Morphism of Diagrams
Throughout this article denotes a category and J is an index category.
Given diagrams , a morphism is a natural transformation .
Thus we have the category of all diagrams in of type J, which we will denote by .
For example if we write D and D’ as tuples:
a morphism is a collection of morphisms such that
In diagram form, we have the following, where all “rectangles” commute.
Suppose any diagram of type J in has a colimit. Prove that we get a functor which takes a diagram in to its colimit. In other words show that a morphism of two diagrams of same type in induces a morphism of their colimits.
Category of Modules
Colimits always exist in the category of A-modules.
Suppose is a diagram of type J. Let with canonical embeddings . Let be the submodule generated by all elements of the form over all and in J. We claim that satisfies our desired universal properties. Define
By definition for all .
Now suppose we have a module N with linear maps such that for any we have . The collection of induce, by definition of direct sum, a unique map such that for each . Hence
so factors through such that . Thus for each we have . ♦
Prove that colimits exist in the categories , and .
Suppose is a morphism of index categories. Composition then gives:
Thus a diagram of type J gives us a diagram of type . If is a subcategory of J, this is just the restriction of D to , denoted by .
In fact we get a functor . Indeed, a morphism between diagrams is a natural transformation . We let F take this T to
where * is a form of “horizontal composition” of natural transformation (see the optional exercise here).
Although the abstract definition looks harrowing, the underlying concept is quite easy when is a subcategory of J, so it helps to keep this special case in mind. We denote the diagrams by the following tuples
so that a morphism is of a collection of morphisms in such that for any we have . Now the new diagrams and are the same tuples but with and running through morphisms in . Hence, is given by the same collection of , except now i runs through .
Let be a morphism of index categories. Then for any diagram in of type J, denoted by the pair and , we have an induced
assuming both colimits exist.
By definition comes with a collection of morphisms such that for all in J.
Similarly comes with a collection of morphisms such that for all in .
From restricting the first colimit, we get a collection such that for all in .
By universal property of the colimit , this induces a unique morphism such that . ♦
By restricting the following diagram
we obtain a morphism , assuming both objects exist. More generally we have .
In fact, the proof of proposition 1 gives us a clue on how to construct general colimits. First take the coproduct, which corresponds to colimit over a diagram of vertices. Next we “add the arrow relations” by taking the coequalizer for each arrow.
Limits are the dual of colimits.
Take a diagram in of type J, written as
The limit of the diagram comprises of the following data:
where is an object, is a morphism in for each , such that for any arrow , we have .
We require the following universal property. For any tuple
where is an object, is a morphism for each , such that for any arrow , we have , there is a unique morphism such that
As before, we have the following special cases.
Example 1: Products
If J is obtained from an index set I, the limit is the product .
Example 2: Pullbacks (Fiber Products)
If J is the following, the resulting limit is the pullback.
Example 3: Equalizers
The equalizer of in a category is the limit of the following diagram.
This is a pair such that and, for any pair such that , there is a unique such that .
Prove that limits always exist in the category of A-modules.
Initial and Terminal Objects
An object is said to be initial (resp. terminal) if for any object , there is a unique morphism (resp. ).
- If A is initial or terminal, there is a unique morphism , i.e. the identity.
- A is initial if and only if it is a (colimit / limit) of the empty diagram. [ Exercise: pick the right option and write the dual statement. ]
Prove that if A and A’ are initial, there is a unique isomorphism . Dually, the same holds for terminal objects. In summary, initial (resp. terminal) objects are unique up to unique isomorphism.
The phrase “unique up to unique isomorphism” has been used multiple times while looking at universal properties. This is not a coincidence, for initial and terminal objects can be used to describe universal properties of various constructions. Here is an example.
Let M, N be A-modules. Consider the category , whose objects are pairs
where P is an A-module, B is an A-bilinear map. The morphisms
are A-linear maps such that . Then is an (initial / terminal) object in . [ Exercise: pick the right option. ]
Follows directly from the definition. ♦