nLab
dense functor

Idea

A subcategory $S$ of category $C$ is dense if every object $c$ of $C$ is a colimit of a diagram of objects in $S$, in a canonical way.

Definition

A functor $i:S\to C$ is dense if it satisfies the following equivalent conditions.

1. every object $c$ of $C$ is the vertex of the following colimit over the comma category $(i/c)$:

   $$\mathrm{colim}((i/c)\stackrel{{\mathrm{pr}}_S}{⟶}S\stackrel{i}{\to }C)$$\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)

2. every object $c$ of $C$ is the $C(i-,c)$-weighted colimit of $i$. This version generalizes more readily to the enriched context.

3. the corresponding nerve functor (or “restricted Yoneda embedding”) $C\to [S^{\mathrm{op}},\mathrm{Set}]$ is fully faithful.

Terminology and History

Dense functors were defined in MR0175954 under the name left adequate (the name was actually applied to the subcategory). The dual notion of right adequate was also introduced and subcategories satisfying both were called adequate. It was also shown that while the relation of being left (or right) adequate is not transitive, being adequate is transitive.

Also, in arXiv v4 of HTT this notion (for (∞,1)-categories) is referred to as strongly generating, but that term actually means something different.

Examples

1. Let $V$ be a category of algebras and $n\in \mathbb{N}$ such that $V$ has a presentation with operations of at most arity $n$. Let $v$ be the free $V$-algebra on $n$ generators. Then the full subcategory with object $v$ is dense in $V$.

2. In $Set$, a singleton space is dense.

Warning

There is a different notion of a dense subcategory, often used in shape theory, which has a bit of the same spirit. A full subcategory $D\subset C$ is dense in this second sense, if every object in $C$ admits a $D$-expansion.

A $D$-expansion of an object $X$ in $C$ is a morphism $X\to X$ in $\mathrm{pro}C$ such that $X$ is in $\mathrm{pro}D$ and $X$ is the rudimentary system (constant inverse system) corresponding to $X$; moreover one asks that the morphism is universal among all such morphisms $X\to Y$.

Given a dense subcategory $D\subset C$ one defines an abstract shape category $\mathrm{Sh}(C,D)$ which has the same objects as $C$, but the morphisms are the equivalence classes of morphisms in $\mathrm{pro}D$ of $D$-expansions.

References

• Isbell, J. R. (1960). Adequate subcategories. Illinois J. Math., 4, 541–552. MR0175954