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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?

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    $\begingroup$ In my view, the morally correct definition is that a dense functor is one such that the induced Yoneda representation is fully faithful. It is a consequence of the fact that every presheaf is a colimit of a canonical diagram that we obtain the equivalent characterisation in terms of colimits. $\endgroup$
    – Zhen Lin
    Commented Jun 12, 2016 at 20:36
  • $\begingroup$ Very late to the party. If you would like to have something analogue to the density of the image of a function $f:C\to D$ (so that continuous functions $g$ on $D$ are determined by their compositions $g\circ f$) the notion of initial functors comes close: $F:C\to D$ is initial if and only, for every set-valued functor $G:D\to Set$, $G$ and $G\circ E$ have the same limit. You can characterize this by the fact that the comma category $(F/d)$ is non-empty and connected for every $d\in D$. Unfortunately, the nLab entries for dense and initial functors don't mention any relations. $\endgroup$
    – Jochen Wengenroth
    Commented Dec 13, 2020 at 11:32
  • $\begingroup$ the comma category is natural in the top object, a particulate diagram no, Ind-categories theory is a way to have a surrogate of naturality for particular diagrams. $\endgroup$
    – Buschi Sergio
    Commented Jul 30, 2022 at 14:14

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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; 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, although it doesn't discuss dense generation.

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