Contents

category theory

# Contents

## Idea

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

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

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 $(i/c)$:

$\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)$
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 restricted Yoneda embedding $C \to [S^{op},Set]$ is fully faithful.

## 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.

## 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 $X\to \mathbf{X}$ in $\mathrm{pro}C$ such that $\mathbf{X}$ 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 $X\to\mathbf{Y}$.

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.

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

• 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)

• 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.