nLab dense functor

Idea

In topology, a not necessarily continuous function $f \colon X \to Y$ between Hausdorff spaces is dominant, or dense, in the sense that the image of $f$ is a dense subspace of $Y$, precisely if every continuous function $g \colon Y \to Z$ to any Hausdorff space $Z$ is uniquely determined by the composition $g \circ f$.

In category theory, the concept of a dense functor is an analogy of this concept for functors.

An important special case that was also historically the source of the concept, is the case of a dense subcategory inclusion: 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.

Remark

(Terminology and History)

Isbell 1960 introduced dense subcategories under the name left adequate subcategories. He showed that while the relation of being left (or right) adequate is not transitive, being adequate (left and right) is transitive. This brought out interesting connections with set theory and measurable cardinals.

Later, Ulmer 1968 considered the concept for more general functors $F:C\to D$, not only inclusions $I:C\hookrightarrow D$, and introduced the name dense for them. In Gabriel & Ulmer 1971 this gave rise to the notion of locally presentable categories.

Compare also Lazard's criterion in the abelian context, due to Lazard 1964.

More recently, Jacob Lurie has referred to the analogue notion for (∞,1)-categories as strongly generating in a version (arXiv v4) of his HTT, but that term normally means something different.

Definition

Definition

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

1. every object $c$ of $C$ is the colimit

   $\underset{\longrightarrow}{\lim} \Big( (i/c) \overset{\mathrm{pr}_S}{ \longrightarrow } S \overset{i}{\longrightarrow} C \Big)$

   over the comma category $(i/c)$:

2. every object $c$ of $C$ is the $C(i-,c)$-weighted colimit of $i$.

   (This version generalizes readily to the enriched category theory).

3. the corresponding restricted Yoneda embedding $C \to [S^{op},Set]$ is fully faithful.

4. the left Kan extension $lan_i i$ exists, is pointwise, and is isomorphic to the identity.

5. $C$ is the closure of $S$ under colimits of a family of diagrams, by which is meant a class of pairs $(F \colon L^{op} \to \mathcal{V}, P \colon L \to C)$ consisting of a weight and a diagram for that weight, and these colimits are $i$-absolute (i.e. preserved by the nerve $N_i$ of $i$), cf. Kelly 1982 Thm 5.19.

6. For every category $B$, precomposition by $i$ induces a full and faithful functor

   $$[i, 1] : S\text{-Cocts}[C, B] \to [S, B]$$

   where $S\text{-Cocts}[C, B]$ is the category of functors preserving weighted colimits whose weights are presheaves $S^{op} \to V$.

Properties

Remark

(dense functors not closed under composition)
Beware that the class of dense functors (Def. ) is not closed under composition of functors.

For a counter-example see Example below.

Examples

Example

The inclusion of the discrete category on the singleton set into all of Sets is a dense subcategory inclusion.

More generally, if $C$ is an essentially small category, then the Yoneda embedding $C \to [C^{op},Set]$ is dense.

Example

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 $V$.

More generally, if $V$ is a $\kappa$-accessible category, then the full subcategory inclusion $V_\kappa \subseteq V$ of $\kappa$-presentable objects is dense. This means in particular that if $C$ is a small category, then the canonical inclusion $C \to Ind(C)$ into its Ind category is dense, and that categories of sheaves have small dense subcategories.

Example

Consider the simplex category $\Delta$, regarded in the usual way as a subcategory of Cat. Let $\Delta_{\leq [1]} \subset \Delta$ be the full subcategory with object set $\{[0],[1]\}$ – i.e. comprising the 0-dimensional simplex and the 1-dimensional simplex.

Then

1. $\Delta_{\leq [1]} \hookrightarrow \Delta$ is a dense subcategory inclusion;

2. $\Delta \hookrightarrow \mathbf{Cat}$ is a dense subcategory inclusion (Kerodon 8.4.1.3: 03VB),

3. but the composite $\Delta_{\leq [1]} \hookrightarrow \Delta \hookrightarrow Cat$ is not dense in $\mathbf{Cat}$ (see Remark ).

Example

The category $Top$ of topological spaces does not have any small full subcategory which is dense. Indeed, $Top$ is not generated under colimits by any small subcategory.

Example

The full subcategory of $Set^{op}$ on a countable set is dense if and only if there is not a proper class of measurable cardinals. (Isbell ‘60, Section 2.5.) In fact, $Set^{op}$ admits a small dense subcategory if and only if there is not a proper class of measurable cardinals. (Adámek–Rosický, Theorem A.5.)

Example

Let $V$ be a Bénabou cosmos. Then the inclusion of the 2-category of small $V$-categories into the 2-category of large $V$-categories is dense as a 2-functor (Lemma 2.16 of Przybyłek (2014)).

Example

In the category of schemes the subcategory of affine schemes is dense, see schemes as sheaves on affine schemes.

Example

In the category of smooth manifolds, the full subcategory whose objects are given by $\mathbb{R}^n$ for $n\geq 1$ with the standard smooth structure, is dense.

Example

In the category of modules over a ring $R$, the full subcategory spanned by $\{R,R^2\}$ is dense.

Proof

The restricted Yoneda to the full subcategory spanned by $\{R\}$ is faithful, hence so is when restricted to the full subcat spanned by $\{R,R^2\}$. To see fullness when restricted to the full subcat $\mathcal{C}$ spanned by $\{R,R^2\}$, suppose we are given a morphism $\operatorname{Hom}(-,M)\Rightarrow\operatorname{Hom}(-,N)$ of presheaves on $\mathcal{C}$. Given $x\in M$, write $z_x\in N$ for the image of $1\in R$ of the induced map $R\to N$ from $R\to M,1\mapsto x$. Write $[x,y]:R^2\to M$ for the map sending $(1,0)$ and $(0,1)$ to $x$ and $y$, respectively, and write $z_{[x,y]}:R^2\to N$ for the induced map. Then $z_{[x,y]}=[z_x,z_y]$ by compatibility with the $i$-th component map $c_i:R\to R^2$, $i=1,2$. Thus, compatibility with the diagonal map $\Delta:R\to R^2$ gives $z_{x+y}=\Delta^*z_{[x,y]}=\Delta^*[z_x,z_y]=z_x+z_y$. On the other hand, given $r\in R$, compatibility with $r: R\to R$ gives $z_{r x}=rz_x$. Thus $x\mapsto z_x$ defines a map $f:M\to N$ of $R$-modules such that $f^*:\operatorname{Hom}(-,M)\Rightarrow\operatorname{Hom}(-,N)$ is the morphism we began with.

Related entries

• dense

• codense functor

• absolutely dense functor

• dense subcategory

• codensity monad

• space and quantity

• dominant geometric morphism

References

• John Isbell, Adequate subcategories, Illinois J. Math. 4 (1960) pp.541-552. MR0175954. (euclid)

• John Isbell, Subobjects, adequacy, completeness and categories of algebras, Rozprawy Mat. 36 (1964) pp.1-32. (toc, pdf)

• Friedrich Ulmer: Properties of dense and relative adjoint functors, J. of Algebra 8 (1968) 77–95 [doi:10.1016/0021-8693(68)90036-7]

• Horst Schubert; section 17.2, pp.29 in: Kategorien II, Springer (1970)

• Peter Gabriel, Friedrich Ulmer; § 3 in: Lokal präsentierbare Kategorien, LNM 221, Springer (1971) 38–44

• Saunders MacLane; section X.6, pp. 245–250 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

• William Lawvere: John Isbell’s Adequate Subcategories, TopCom 11 1 (2006) [archive]

• Giacomo Tendas: Dense functors and density presentations, unpublished notes [pdf, pdf]

• Jiří Adámek, Jiří Rosický: Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series 189, Cambridge University Press (1994) [doi:10.1017/CBO9780511600579]

• Michał Przybyłek: Analysis and construction of logical systems: a category-theoretic approach, PhD thesis (2014) [pdf]

• Tom Avery, Tom Leinster: Isbell conjugacy and the reflexive completion [arXiv:2102.08290]

• Kerodon, 8.4.1.3 (03VB)