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

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A lax monoidal functor induces change of base on enriched categories

Statement

Let $F : V \to W$ be a lax monoidal functor. Every $V$ -category $A$ determines a $W$ -category $F_{*} A$ with the same objects and hom-objects $(F_{*} A) (A, B) : = F (A (A, B))$ .

Facts & Assumptions

Given: A lax monoidal functor $F : V \to W$ and a $V$ -category $A$ .

[L1]

A lax monoidal functor provides comparison morphisms $F (X) \otimes F (Y) \to F (X \otimes Y)$ and $1_{W} \to F (1_{V})$ compatible with associativity and the unitors (Lax, strong, and strict monoidal functors).

[L2]

A $V$ -category is specified by hom-objects, composition maps, and identity maps satisfying the usual enriched diagrams (Enriched category over a monoidal base).

Proof

technique · direct

1.1L2given

Keep the same object set as $A$ and define the new hom-object from $A$ to $B$ to be $F (A (A, B))$ .

1.2L1L2construct

Use the laxity morphism of [L1] to send $F (A (B, C)) \otimes F (A (A, B))$ into $F (A (B, C) \otimes A (A, B))$ , then follow with $F (M_{A, B, C})$ to obtain the new enriched composition map. Use the unit morphism $1_{W} \to F (1_{V})$ followed by $F (j_{A})$ for the new enriched identity.

2.1L1L2step 1.2∎

The coherence axioms of [L1] ensure that applying $F$ to the associativity and unit diagrams of [L2] yields the corresponding diagrams in $W$ . Thus the data of steps 1.1 and 1.2 define a $W$ -category.

Depends on

Used by

Change of base extends to enriched functors and natural transformations as a 2-functor Theorem

Dependency tree · two levels

6 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

Emily Riehl, Categorical Homotopy Theory, Lemma 3.4.3 (standard reference, not scraped)
Geoffrey Cruttwell, Normed Spaces and the Change of Base for Enriched Categories, Proposition 4.2.1 (standard reference, not scraped)