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Category-theoretic limit related to topological limit?
I was wondering if there exists some way to re-interpret "analytical" limits $$ \lim_{x\to c}\; f(x)$$ as categorical limits/colimits of a functor $F\colon J\to \bf C$.
This could be done following a naive point of view, the analogy $$\text{cnts function}\iff \text{cnts functor}$$ but I feel there are some evident problems: for a function of top spaces $f\colon X\to Y$, just assume $Y$ to be non-Hausdorff and uniqueness of limit $\lim_{x\to c}$ is lost. Is there some way to link these kind of limits to weak limits in Category Theory?
In fact a deeply linked question is: how to categorify the notion of limit of a sequence on a topological space? This turns out to be easier; just consider $\mathbb N$ as a discrete category, a sequence $f\colon \mathbb N\to X$ is then a functor from $\mathbb N$ to... well, the bare problem I can't get what is the right way to categorify a topological space: discrete category? Or maybe I should exploit in some way the definition of a Grothendieck topology.
To tell the truth, I hardly believe that nobody studied this kind of problem: maybe I'm not able to find good references.
Thanks a lot.
