**Definition.** A functor $F:\mathsf C\rightarrow \mathsf D$ is *dense* if every $D\in \mathsf D$ is the vertex of the following colimit $$\varinjlim \left(F\downarrow D\rightarrow\mathsf C\rightarrow \mathsf D \right).$$


I would like to understand the idea behind this definition better. It seems that to imitate the spatial situation of dense subsets (or dominant functions), we could just ask every $D\in\mathsf D$ to be *a* colimit of *some* diagram in the image of $F$. Instead, we ask every object to be the colimit of an enormous diagram, because it is canonical.


Being canonical is nice and all, but is there a more intuitive justification behind this definition? (Having the truncated Yoneda embedding full and faithful, in my view, is not a justification, but a consequence.) The only thing I can come up with is that in spaces, nets are hugely redundant because only their tail matters, and the real approximation using dense subsets is by "close points", whatever that means. The analogue of this is somehow looking at colimits only on objects equipped with a map from the essential image of $F$.


Why is the primitive imitation of the spatial situation not interesting?



Definition of dense functors
I don't think the topological analogue is worth paying attention to. Here's a basic reason to care: suppose $G : C \to E$ is another functor, and you'd like to compute the left Kan extension $\text{Lan}\_F(G) : D \to E$. If enough colimits exist, the left Kan extension is given pointwise by


$$\text{Lan}\_F(G)(d) = \text{colim}\_{f : F(c) \to d} G(c)$$


which can be interpreted as follows. For fixed $d$, the canonical diagram of maps $f : F(c) \to d$ is the universal way to approximate $d$ by a colimit of objects $c \in C$, and the left Kan extension figures out how to apply $G$ to $d$ by applying $G$ to this diagram, then taking the colimit in $E$. 


In particular, $F$ is dense iff $\text{Lan}\_F(F)$ is the identity functor. The left Kan extension $\text{Lan}\_F(F)$ of a functor along itself is the density comonad, dual to the [codensity monad](https://ncatlab.org/nlab/show/codensity+monad); it measures the extent to which $F$ fails to be dense. 


There are various notions of what it might mean for (the image of) $C$ to generate $D$; this is one of them, and there are others. See, for example, [this blog post](https://qchu.wordpress.com/2015/05/17/generators/), although it doesn't discuss dense generation. 


