TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Ab-enriched categories are exactly preadditive categories

Statement

A category enriched in $A b$ with tensor product $\otimes_{Z}$ is exactly a preadditive category.

Facts & Assumptions

Given: A category or enriched category with object set $Ob$ .

[L1]

An $A b$ -enriched category has abelian-group hom-objects, composition morphisms $A (B, C) \otimes_{Z} A (A, B) \to A (A, C)$ , and identity morphisms $Z \to A (A, A)$ (Enriched category over a monoidal base).

[L2]

A preadditive category is a category whose hom-sets are abelian groups and whose composition is bilinear (Preadditive category).

[L3]

$A b$ is monoidal under $\otimes_{Z}$ , and morphisms out of $X \otimes_{Z} Y$ are exactly bilinear maps out of $X \times Y$ (Abelian groups are monoidal under the tensor product).

[L4]

In a preadditive category, the hom-bifunctor already takes values in abelian groups (The hom-bifunctor of a preadditive category takes values in abelian groups).

Proof

technique · direct

1.1L1L2L3given

Assume $A$ is $A b$ -enriched. By [L1], each hom-object $A (A, B)$ is an abelian group. The composition morphism is a morphism in $A b$ out of the tensor product, so by [L3] it is exactly a bilinear map $A (B, C) \times A (A, B) \to A (A, C)$ . The identity map $Z \to A (A, A)$ selects the identity element in the endomorphism group. Thus the underlying ordinary category has abelian-group homs and bilinear composition, so [L2] makes it preadditive.

1.2L2L3L4algebra

Conversely, let $C$ be preadditive. By [L4], each hom-set is an abelian group and the two-variable hom-assignment is additive in each variable. So for every triple $A, B, C$ , the bilinear composition map $C (B, C) \times C (A, B) \to C (A, C)$ transposes uniquely, by [L3], to a group homomorphism $C (B, C) \otimes_{Z} C (A, B) \to C (A, C)$ . Sending $1 \in Z$ to the identity morphism of $A$ gives the required unit map $Z \to C (A, A)$ . The ordinary associativity and unit laws are exactly the enriched ones after this transposition.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 show that the two notions encode the same data.

Depends on

Used by

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.2 (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 3.3 (standard reference, not scraped)