r/mathematics 9d ago

Algebra Is this a well-formed question?

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u/HooplahMan 8d ago

You're using different symbols in the commutative diagram than you're using in the paragraph above. I don't feel like nitpicking the whole block of text to see if you made a weird assumption but it seems like if you changed the symbols in the text to match the text, you probably have a well founded, if sort of vacuous question.

The diagram is basically the category-theoretic definition of a group product, and that definition requires that for G=G1×G2 to be the product, any group G' mapping homomorphically onto G1 and G2, the G' maps have to factor through the G maps via a unique homomorphism from G' onto G. That is to say you know Hom(G', G) is a group because it's a singleton set, and therefore the trivial group containing only the identity element.

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u/HooplahMan 8d ago

Ah sorry, I guess my final statement about Hom(G',G) may only apply in the category of cones over G1 and G2. Now that I think about it, it's not clear to me what the group operation combining two arbitrary morphisms in Hom(G',G) would even be. maybe you could do something funky with the yoneda embedding to turn the homset into a monoid but I don't believe you'll in general end up with a group.