Franklin's Notes

Cones and limits

A diagram of shape $\mathsf J$ in a category $\mathsf C$ is defined to be a functor $F:\mathsf J\to\mathsf C$. A cone over the diagram $F$ with summit $c\in\mathsf C$ is a natural transformation $\lambda:c\implies F$ whose domain is the constant functor at $c$, which sends every object of $\mathsf J$ to $c$, and every morphism to $1_c$. The components of $\lambda$ are called the legs of the cone. We can think of a cone $\lambda:c\implies F$ as a collection of morphisms from $c$ to the objects in the image of $F$ which are all "consistent" with each other with respect to the morphisms in the image of $F$, as per the naturality condition:

On the other hand, a cone under $F$ with nadir $c$ is a natural transformation $\lambda:F\implies c$, whose legs are again the components of $\lambda$. These are also called cocones, and are a dual concept to cones.

Now, for any diagram $F:\mathsf J\to\mathsf C$, we can define a functor $\mathrm{Cone} (-,F):\mathsf C^\text{op}\to\mathsf{Set}$ which sends $c\in\mathsf{C}$ to the set of cones over $c$ (assuming $\mathsf C$ is locally small). This construction works because if we have a function $f:d\to c$, we can turn a cone over $F$ with summit $c$ into a cone over $F$ with summit $d$ by pre-composing $f$ with each of the legs. That is, if $\lambda:c\implies F$ is a cone, then $\mu:d\implies F$ is a cone, where the components of $\mu$ are defined by $\mu_x = f\lambda_x$. Finally, we say that $\text{lim} F\in\mathsf C$ is a limit of $F$ if it represents the functor $\mathrm{Cone} (-,F)$, and the cone with $\text{lim} F$ at its summit is called the limit cone. Similarly, a colimit for $F$, denoted $\text{colim} F\in\mathsf C$, is an object representing $\mathrm{Cone} (F,-)$, the functor sending each object $c\in\mathsf C$ to the set of cocones with nadir $c$. The colimit cone is defined analogously.

A (slightly less confusing, in my opinion) way of equivalently defining a limit is as a universal arrow from the diagonal functor $\Delta:\mathsf C\to\mathsf C^\mathsf J$ to some object $F\in \mathsf C^\mathsf J$. Analogously, a colimit is a universal arrow in the opposite direction, from $F$ to $\Delta$. (This is how MacLane introduces colimits and limits, whereas the above is how Riehl introduces them.)

Here are some examples:

• If $F:\mathsf J\to\mathsf C$ is a diagram in an abstract poset category $\mathsf C$, then a limit for $F$ is a greatest lower bound for objects in the image of $\mathsf J$, and a colimit is a least upper bound. Therefore, if $\mathsf C$ is a lattice, it has limits and colimits.

• A diagram of shape $\omega$ in $\mathsf C$ may have several different limits depending on the morphisms used in the diagram, even if the objects used are the same. Consider the category $\mathsf{LinOrd}$, and the following three diagrams of shape $\omega$, where $X_i$ denotes $F(i)$ and $f_i$ denotes $Fg_i$ where $g_i$ is the unique morphism $g_i:i\to (i+1)$ in $\omega$:

1. $X_i=\mathbb N$, $f_i=1_\mathbb N$
2. $X_i=\mathbb N$, $f_i(n)=n+1$ for all $n\in\mathbb N$
3. $X_i=\mathbb N$, $f_i(n)=2n$ for all $n\in\mathbb N$

The first cone has limit $\mathbb N$, the second cone has limit $\mathbb Z$, and the third cone has limit $\mathbb Q$. In fact, for any countable ordering $\alpha$ with no greatest element, there exists a choice of morphisms $f_i:\mathbb N\to\mathbb N$ such that the cone has limit $\alpha$.

• In the category of rings, the p-adic integers can be defined as the limit of the diagram of shape $\omega^\text{op}$ consisting of

• If $F:\mathsf J\to\mathsf C$ and is a diagram and $\mathsf J$ has an initial object $c$, then $Fc$ is the summit of a limit cone $\lambda$ over $F$ whose leg $\lambda_x$ is equal to $Ff$, where $f$ is the unique morphism $f:c\to x$. One can therefore think of the problem of finding a limit of a cone $\lambda$ as equivalent to the problem of augmenting a subcategory of $\mathsf C$ (namely, the subcategory given by the image of $\mathsf J$ under $F$) with a single object such that that object serves as an initial object in the augmented category.

We can also define a category of cones over a diagram $F$, which has cones as its objects, so that a morphism $f:\lambda\to\mu$ from the cone $\lambda:c\implies F$ to the cone $\mu:d\implies F$ is a morphism $f:c\to d$ that commutes with the legs of the cones, so that $\mu_x f=\lambda_x$ for all objects $x\in\mathsf J$. A limit of $F$ corresponds to a terminal object in the category of cones over $F$, and a colimit of $F$ corresponds to an initial object in the category of cones under $F$.

Proposition 1. A limit is a terminal object in the category of cones.

Proof. Let $\lambda:l\implies F$ be a limit over $F$ in $\mathsf C$, so that $\alpha:\text{Hom}(-,l)\cong \text{Cone}(-,F)$ as representable functors , where $\alpha$ is a natural bijection. We know from our exploration of the Yoneda lemma that $\alpha$ is completely determined by $\alpha_l(1_l)$, whose value can be taken to be any element of $\text{Cone}(l,F)$. Let $\alpha_l(1_l)=\lambda\in \text{Cone}(l,F)$. Then, as we have shown here , the values of $\alpha$ at each of its components are given as follows, where $f\in\text{Hom}(x,l)$ is arbitrary: which is equal to the cone obtained by precomposing each leg of $\lambda$ by $f$, by the definition of the functor $\text{Cone}(-,F)$.

Thus, there is a bijection between morphisms $f\in\text{Hom}(x,l)$ and cones in $\text{Cone}(x,F)$, for all $x\in\mathsf C$, and this bijection is given by precomposition of the legs of $\lambda$ by $f$. Therefore, for any cone $\mu$ over $F$ with summit $x$, there exists exactly one morphism $f:x\to l$ such that precomposition by $f$ respects the legs of the cones $\lambda$ and $\mu$, meaning that there is a unique morphism $\mu\to\lambda$ in the category of cones, and $\lambda$ is a terminal object in this category. $\blacksquare$

The following is a kind of uniqueness theorem for limits. Note that it is stronger than simply stating that limits are unique up to isomorphism: not only are they unique up to isomorphism, but the isomorphism itself (between the summits) is also uniquely determined.

Proposition 2. If $\lambda: l\implies F$ and $\lambda':l'\implies F$ are limit cones over the common diagram $F$, then there is a unique isomorphism $i:l\cong l'$ that respects the legs of the limit cones, so that $i\lambda'_x=\lambda_x$ for all components $\lambda_x,\lambda'_x$.

Proof. Since $\lambda$ and $\lambda'$ are terminal objects in the category of cones over $F$, there must exist a unique isomorphism between them in the category of cones. An isomorphism in the category of cones corresponds to an isomorphism that respects cone legs, via the natural isomorphism $\text{Cone}(-,F)\cong\text{Hom}(-,l)$. In particular, if $\alpha:\text{Cone}(-,F)\cong\text{Hom}(-,l)$ is this natural isomorphism, then $i^{-1}=\alpha_{l'}(\lambda')$ or $i=\alpha_{l'}(\lambda')^{-1}$ gives us the isomorphism $i:l\cong l'$ that we seek. $\blacksquare$