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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.

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    I think [this thread here](http://math.stackexchange.com/q/60590/5363) is better than the linked ones.2012-03-31

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