The categorical role of $\overline{\mathbb{N}}$ and sequences in the category of topological spaces If you have a nice topological space, say, one in which the topology comes from a metric, then every topological property (like being continuous, closed, open,...) can be described in terms of sequences. As sequences in $X$ correspond to maps from the one point compactification $\overline{\mathbb{N}}=\mathbb{N}\cup\{\infty\}$ to $X$, this means that the space $\overline{\mathbb{N}}$ is somehow import in the overall category. Hence, my question is

Is there a categorical notion which describes the role of $\overline{\mathbb{N}}$ in the category of "nice" topological spaces?

For example, the functor of points $\operatorname{Hom}(\overline{\mathbb{N}},-)$ reflects isomorphisms, which is nice but I don't know if it is enough to recover the important properties about sequences or to characterize $\overline{\mathbb{N}}$ (maybe it is enough).
 A: An object $S$ such that $\textrm{Hom} (S, -)$ is faithful is said to be a separator or generator.
If the functor $\textrm{Hom} (S, -)$ is (in addition) conservative then $S$ is an extremal generator.
The reason for this is that the family of all morphisms $S \to X$, for a given $X$, is jointly extremally epimorphic, meaning that if $X' \hookrightarrow X$ is a monomorphism such that every morphism $S \to X$ factors through $X' \hookrightarrow X$, then $X' \hookrightarrow X$ is an isomorphism.
The property of being an extremal generator is not a universal property by any means: for instance, if $T$ is any object admitting a morphism $T \to S$ and $S$ is an extremal generator, then $S \amalg T$ is also an extremal generator.
It seems clear that $\overline{\mathbb{N}}$ is somehow minimal among extremal generators in the category of sequential topological spaces.
Perhaps this could be formalised as follows: if $S$ is any extremal generator in the category of sequential topological spaces then there is a split monomorphism $\overline{\mathbb{N}} \to S$.
(I think this is true, but I haven't checked the details.)
A: Not a complete answer yet, but certainly too long for a comment.

The category of "nice" spaces is generated by the topology on $\bar N$ in the following sense (taken from 1.1. in Kriegl and Frolicher's book Linear Spaces and Differentiation Theory).
The topology on $\bar N$, e.g. its family of open subsets, is equivalently given by the set $\mathcal T$ of functions $T\colon\bar N\to\Sigma$ where $\Sigma=\{o,c\}$ for which $T^{-1}(o)$ is open, i.e. for which $\infty\in T^{-1}(o)$ implies $n\in T^{-1}(o)$ for all but finitely many $n$.
We can define a sequential space to be set $X$ equipped with a set $S_X$ of functions $x_-\colon\bar N\to X$, called (convergent) sequences, and a set $F_X$ of functions $f\colon X\to\Sigma$, such that $x_-\in S_X$ if and only if $f\circ x_-\in\mathcal T$ for every $f\in F_X$, and $f\in F_X$ if and only if $f\circ x_-\in\mathcal T$ for every $x_-\in S_X$.
Explicitly, the above says that $f\in F_X$ if and only if for every $x_-\in S_X$, $x_\infty\in U=f^{-1}(o)$ implies $x_n\in U$ for all but finitely many $n$. In fact, the subsets of $X$ corresponding to $f^{-1}(o)$ for $f\in F_X$ form a topology: the finest topology for which the sequences are continuous, i.e. convergent.
To see that open subsets are closed under finite intersection, note that $x_\infty\in U\cap V$ implies $x_\infty\in U=f^{-1}(o)$ and $x_\infty\in V=g^{-1}(o)$, whence $x_n\in U$ for all but finitely many $n$ and $x_n\in V$ for all but finitely many $n$. Since $x_n\not\in U\cap V$ only if $x_n\not\in U$ for $x_n\not\in V$, we have $x_n\in U\cap V$ for all but finitely many $n$, whence $U\cap V=h^{-1}(o)$ for some $h\in F_X$.
To see open subsets are closed under arbitrary unions, note that if $U_i=f_i^{-1}(o)\subseteq X$ for $f_i\in F_X$, then $x_\infty\in\bigcup_i U_i$ is equivalent to $x_\infty\in U_j$ for some $j$, hence implies $x_n\in U_j$ for all but finitely many $n$, and thus implies $x_n\in\bigcup_iU_i$ for all but finitely many $n$. Thus $\bigcup_i U_i=f^{-1}(o)$ for some $f\in F_X$.
Finally, a function $g\colon X\to Y$ is a morphism from a sequential space $X$ to a sequential space $Y$ if and only if one of the equivalent conditions holds:

*

*$g\circ x_-\in S_Y$ for every $x_-\in S_X$


*$f\circ g\in F_X$ for every $f\in F_Y$


*$f\circ g\circ x_-\in\mathcal T$ for every $f\in F_Y$, $x_-\in S_X$.
In particular, 2. asserts that morphisms of sequential spaces are the continuous ones for the topology.

Evidently, the above is a special case of a general construction of a category beginning with any set of morphisms $S\to R$ between two objects. One can ask whether there is an intrinsic way of characterizing within that category the object $S$ with the structure of the set of morphisms $S\to R$, and the set of endomorphisms such that every composite with $S\to R$ lands in $S\to R$.
Note that you can recover the underlying set via the functor represented by the terminal object (when all constant functions $S\to R$ are included), the functions $S\to X$ from the functor represented by $S$, and the functions $X\to R$ from the contravariat functor represented by $R$.
In particular, the question reduces to the analysis of when two pairs of sets of morphisms $S_1\to R_1$ and $S_2\to R_2$ generate the "same" category.
