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 $C$. This consists of a family of objects in $C$ together with certain arrows between them.
• We consider all objects $c\in C$ with arrows "to the diagram" (or dually, "from the diagram"). In other words, for every object in our family we have an arrow from $c$ 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 $c_0$ 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 $C$ as simply a constellation of dots (the objects) and arrows:

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 $c\in C$ as dots floating above this plane. Now imagine we had arrows connecting an object $c$ to every object in the diagram. Visually, this would look like a cone (of light, say) with vertex at $c$ and rays shooting down from $c$ to every object in the diagram.

Assuming this picture we've drawn represents a commutative diagram in $C$, this is what we mean when we say we have arrows from $c$ to the original diagram. It is also because of this picture that we often describe such an object (with a family of maps) as a 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:

Observe that this short definition really does encode the idea of a diagram in $C$ of "shape" $J$. If we visualize $J$ as some dots connected by some arrows, then a functor $F:J\to C$ assigns to each dot some object of $C$, and to each arrow between those dots some arrow in $C$. The commutativity of the resulting diagram is encoded in the fact that $J$ is a category and $F$ is a functor.

Examples of diagrams of various shapes

The empty diagram

Suppose $J$ is the empty category, usually denoted $\mathbf{0}$. For any category $C$, there is a unique functor $\mathbf{0}\to C$, namely the empty functor^[1]. This functor corresponds to the "empty diagram" in $C$.

The one-object category

If $J$ is the category $\mathbf{1}$ (with a single object and only the identity arrow), then there is a bijection between diagrams in $C$ of shape $\mathbf{1}$ and the objects of $C$.

The two-object, one-arrow category

If $J$ is the category $\mathbf{2}$ (with two objects and a single non-identity arrow between them), then there is a bijection between between diagrams in $C$ of shape $\mathbf{2}$ and arrows in $C$. Indeed, the object function of a functor $F:\mathbf{2}\to C$ corresponds to a choice of two objects in $C$, and the arrow function corresponds to a choice of arrow in $C$ between those objects.

Two arrows with a common codomain

Suppose $J$ is the category with three objects and two nonidentity arrows with common codomain, as illustrated below:

Each functor $F:J\to C$ corresponds to a diagram in $C$ of the form

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 $c$ in $C$ can be "turned into" a diagram in $C$ of shape $J$, using a very simple functor.

This sounds like a mouthful, but it simply boils down to labeling a diagram of "shape" $J$ with the object $c$ at every dot, and the identity arrow $1_c$ on every arrow.

Example of a constant diagram

Suppose $J$ is the category $\mathbf{\text{3}}$, shown below:

For each $c\in C$ we then have the constant diagram of shape $J$, defined by the constant functor ${\mathrm{\Delta }}_c:\mathbf{3}\to C$:

When $f:c\to c^{\prime }$ is an arrow in $C$, we can "connect" two such constant diagrams by simply connecting the corresponding dots everywhere with the arrow $f:c\to c^{\prime }$:

This connects the image of ${\mathrm{\Delta }}_c$ to the image of ${\mathrm{\Delta }}_{c^{\prime }}$, which is exactly the info needed to define a natural transformation ${\mathrm{\Delta }}_c\Rightarrow {\mathrm{\Delta }}_{c^{\prime }}$.

Cones to/from diagrams

We can now officially define a cone from an object $c$ to a diagram of shape $J$ in $C$.

In other words, a cone from $c$ to $F$ consists of an arrow ${\tau }_j:c\to F(j)$ for every $j\in J$, satisfying the usual naturality conditions. In this case, that means for every arrow $f:j\to j^{\prime }$ in $J$ we have a commutative diagram

Observe that if we merge the two copies of $c$ (connected by the identity arrow), we can redraw the above diagram to look a lot more like the cone we were expecting:

Limits (and colimits) are universal cones

At last we can formalize the notion of a single object $r\in C$ "representing" (or "encoding") all of the information in a given commutative diagram in $C$. In terms of the language we've now established, we are really looking for a cone to the diagram through which all other cones uniquely factor. To be more precise, we define the following:

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 $C$ corresponds to the empty functor $\mathbf{\text{0}}\to C$. What is a cone to/from the empty diagram? It's simply an object $c\in C$. A universal cone to the empty diagram is then an object through which all other objects uniquely factor, i.e., an object $t\in C$ such that for every $c\in C$ there is a unique arrow $c\to t$. Such an object $t$ is called a terminal object of the category $C$.

Dually, a universal cone from the empty diagram is an object $i\in C$ such that for every object $c\in C$ there is a unique arrow $i\to c$. Such an object $i$ is called an initial object of the category $C$.

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 $\mathbf{\text{1}}\to C$ correspond to objects of $C$, and hence to to single-object diagrams (with no nonidentity arrows). Convince yourself that both the limit and colimit of such a one-object diagram is simply that object itself (with the identity arrow to/from that object).

The two-object, no-arrows diagram

Let $J$ be a category with exactly two objects and no non-identity arrows. Each functor $F:J\to C$ then corresponds to a choice of two objects $c_1,c_2\in C$. A limit of this two-object diagram is then a universal cone to the two objects:

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 $c_1$ and $c_2$, and is denoted $c_1\times c_2$.

• In the category $\mathbf{\text{Set}}$, this diagram corresponds to a pair of sets $A$ and $B$. This limit always exists and is called their Cartesian product, denoted $A\times B$.
• In the category $R-\mathbf{\text{Mod}}$, the diagram corresponds to a pair of $R$-modules $M$ and $N$. The limit always exists and is called their direct product, denoted $M\times N$.
• If $M$ is a given $R$-module and $\mathbf{\text{M}}$ is the category of submodules of $M$, the diagram corresponds to a pair of submodules $N_1$ and $N_2$ of $M$. This limit always exists and is called their intersection, denoted $N_1\cap N_2$.

Suggested next note

Universal Properties III - Yoneda's Lemma