Category Theory, Commutative Diagrams

Diagram

Within the context of a category, a diagram is a digraph with objects as vertices and morphisms as edges. The digraph may have infinitely many vertices and edges. As you might expect, an arc from one vertex to another must represent a morphism between the corresponding objects.

In this and subsequent pages, we will often abuse the terminology and call the structure a graph, even though it is a digraph.

The graph allows loops and multiple edges. You can represent a morphism from an object into itself, or several different morphisms between two objects.

You may recall the abstract definition of a category, which is based on morphisms alone. An object is merely an identity morphism. This definition inspires an additional constraint on a diagram: each object includes the identity morphism on itself. Every vertex has an arc that loops around and runs back into itself. Let's assume this additional constraint is in place.

Commutative Diagram

A commutative diagram is a diagram that exhibits path independence. This means there is at most one morphism, implied by the graph, connecting any ordered pair of vertices. A morphism may be implied by transitivity. If the graph contains A → B → C, there is an implied morphism from A to C, produced by composing the two morphisms in the chain. Any other chain from A to C must produce the same morphism.

As a corollary, there is at most one arc directly connecting any two vertices. Multiple edges have just disappeared. And a literal interpretation means there is at most one arc joining a vertex to itself, and we already have that; it's the identity map. There are no other loops.

If a cycle starts and ends at V, the composition of those morphisms has to produce the morphism on V, which is the identity map. If U and V are connected by antiparallel arcs, that forms a cycle of length 2. The composition is the identity map, the morphisms are inverses, and the points are equivalent.

Note that an endomorphism can be included in a commutative diagram; you just need to vertices that represent the same object. Draw an arc between them, and it won't comflict with the identity arc, which is a loop at either vertex. If you want to depict multiple morphisms between two objects, use several vertices to represent those objects.

Category of Morphisms

Given a category C₁, we can construct another category C₂, such that the objects of C₂ are the morphisms of C₁.

The trick is defining the new morphisms. Within C₁, let f take A to B, and let g take C to D. Draw a commutative diagram like this.

A f B
x   y
C g D

x and y are morphisms in C₁, yet to be determined. Arrows point down and to the right. We are only interested in commutative diagrams. Thus y(f(A)) = g(x(A)). Each ordered pair of morphisms x,y that makes the diagram commute becomes a morphism from f to g in C₂.

Let's verify the category axioms, starting with composition. Stack two squares on top of each other, like this.

* f *
w   y
* g *
x   z
* h *

If the two squares form commutative diagrams, use the properties of associativity in C₁ to write wxh = wgz = fyz. In other words, the entire rectangle commutes, and wx,yz is a valid morphism from f to h. Let this be the composition of w,y and x,z.

If I means identity map in C₁, show that I,I is the identity morphism for any function f. Remember, you need to show that I,I joins with any other morphism to leave it unchanged.

* f *
I   I
* f *
x   y
* g *

Since morphism composition in C₂ follows directly from morphism composition in C₁, associativity is inherited. Therefore C₂ forms a category.