TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-31

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A lax monoidal functor carries monoid objects to monoid objects

Statement

Let $F : C \to D$ be a lax monoidal functor. If $(M, μ, η)$ is a monoid object in $C$ , then $F (M)$ is a monoid object in $D$ with multiplication

$F (M) ⊠ F (M) \overset{F_{2; M, M}}{\to} F (M \otimes M) \overset{F (μ)}{\to} F (M)$

and unit

$1^{'} \overset{F_{0}}{\to} F (1) \overset{F (η)}{\to} F (M) .$

Facts & Assumptions

Given: A lax monoidal functor $F : C \to D$ and a monoid object $(M, μ, η)$ in $C$ .

[L1]

A lax monoidal functor has structure maps $F_{2}$ and $F_{0}$ satisfying associativity and unit compatibility (Lax, strong, and strict monoidal functors).

[L2]

A monoid object is defined by multiplication and unit maps satisfying associativity and unit diagrams (Monoid objects and comonoid objects in a monoidal category).

Proof

technique · direct

1.1

Define $μ^{'} : = F (μ) \circ F_{2; M, M}$ and $η^{'} : = F (η) \circ F_{0}$ . These are the only typed maps with the displayed source and target.

givenL1L2construct

2.1

For associativity, paste the lax associativity square for $F$ with the image under $F$ of the monoid-object associativity square for $μ$ . Both composites from $(F (M) ⊠ F (M)) ⊠ F (M)$ to $F (M)$ equal $F (μ \circ (1_{M} \otimes μ) \circ α_{M, M, M})$ with the same inserted $F_{2}$ -maps. Hence $μ^{'}$ is associative.

step 1.1L1L2

2.2

The left and right unit diagrams for $η^{'}$ are obtained in the same way by pasting the lax unit squares with the image under $F$ of the two unit diagrams for $(M, μ, η)$ . Thus $η^{'}$ is a two-sided unit for $μ^{'}$ .

step 1.1L1L2

3.1

Therefore $(F (M), μ^{'}, η^{'})$ is a monoid object in $D$ .

step 2.1step 2.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Chapter 2.4 (standard reference, not scraped)