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A functor $i:S\to C$ is dense if every object $c$ of $C$ is the vertex of the colimit of the following diagram $\varinjlim((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)$.

I would like to understand how this definition captures the intuitive case of limits in topological spaces. I thought of using the "categorical" characterization of convergence via filters, which is just $F\rightarrow x\iff N_x\subset F$ where $F$ is a filter on $X$ and $N_x$ is the neighborhood filter of $x$. Unfortunately, I'm having a lot of difficulty filling in the blanks. What would the colimit above look like in these case? I think the functor $i$ would be the functor $E$ from this MSE question, but I'm not sure about that either.

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  • I'm not sure there is actually a connection of the type you are looking for. – Zhen Lin May 03 '16 at 20:56
  • @ZhenLin expressing points in a space as limits of nets in dense subspaces seems to be inherently arbitrary, so maybe what I want is unreasonable to expect.. Could you perhaps help me get a feel for what exactly the colimit $\varinjlim((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)$ encodes "geometrically"? I tried reading what Lawvere says, but it was too terse for me. – Arrow May 04 '16 at 08:26

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