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dense functor

Contents

Contents

Idea

In topology, a not necessarily continuous function :XY \colon X \to Y between Hausdorff spaces is dominant, or dense, in the sense that the image of ff is a dense subspace of YY, precisely if every continuous function g:YZg \colon Y \to Z to any Hausdorff space ZZ is uniquely determined by the composition gfg \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 SS of category CC is dense if every object cc of CC is a colimit of a diagram of objects in SS, in a canonical way.

Definition

Definition

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

  1. every object cc of CC is the colimit

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

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

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

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

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

  4. the left Kan extension lan iilan_i i exists, is pointwise, and is isomorphic to the identity.

  5. CC is the closure of SS under colimits of a family of diagrams, by which is meant a class of pairs (F:L op𝒱,P:LC)(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 ii-absolute (i.e. preserved by the nerve N iN_i of ii). See (Theorem 5.19 of Kelly), for instance.

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

Example

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

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

Example

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

Then

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

  2. ΔCat\Delta \hookrightarrow \mathbf{Cat} is a dense subcategory inclusion,

  3. but the composite Δ <2ΔCat\Delta_{\lt 2} \hookrightarrow \Delta \hookrightarrow Cat is not dense in Cat\mathbf{Cat} (see Remark ).

Example

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

The category Set opSet^{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.

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:CDF:C\to D, not only inclusions I:CDI: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.

Last revised on July 21, 2021 at 05:22:05. See the history of this page for a list of all contributions to it.