# Universal Properties II - Commutative diagrams, cones and limits

Many of our inspiring examples of universal properties can be described by the following basic situation:

- There is a commutative diagram in a given category
. This consists of a family of objects in together with certain arrows between them. - We consider all objects
with arrows "to the diagram" (or dually, "from the diagram"). In other words, for every object in our family we have an arrow from to that object, and the diagram formed from these arrows together with the initial diagram is commutative. - Among all such objects (equipped with such arrows), there is a "universal" object
through which all other such objects must uniquely factor.

Let's try to make sense of this and gradually discover how to "encode" such information in a compact, categorical way.

# An intuitive way to visualize arrows "to a diagram"

This first step is not strictly necessary but it can definitely help simplify our mental picture. Visually imagine a commutative diagram in a category

We can picture this as existing in some ambient space, even though this is entirely metaphorical. Let's now imagine the diagram has been drawn (or exists) on a horizontal glass plane. We can then think of other objects

Assuming this picture we've drawn represents a commutative diagram in **cone to the diagram**.

Of course, we have the dual notion of a **cone from the diagram**.

For some examples of cones (both to and from diagrams), check out our collection of inspiring examples.

# Formalizing the notion of "commutative diagram"

How do we formally encode the concept of a commutative diagram in a category, without using the clunky phrasing "a collection of objects in the category together with certain arrows, such that all possible compositions of arrows in that collection that have the same domain and codomain agree"? The answer: we use a "helper" category (sometimes called a "diagram category") and a functor from that category to our category.

The general definition is as follows:

Let **diagram in of shape ** is a functor

Observe that this short definition really does encode the idea of a diagram in

In the definition above, there are no conditions on the categories *any* functor to

## Examples of diagrams of various shapes

### The empty diagram

Suppose ^{[1]}. This functor corresponds to the "empty diagram" in

### The one-object category

If

### The two-object, one-arrow category

If

### Two arrows with a common codomain

Suppose

Each functor

This functor category is useful when studying pullbacks and pushforwards.

# Formalizing cones to diagrams

Now that we have a formal way to define a diagram of a given shape in our favorite category, we can repackage the notion of a cone to that diagram. We first observe that any single object

Suppose

If

These functions together define a functor **diagonal functor**.

This sounds like a mouthful, but it simply boils down to labeling a diagram of "shape"

## Example of a constant diagram

Suppose

For each

When

This connects the image of

## Cones to/from diagrams

We can now officially define a cone from an object

Suppose **cone from to ** is an arrow

Similarly, a **cone from to ** is an arrow

In other words, a cone from

Observe that if we merge the two copies of

# Limits (and colimits) are universal cones

At last we can formalize the notion of a single object

Suppose **limit of ** is a cone

**limit**object of

Dually, a **colimit of ** is a cone

**colimit**object of

Note that we are not claiming that limits and colimits always exist. We'll delay that question until the distant future and instead focus on concrete examples.

# Examples of limits and colimits

## The empty diagram

Recall that the empty diagram in a category **terminal object** of the category

Dually, a universal cone from the empty diagram is an object **initial object** of the category

So, a limit of the empty diagram is a terminal object, and a colimit of the empty diagram is an initial object.

## The one-object, no-arrows diagram

Recall that functors

## The two-object, no-arrows diagram

Let

Universality means that any other such cone factors uniquely through this one:

When this limit exists, the limit object is usually called the **product** of

- In the category
, this diagram corresponds to a pair of sets and . This limit always exists and is called their **Cartesian product**, denoted. - In the category
, the diagram corresponds to a pair of -modules and . The limit always exists and is called their **direct product**, denoted. - If
is a given -module and is the category of submodules of , the diagram corresponds to a pair of submodules and of . This limit always exists and is called their **intersection**, denoted.

This section is currently under construction. More examples will be added soon, including:

- products and coproducts
- powers and copowers
- pullbacks and pushforwards
- equalizers and coequalizers
- kernels and cokernels

## Suggested next note

Universal Properties III - Yoneda's Lemma

Observe that in our functor category notation, the functor category

is equivalent to the one-object category, . ↩︎