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[curry-howard] Re: unicity of arrows #1

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glmxndr opened this Issue Feb 7, 2019 · 3 comments

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glmxndr commented Feb 7, 2019

Hi, nice blog post.

Two nitpicks though.

Similarly to the step before, by definition of product, since we know (A × B) × C is a product of A × B and C, and since we have the arrows l : T → A × B and q' ∘ q ∘ t : T → C, then there must exist an unique arrow t' : (A × B) × C.

Nitpick 1: t' should probably be a T -> (AxB)xC since it is called an arrow.

Nitpick 2: t' is not necessarily unique: there are as many arrows from T to (AxB)xC as there are members of this object. There is only one that makes some diagram (involving t) commute, though. I get what you meant but the wording is maybe a bit misleading to category theory newcomers.

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BartoszMilewski commented Feb 7, 2019

Nitpick 1: You are correct.
Nitpick 2: There is a unique arrow that factorizes the two arrows l and the other, q' ∘ q ∘ t, which should probably be given a name too.

You're right that there are as many arrows from T to (AxB)xC as there are elements in it, but that might be zero. It would be the case if, for instance, C were empty. But then Ax(BxC) would be empty too, and we wouldn't have the starting arrow for the proof.

@vladciobanu

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vladciobanu commented Feb 7, 2019

Thank you for your comments! Indeed, 1 is absolutely correct.

Would you agree with the statement that t' is the unique arrow T -> (A x B) x C that can be derived from the initial arrow t : T -> A x (B x C)?

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BartoszMilewski commented Feb 7, 2019

I think you should modify the definition of the product. There exist a unique arrow m such that p . m = p' and q . m = q'.

vladciobanu added a commit that referenced this issue Feb 7, 2019

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