Two Kinds of One

At a glance

Every product unit has exactly one element. The only tagged-union unit is the empty set. Terminality depends on the arrows in the category, while a monoidal unit depends on the chosen combining operation.

Key idea: test a law that must work for every set

A unit law must work for every set 𝑋. A carefully chosen test set can strip away 𝑋 and expose the proposed unit itself. A singleton does this for Cartesian product; the empty set does it for tagged union.

1. See both rules

For 𝑋 ={𝑎,𝑏},

Deleting the ⋆-entry gives the evident bijection back to 𝑋.

For tagged union,

Deleting the tag gives the evident bijection back to 𝑋.

2. The singleton is the product unit

For every set 𝑋, define

Composing either displayed map with its displayed inverse returns the original element in both directions. Therefore both unit maps are bijections.

3. The empty set is the tagged-union unit

Every element of ∅ +𝑋 has the form (𝑅,𝑥), and every element of 𝑋 +∅ has the form (𝐿,𝑥). Define

Again, each map and its stated inverse undo one another in both directions, so the maps are bijections.

4. Verify naturality

Let ℎ :𝑋 →𝑌.

For the product left unit, start with ( ⋆,𝑥). Removing the unit and then applying ℎ gives

Applying ℎ to the 𝑋-entry first and then removing the unit gives

Thus both routes agree.

For the tagged-union left unit, start with (𝑅,𝑥). The two routes give

and

The right-unit calculations use (𝑥, ⋆) for products and (𝐿,𝑥) for tagged union. In each case, applying ℎ and deleting the fixed unit entry commute for the same reason.

5. Prove both classifications

5a. Every product unit is a singleton

Suppose 𝐼 is a unit for ×. The unit law must hold when 𝑋 ={ ⋆}, so there is a bijection

The map 𝑢 ↦(𝑢, ⋆) is itself a bijection 𝐼 ≅𝐼 ×{ ⋆}. Therefore 𝐼 is in bijection with a singleton and must have exactly one element.

Conversely, every singleton works by the formulas in Part 2. This proves the classification up to bijection.

5b. Every tagged-union unit is empty

Suppose 𝐽 is a unit for +. The unit law must hold when 𝑋 =∅, so

If 𝐽 contained an element 𝑗, then (𝐿,𝑗) would belong to 𝐽 +∅. A nonempty set cannot be in bijection with the empty set. Therefore 𝐽 =∅.

Conversely, the empty set works by the formulas in Part 3. This proves the classification.

6. Complete the table

A singleton is terminal because every set has exactly one function to it. The empty set is not terminal because there is no function from a singleton to the empty set.

A singleton is the product unit, while the empty set is not: for a nonempty 𝑋, the product ∅ ×𝑋 is empty and cannot be in bijection with 𝑋.

The empty set is the tagged-union unit, while a singleton is not: for 𝑋 =∅, the set { ⋆} +∅ still has one element.

These facts give the table shown at the top.

7. Separate the meanings

A model response is:

Terminality is determined only by functions into an object, so it does not change when the combining rule changes. A monoidal unit is determined by how objects are combined, so × selects a singleton while tagged union selects the empty set.

Check against the definitions

The proposed unit maps were given for every set 𝑋, explicit inverses proved that they are bijections, and the naturality calculations showed that they respect every function. The singleton and empty test sets then ruled out every other possible unit object. The terminal-object claims were checked independently from both combining operations.

Another facet

A neutral object is not neutral by itself. It is neutral for a specified way of combining objects. Changing that operation can change the unit even when the objects and arrows of the category do not change at all.

Teaching notes

• The symbol + denotes tagged union in this worksheet, not arithmetic addition or ordinary untagged union.
• “Unit” means unchanged up to a specified natural bijection, not necessarily literal set equality.
• The singleton’s two roles for × coincide because Cartesian product is the categorical product in 𝐒𝐞𝐭; that coincidence is special, not part of the definition of monoidal unit.