Two Kinds of One
One word, two jobs
An Archive folio fixes the category but places two combination instruments beside it. The sets and functions stay fixed; only the chosen operation changes.
The two definitions
Recall that a set is terminal when every set has exactly one function . A bijection is an isomorphism in .
Now fix a way to combine sets. Both combinations below have their standard associative rebracketing and coherence; you may use those facts without proof.
A unit for is a set with bijections
for every set , using one rule that respects every function.
Precisely, if , then naturality means
For these standard operations, the natural left and right unit maps are the usual monoidal-unit data. On the combined sets, changes only the -entry.
Two combining rules
Pairs
is the set of ordered pairs .
Tagged choice
The tags keep the two sides separate, even if . Here means tagged union, not arithmetic addition.
1. See both rules
Use separate paper for your lists.
Let .
- List the elements of and .
- List the elements of and .
- In each list, describe the evident bijection back to .
What a complete unit proof must show
For each combining rule, give formulas for both unit maps and their inverses. Then check that changing the name of an -element before or after removing the unit has the same result.
Naturality diagrams
Pairs
Tagged choice
2–4. Prove the unit laws
Use separate paper for your formulas and verification.
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For every set , prove that is a unit for . Give formulas for , , and both inverses.
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For every set , prove that is a unit for . Give formulas for , , and both inverses.
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Let . Verify the left naturality equation once for the product unit and once for the tagged-union unit. State why the right equations work the same way.
5–7. Separate the two meanings
Complete the table here. Use separate paper for the proofs.
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Prove both classifications.
- If is any unit for , prove that must have exactly one element. Use the test set .
- If is any unit for , prove that must be empty. Use the test set .
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Complete the comparison table.
Representative set Terminal in ? Unit for ? Unit for ? -
The arrows of stayed the same throughout the problem. Only the combining rule changed. In two precise sentences, explain why terminal object and monoidal unit are different meanings of one.