of arrows/ morphisms.”
  • Arrow composition is associative.
  • For any object a, there is a morphism 1_a : a \to a such that for any other arrow f: a \to b, we have f \circ 1_a = f = 1_a \circ f.

So that’s a bit of a mouthful.  Unwinding all these criteria basically yields the above “intuitive” explanation.  The criteria concerning arrows simply axiomatize this intuition (i.e. arrows basically act like we think functions “should” act).

We shall encounter many examples of categories in future posts.  The ones that will come up quite often (as they contain a host of interesting examples) are AbR-mod, and Top (which are the categories of abelian groups, left R-modules, and topological spaces, respectively).