Mathematics.

advanced category theory

Enriched Categories

Category Theory90 minDifficulty9 out of 10

You should know: categories and morphisms, functors, natural transformations, monads

Overview

In an ordinary category, the hom-sets Hom(A, B) are just sets. In an enriched category, each hom-set is replaced by an object in some monoidal category V — called the enriching category. When V = Ab, you get additive categories (hom-sets are abelian groups). When V = [0,∞], you get generalised metric spaces. When V = Cat, you get strict 2-categories. Enriched category theory provides a unified framework for all these structures.

Intuition

Ordinary categories track 'what morphisms exist'. Enriched categories also track 'how morphisms relate to each other' — e.g., whether you can continuously deform one into another (topological enrichment), or add them (linear enrichment). The enrichment replaces a bare set of arrows with a structured object that remembers richer relationships.

Formal Definition

Definition

Let (V, ⊗, I) be a monoidal category. A V-enriched category C consists of: objects Ob(C); for each A,B ∈ Ob(C) a hom-object C(A,B) ∈ V; composition morphisms and identity morphisms in V satisfying associativity and unit coherence.

∘A,B,C:C(B,C)⊗C(A,B)→C(A,C)\circ_{A,B,C}: \mathcal{C}(B,C) \otimes \mathcal{C}(A,B) \to \mathcal{C}(A,C)

Composition in V

composition
jA:I→C(A,A)j_A: I \to \mathcal{C}(A,A)

Identity morphism in V

identity
∘∘(id⊗∘)=∘∘(∘⊗id)∘α\circ \circ (\mathrm{id} \otimes \circ) = \circ \circ (\circ \otimes \mathrm{id}) \circ \alpha

Enriched associativity coherence

associativity

Properties

V = Set recovers ordinary categories

ASet-enriched category is an ordinary locally small category.A \text{Set}\text{-enriched category is an ordinary locally small category.}

V = Ab gives pre-additive categories

AnAb-enriched category has abelian group structure on each hom-set with bilinear composition.An \text{Ab}\text{-enriched category has abelian group structure on each hom-set with bilinear composition.}

V = [0,∞] gives generalised metric spaces

A [0,∞]-enriched category is a generalised metric space: d(A,C)≤d(A,B)+d(B,C) (enriched associativity = triangle inequality).\text{A } [0,\infty]\text{-enriched category is a generalised metric space: } d(A,C) \leq d(A,B) + d(B,C) \text{ (enriched associativity = triangle inequality).}

V = Cat gives strict 2-categories

A Cat-enriched category is a strict 2-category with objects, 1-cells, and 2-cells with strict interchange laws.\text{A Cat-enriched category is a strict 2-category with objects, 1-cells, and 2-cells with strict interchange laws.}

Worked Examples

  1. Define a category with objects = points of X. Set hom-object C(x,y) = d(x,y) ∈ [0,∞].

  2. The monoidal structure on [0,∞] is addition (+) with unit 0. There is a morphism a → b in [0,∞] iff a ≥ b (reversed order).

  3. Composition C(y,z)⊗C(x,y) → C(x,z) means d(y,z)+d(x,y) ≥ d(x,z), i.e., the triangle inequality.

    d(y,z)+d(x,y)≥d(x,z)d(y,z) + d(x,y) \geq d(x,z)

  4. Identity j_x: 0 → C(x,x) = d(x,x) = 0 exists since d(x,x)=0.

Answer: A generalised metric space is exactly a [0,∞]-enriched category where morphisms 'measure distance' and composition is the triangle inequality.

Practice Problems

Difficulty 7/10

A preorder (P, ≤) can be viewed as an ordinary category. What monoidal category V makes this a V-enriched category, and what are the hom-objects?

Difficulty 8/10

What is a V-enriched functor F: C → D between V-enriched categories?

Difficulty 9/10

Explain why Cat-enriched categories are called 'strict 2-categories'.

Common Mistakes

Common Mistake

Thinking enriched categories are just ordinary categories in disguise.

Enriched categories carry strictly more structure. A Top-enriched category (where hom-sets are topological spaces) records the topology of paths between morphisms, which is invisible to the underlying ordinary category.

Common Mistake

Confusing 2-categories with Cat-enriched categories.

Cat-enriched categories are strict 2-categories (composition is strictly associative). Weak 2-categories (bicategories) allow associativity only up to coherent isomorphism and require a different formalism.

Quiz

In enriched category theory, what replaces the hom-set Hom(A,B) of an ordinary category?
A [0,∞]-enriched category corresponds to:
Kelly's Enriched Category Theory (1982) is important because:

Historical Background

Eilenberg and Kelly developed enriched category theory in the 1960s, motivated by the observation that many categories (modules, chain complexes, sheaves) naturally have hom-sets carrying extra algebraic structure. Lawvere's 1973 insight that metric spaces are enriched categories over ([0,∞], +) showed the framework's surprising breadth.

  1. 1965

    Eilenberg–Kelly introduce enriched categories

    Samuel Eilenberg, G. Max Kelly

  2. 1973

    Lawvere observes that metric spaces are [0,∞]-enriched categories

    William Lawvere

  3. 1982

    Kelly's monograph 'Basic Concepts of Enriched Category Theory' becomes standard reference

    G. Max Kelly

Summary

  • An enriched category replaces hom-sets with hom-objects in a monoidal category V (the enriching category).
  • V = Set gives ordinary categories; V = Ab gives pre-additive categories; V = [0,∞] gives generalised metric spaces; V = Cat gives strict 2-categories.
  • The composition morphisms in V must satisfy associativity and unit coherence diagrams.
  • Lawvere's insight that metric spaces are [0,∞]-enriched categories unified metric geometry with category theory.
  • Enriched functors require their action on hom-objects to be morphisms in V, compatible with composition and identities.

References

  1. WebsiteWikipedia — Enriched category