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 a generalization of this concept to 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.

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 $i$). See (Theorem 5.19 of Kelly), for instance.

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,

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.

The category $Set^{op}$ has a small full subcategory which is dense if and only if there is not a proper class of measurable cardinals, a result due to Isbell.

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)).

Related entries

• dense

• codense functor

• absolutely dense functor

• dense subcategory

• codensity monad

• space and quantity

• dominant geometric morphism

Terminology and History

John Isbell introduced dense subcategories in a seminal paper (Isbell 1960) under the name left adequate. 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. He also brought out interesting connections with set theory and measurable cardinals.

Later in the mid 60s, Friedrich Ulmer 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.

Independently, Pierre Gabriel worked on this concept and their work coalesced to what was to become the concept of a locally presentable category of their 1971 monograph. It is also good to keep in mind the ‘Abelian’ subcontext in the background, in particular the developments in module theory e.g. Lazard’s (1964) characterization of flat modules as filtered colimits of finitely generated free modules.

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.

References

• Tom Avery, Tom Leinster, Isbell conjugacy and the reflexive completion, arXiv:2102.08290 (2021). (abstract)

• Peter Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien , LNM 221 Springer Heidelberg 1971. (§ 3, pp.38-44)

• 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)

• Max Kelly, Basic Concepts of Enriched Category Theory , Cambridge UP 1982. (Reprinted as TAC reprint no.10 (2005); chapter 5, pp.85-112)

• William Lawvere, John Isbell’s Adequate Subcategories, TopCom 11 no.1 2006. (link)

• Saunders Mac Lane, Categories for the Working Mathematician , Springer Heidelberg 1998². (section X.6, pp.245ff, 250)

• Horst Schubert, Kategorien II , Springer Heidelberg 1970. (section 17.2, pp.29ff)

• Friedrich Ulmer, Properties of dense and relative adjoint functors , J. of Algebra 8 (1968) pp.77-95.

• Giacomo Tendas, Dense functors and density presentations, unpublished notes (pdf)

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