# Morphism of Diagrams

Throughout this article denotes a category and *J* is an index category.

DefinitionGiven diagrams , a

morphismis 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.

**Exercise A**

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

Proposition 1.Colimits always exist in the category of A-modules.

**Proof**

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 . ♦

**Exercise B**

Prove that colimits exist in the categories , and .

# Another Functoriality

Definition.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 .

Proposition 2.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.

**Proof**

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 . ♦

**Example**

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

Limits are the dual of colimits.

Definition.Take a diagram in of type J, written as

The

limitof 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

Definition.The

equalizerof 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 .

**Exercise C**

Prove that limits always exist in the category of *A*-modules.

# Initial and Terminal Objects

Definition.An object is said to be

initial(resp.terminal) if for any object , there is a unique morphism (resp. ).

**Note**

- 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. ]

**Exercise D**

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.

Lemma 1.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. ]

**Proof**

Follows directly from the definition. ♦