Is there a connection between the concepts of limits in ordinals, functions and categories? In set theory there is the concept of a limit ordinal: Nonzero ordinals that are the supermum of all ordinals below them.
In functional analysis there are the concepts of limits of functions (and sequences) a value that the function comes arbitrarily close to at a point.
And in category theory there is a concept of a limit which is a universal cone.
Is there something common about all these ideas that justifies them all being called limits or is it a coincidence of language ?
 A: To connect ordinals and analysis, one puts the order topology on ordinals. Then, given a metric space (or topological space) $X$, a sequence is just a function $f: \mathbb{N}\rightarrow X$, or, using the first countable ordinal $f: \omega\rightarrow X$. It now happens that a sequence converges iff it can be extended to a function $f':\omega +1 \rightarrow X$, such that $f'(n)=f(n)$ for all $n \in \omega$ and $f'$ is continuous with respect to the order topology on $\omega + 1$.
Then, to connect ordinals and category theory, take the category of all ordinals and, say, increasing maps between them. We use the fact that for any two ordinals $\alpha$ and $\beta$, one of them embeds into the other as an initial segment. For any set $A$ of ordinals form a diagram with this embedding maps. Then, the categorical limit of this diagram is precisely the limit in the sence of ordinals: it will be the supremum (limit) of all ordinals from $A$, and the limit maps will embed ordinals from $A$ into this limit.
A: They are all special cases of limits in the category-theoretic sense.  
Limit ordinals are a special case of least upper bounds in partially ordered sets.  Given a partially ordered set $(X,\le)$, we may form a category whose objects are elements of $X$ where there is a single morphism from $x$ to $y$ whenever $x\le y$.  Transitivity gives us composition and reflexivity gives us identity morphisms.  In that case, the least upper bound of some subset $Y\subset X$ is precisely the limit of the diagram spanned by $Y$.  
The limits of functions and sequences that we study in functional analysis and, more generally, in topology are in fact also a special case of least upper bounds in partially ordered sets, so they are also generalized by category-theoretic limits.  If $X$ is a (topological) space, a filter on $X$ is a set $\mathcal F$ of subsets of $X$ such that:


*

*$\emptyset\not\in\mathcal F$

*If $Y\in\mathcal F$ and $Y\subset Z$ then $Z\in\mathcal F$

*If $Y,Z\in\mathcal F$ then $Y\cap Z\in\mathcal F$


As an example, if $x\in X$ then the set of all neighbourhoods of $x$ (i.e., subsets of $X$ that contain some open neighbourhood of $x$) is a filter on $X$, called the neighbourhood filter $\mathcal N_x$.  We say a filter $\mathcal F$ converges to $x$, and write $\mathcal F\to x$, if $\mathcal N_x\subset\mathcal F$.  
What has this to do with convergence of sequences and functions?  Well, suppose that $(x_n)$ is a sequence in $X$.  Then we can define a filter $\mathcal S_{(x_n)}$ by:
$$
S_{(x_n)} = \left\{Y\subset X\;\colon\;\exists N \;.\;\textrm{if }n\ge N\textrm{ then }x_n\in Y\right\}
$$
the set of all subsets of $X$ that eventually contain every term of the sequence.  You can check for yourself that $x_n\to x$ if and only if $\mathcal N_x\subset S_{(x_n)}$.  
Limits of functions can be handled in a similar way.  Now, given some space $X$, we may define a partially ordered set $F$ whose elements are the filters on $X$, ordered by inclusion.  Let $\mathcal F$ be a filter whose limit we want to find.  For example, we might have $\mathcal F=S_{(x_n)}$ for some sequence $(x_n)$.  Given $x\in X$, define
$$
\mathcal L_{\mathcal F,x}=\left\{\mathcal G\in F\;\colon\; \mathcal G\subset\mathcal F, \mathcal G\to x\right\}
$$
Then $\mathcal F\to x$ if and only if $\mathcal F$ is the least upper bound in $F$ for $\mathcal L_{\mathcal F, x}$.
A: I like this question and I look forwards to seeing other answers.
Until then, here's a thin connection.
In elementary calculus, one is taught that the limit of a monotonic function can
be obtained by considering its supremum. Then when one peaks into the world of
order theory (eg probability theory) then the notion of a continous function is
defined to be one that preserves certain suprema and suprema are even sometimes
called limits in such contexts. Now, the notion of a product, in set theory,
can also be construed as a limit, as a supremum: given sets $A, B$, define
P₀ ≔ ∅
Pₙ₊₁ ≔ P ∪ {(a,b)} where a ∈ A are b ∈ B selected such that (a,b) ∉ Pₙ

Then we have $A×B = limₙ Pₙ = ⋃ₙ Pₙ = ⋃_{a ∈ A, b ∈ B} \{(a,b)\}$.
Hope this helps a bit.
