Limit in Analysis and Category Theory

$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n\end{array}$$\lim_{n\to\infty}\alpha_n\\$ is well-known in Mathematical Analysis,

In Category Theory, we also have $\begin{array}{c}\underset{⟶}{lim}\alpha \end{array}$$\lim_{\longrightarrow}\alpha\\$ and $\begin{array}{c}\underset{⟵}{lim}\alpha \end{array}$$\lim_{\longleftarrow}\alpha\\$. (direct limit and inverse limit)

A natural question is, why do we use the term limit?

What is the connection between the limit in Analysis and the limit in Category Theory?

This article aims to see the connection.

 

Let $(\mathbb{R},\le )$$(\mathbb R,\le)$ be a category, call it $\mathcal{R}$$\mathcal R$.

The objects are real numbers and the morphism between two real numbers is: ${\mathrm{Hom}}_{\mathcal{R}}(a,b)=a\le b$$\mathrm{Hom}_{\mathcal R}(a,b)=a\le b$.

Consider a bounded increasing sequence sequence $\alpha (i)\in {\mathrm{Hom}}_{\mathrm{Set}}(\mathbb{N},\mathbb{R})$$\alpha(i)\in\mathrm{Hom}_{\mathrm{Set}}(\mathbb N,\mathbb R)$.

i.e. $\alpha :$$\alpha:$ is a functor from $\mathcal{N}:=(\mathbb{N},\le )$$\mathcal N:=(\mathbb N,\le)$ to $\mathcal{R}:=(\mathbb{R},\le )$$\mathcal R:=(\mathbb R,\le)$.

Then we have

$$\begin{array}{}\text{(1)}& \alpha (0)\le \alpha (1)\le \alpha (2)...\end{array}$$

i.e.

$$\begin{array}{}\text{(2)}& \alpha (0)⟶\alpha (1)⟶\alpha (2)...\end{array}$$

Then $\begin{array}{c}sup\alpha (i)=\underset{i\to \mathrm{\infty }}{lim}\alpha (i)\end{array}$$\sup \alpha(i)=\lim_{i\to\infty}\alpha(i)\\$

Which is exactly the $\begin{array}{c}\underset{⟶}{lim}\alpha \end{array}$$\lim_{\longrightarrow}\alpha\\$ in $\mathcal{R}$$\mathcal R$!

By duality, we could consider any bounded decreasing sequence. $\beta (i)\in {\mathrm{Hom}}_{\mathrm{Set}}(\mathbb{N},\mathbb{R})$$\beta(i)\in\mathrm{Hom}_{\mathrm{Set}}(\mathbb N,\mathbb R)$

i.e. $\beta$$\beta$ is a functor from $\mathcal{N}:=(\mathbb{N},\ge )$$\mathcal N:=(\mathbb N,\ge)$ to $\mathcal{R}:=(\mathbb{R},\le )$$\mathcal R:=(\mathbb R,\le)$.

$$\begin{array}{}\text{(3)}& \beta (0)\ge \beta (1)\ge \beta (2)...\end{array}$$

i.e.

$$\begin{array}{}\text{(4)}& \beta (0)⟵\beta (1)⟵\beta (2)...\end{array}$$

Then $\begin{array}{c}inf\beta (j)=\underset{j\to \mathrm{\infty }}{lim}\beta (j)\end{array}$$\inf\beta(j)=\lim_{j\to\infty}\beta(j)\\$

Which is exactly the $\begin{array}{c}\underset{⟵}{lim}\beta \end{array}$$\lim_{\longleftarrow}\beta\\$ in $\mathcal{R}$$\mathcal R$!

A more natural way to consider the $\begin{array}{c}\underset{⟶}{lim}\end{array}$$\lim_{\longrightarrow}\\$ and $\begin{array}{c}\underset{⟵}{lim}\end{array}$$\lim_{\longleftarrow}\\$ as the initial and final object in the comma category.

In that category, the object is the bounded monotone sequence...