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

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A poset with finite meets is a strict monoidal category

Statement

Let $P$ be a poset with a top element $⊤$ and binary meets. Then the category associated to $P$ is a strict monoidal category with tensor product $x \otimes y : = x \land y$ and unit object $⊤$ .

Facts & Assumptions

Given: A poset $P$ with top element $⊤$ and binary meet operation $\land$ .

[L1]

A lattice supplies binary meets, written $\land$ (Lattices, distributive lattices, and order ideals).

[L2]

A preorder, and hence a poset, determines a category with at most one morphism between two objects (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[L3]

A strict monoidal category has literal associativity and unit equalities and identity constraints (Strict monoidal category).

Proof

technique · direct

1.1

By [L2], $P$ becomes a category whose objects are the elements of $P$ and whose morphisms are the order relations.

givenL2

2.1

Define $x \otimes y : = x \land y$ on objects. Because meet is monotone in each variable, the unique arrows $x \to x^{'}$ and $y \to y^{'}$ induce the unique arrow $x \land y \to x^{'} \land y^{'}$ , so $\otimes$ is a bifunctor.

step 1.1L1L2

3.1

The universal property of meet gives $(x \land y) \land z = x \land (y \land z)$ and $⊤ \land x = x = x \land ⊤$ as equalities of objects in the poset. Since each hom-collection has at most one arrow, the associator and unitors are automatically identity morphisms.

step 2.1L1L2L3

4.1

Therefore the associated category is strict monoidal.

step 3.1L3∎

Depends on

Used by

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Dependency tree · two levels

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Example 2.3.12 (standard reference, not scraped)