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Stream: learning: questions

Topic: Question about a proof from 'Non-Canonical Isomorphisms'

Nayan Rajesh (May 28 2025 at 14:17):

In Lack's Non-Canonical Isomorphisms, there is a claim that any category with finite products and finite coproducts and any natural family of isomorphisms, $Y+Z \cong Y\times Z$ becomes semi-additive. This is part of the proof:

image.png

The isomorphism $\psi_{Y,0}$ goes from $Y+0$ to $Y\times 0$, so I presume that the square on the left is shorthand for

image.png

This seems to assume that the square on the top commutes which I'm not able to derive from the rest of the assumptions. I have a similar problem with the square on the right in the first screenshot.

I'd appreciate any help with this: either I've decomposed the diagram incorrectly, or I'm missing something that makes the square on top of the second diagram commute. Thanks!

John Baez (May 28 2025 at 14:31):

If we know that $0$ is both the initial and terminal object then I think we can prove the top square commutes. Are we allowed to know that at this point in the argument?

Ralph Sarkis (May 28 2025 at 14:36):

Put $Y = 0$ and $Z = 1$ in the natural isomorphism above:
$1 \cong 0 + 1 \cong 0 \times 1 \cong 0$
Oh, that is already in the first screenshot. (Pointed means initial and terminal object coincide.)

Nayan Rajesh (May 28 2025 at 14:40):

Thanks @Ralph Sarkis! In this case, I don't think we have $0\times 1\cong 0$, but just the projection $0\times 1\xrightarrow{\pi} 0$, which gives a map $1\to 0$, which forces $1\cong 0$

Ralph Sarkis (May 28 2025 at 14:45):

The terminal object is always the unit of the product: $X \times 1 \cong X$ for any $X$. Dually, the initial object is always the unit of the coproduct: $X + 0 \cong A$ for any $X$. The first sentence of the proof in the screenshot seems to make this more complicated than it is.

Nayan Rajesh (May 28 2025 at 14:51):

You're right, sorry. For some reason I skipped the $1$ in $0\times 1$ and thought that was using $0\times X \cong 0$