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Limits and Colimits, Part 2 (Definitions)

Welcome back to our mini-series on categorical limits and colimits! In Part 1 we gave an intuitive answer to the question, "What are limits and colimits?" As we saw then, there are two main ways that mathematicians construct new objects from a collection of given objects: 1) take a "sub-collection," contingent on some condition or 2) "glue" things together. The first construction is usually a limit, the second is usually a colimit. Of course, this might've left the reader wondering, "Okay... but what are we taking the (co)limit of?" The answer? A diagram. And as we saw a couple of weeks ago, a diagram is really a functor.

We are now ready to give the formal definitions (along with more intuition). First, here's a bit of setup.

The Setup

In what follows, let $\mathsf{C}$ denote any category and let $\mathsf{I}$ be an indexing category. Note that given any object $X$ in $\mathsf{C}$, there is a constant functor- let's also call it $X$ - from $\mathsf{I}$ to $\mathsf{C}$. This functor sends every object to $X$ and every morphism to the identity of $X$.

Therefore, given any functor (ahem, diagram) $F:\mathsf{I}\to\mathsf{C}$, we can make sense of a natural transformation between $X$ and $F$. Such a natural transformation consists of a collection of morphisms between $X$ and the objects in the diagram $F$. Moreover, these morphisms must commute with all the morphisms that appear in diagram. If the arrows point from $X$ to the diagram $F$, then the setup is called a cone over $F$, as we previously discussed here. If, on the other hand, the arrows point from the diagram $F$ to $X$, then it's called a cone under $F$ (or sometimes a cocone).

Notice that a cone is comprised of two things: an object AND a collection of arrows to or from it. Now here's the punchline:

The limit of a diagram $F$ is a special cone over $F$.

The colimit of $F$ is a special cone under $F$.

Let's take a look at the formal definitions. I'll give a lite version first, followed by the full version.

Definitions (Lite Version)

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Definition (limit): The limit of a diagram $F$ is the "shallowest" cone over $F$.

By "shallowest" (not a technical term) I mean in the sense of the picture to the right. There may be many cones -- many objects with maps pointing down to the diagram $F$ (depicted as a blob) -- over $F$, but the limit is the cone that is as close as possible to the diagram $F$. Perhaps this is why "limit" is a good choice of terminology. You might imagine all the cones over $F$ as cascading down to the limit.

If we let gravity pull all the arrows down, then we obtain the dual notion: a colimit.

Definition (colimit): The colimit of a diagram $F$ is the "shallowest" cone under $F$.

Again, by "shallowest" (not a technical term) I mean in the sense of the picture on the left. There may be many cones under $F$, but the colimit is the one that's closest to the diagram. It's the shallowest.

Okay, this is all very handwavy and not very informative. To capture the mathematics behind "shallowest," we'll use a universal property. I'll comment on intuition below.

Definition: Limit (Full Version)

The definitions themselves can be stated very succinctly. But like little onions, they have several layers, which we will peel away slowly to minimize the shedding of tears.

Definition (1): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is the universal cone over $F$.

Let's unwind this a bit...

Definition (2): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{lim }F$ in $\mathsf{C}$ together with a natural transformation $\eta:\text{lim }F\Rightarrow F$ with the following property: for any object $X$ and for any natural transformation $\alpha\colon X\Rightarrow F$, there is a unique morphism $f\colon X\to \text{lim }F$ such that $\alpha=\eta\circ f$.

Let's unwind this a bit more...

Definition (3): The limit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{lim }F$ in $\mathsf{C}$ together with morphisms $\eta_A:\text{lim }F\to A$, for each $A$ in the diagram, satisfying $\eta_B=\phi_{AB}\circ \eta_A$ for every morphism $\phi_{AB}\colon A\to B$ in the diagram. Morever, these maps have the following property: for any object $X$ and for any collection of morphisms $\alpha_A:X\to A$ satisfying $\alpha_B=\phi_{AB}\circ \alpha_A$, there exists a unique morphism $f\colon X\to\text{lim }F$ such that $$\alpha_A=\eta_A\circ f \qquad\text{for all objects $A$ in the diagram.}$$

In summary, for all objects $X$ in $\mathsf{C}$:

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Definition: Colimit (Full Version)

Definition (1): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ the universal cone under $F$.

Let's unwind this a bit...

Definition (2): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{colim }F$ in $\mathsf{C}$ together with a natural transformation $\epsilon:F\Rightarrow \text{colim }F$ with the following property: for any object $X$ and for any natural transformation $\beta\colon F\Rightarrow X$, there is a unique morphism $g\colon \text{colim }F\to X$ such that $\beta= g\circ \epsilon$.

Let's unwind this a bit more...

Definition (3): The colimit of a diagram $F:\mathsf{I}\to\mathsf{C}$ is an object $\text{colim }F$ in $\mathsf{C}$ together with morphisms $\epsilon_A:A\to\text{colim }F$, for each $A$ in the diagram, satisfying $\epsilon_A=\epsilon_B\circ \phi_{AB}$ for every morphism $\phi_{AB}\colon A\to B$ in the diagram. Morever, these maps have the following property: for any object $X$ and for any collection of morphisms $\beta_A: A\to X$ satisfying $\beta_A=\beta_B\circ \phi_{AB}$, there exists a unique morphism $g\colon \text{colim }F\to X$ such that $$\beta_A= g \circ \epsilon_A\qquad\text{for all objects $A$ in the diagram.}$$

In summary, for all objects $X$ in $\mathsf{C}$:

a little intuition + a little exercise

I once heard (or read?) Eugenia Cheng refer to a universal property as a way to describe a special role than an object—or in our case, a cone—plays. I like that analogy, and it's exactly what's going on with limits. (Similar sentiments hold for colimits.) Let me elaborate:

Out of all the cones over a diagram $F$, there is exactly one that plays the role of limit, namely the pair $(\text{lim}F,\eta)$. Of course you might come across another cone $(X,\alpha)$ that plays a very similar role. Perhaps $\alpha$ behaves very similarly to $\eta.$ BUT -- and this is the punchline -- this behavior is no coincidence! The natural transformation $\alpha$ "behaves" like $\eta$ because it is built up from $\eta$! More precisely, it has $\eta$ as a factor: $\alpha=\eta\circ f$ for some unique morphism $f$!

By way of analogy, think of the role that the number 2 plays among the integers. Out of all the integers, we might say that 2 is the quintessential candidate for "an integer which possesses the quality of 'two-ness,'" that is, of being even. Of course there are other integers $a$ that play a similar role. In particular, if $a$ is an even integer, then it also possesses the quality of "two-ness." But this is no coinicidence! An even integer is even because it is built up from 2! More precisely, it has 2 as a factor: $a=2k$, for some unique integer $k$!

These two equations $a=2k$ and $\alpha=\eta\circ f$ are analogous. In fact, they're more than analogous....

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EXERCISE

Let $\mathsf{C}$ be the category $2\mathbb{Z}$ of even integers. A morphism $n\to m$ in this category is an integer $k$ such that $n=mk$. For example, 3 defines an arrow $6\overset{3}{\longrightarrow} 2$ because $6=2\times 3$. On the other hand, there is no arrow $8\longrightarrow 6$. Show that $2$ is the limit of a particular diagram in $2\mathbb{Z}$.

That is, come up with an indexing category $\mathsf{I}$ and a functor $\mathsf{I}\to2\mathbb{Z}$

(Next question: does this diagram have a colimit? If so, what is it?)

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I'll close with one final thought. Once we get used to the ideas/definitions above, we discover that limits and colimits have very familiar names, depending on the shape of the indexing category $\mathsf{I}$!