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The category of binary words is monoidal

Statement

The category $W$ of The category of binary words is monoidal with tensor product $v \otimes w : = v □ w$ , unit object $e_{0}$ , and structural isomorphisms given by the unique arrows between binary words of the same length.

Facts & Assumptions

Given: The category $W$ of binary words.

[L1]

In $W$ , there is exactly one morphism $v \to w$ when $∣ v ∣ = ∣ w ∣$ , and none otherwise (The category of binary words).

[L2]

A monoidal category needs a bifunctor, a unit object, natural isomorphisms $α, λ, ρ$ , and the pentagon and triangle identities (Monoidal category).

Proof

technique · direct

1.1givenL1L2

If $f : v \to v^{'}$ and $g : w \to w^{'}$ exist in $W$ , then $∣ v ∣ = ∣ v^{'} ∣$ and $∣ w ∣ = ∣ w^{'} ∣$ , so $∣ v □ w ∣ = ∣ v^{'} □ w^{'} ∣$ . Define $f \otimes g$ to be the unique morphism $v □ w \to v^{'} □ w^{'}$ . Because all such morphisms are unique when they exist, this tensor preserves identities and composition automatically.

1.2L1construct

For words $u, v, w$ , the source and target of $((u □ v) □ w) ⟶ u □ (v □ w), e_{0} □ u ⟶ u, u □ e_{0} ⟶ u$ have the same lengths, so [L1] gives unique arrows $α_{u, v, w}$ , $λ_{u}$ , and $ρ_{u}$ . Their reverse arrows also exist uniquely, so these are isomorphisms.

2.1step 1.2L1

The two sides of the pentagon are morphisms in $W$ from one fourfold word to another of length $∣ u ∣ + ∣ v ∣ + ∣ w ∣ + ∣ z ∣$ . By [L1] there is only one such morphism, so the pentagon commutes. The same uniqueness argument gives the triangle identity.

3.1L2step 1.1step 2.1∎

Steps 1.1-2.1 supply exactly the data required in [L2], so $W$ is monoidal.

Depends on

Used by

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

S. Mac Lane, Categories for the Working Mathematician, Chapter VII.2 (standard reference, not scraped)