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James Deikun (Jan 30 2023 at 02:14):What happens to a coproduct of morphisms when you push it out along another morphism? It clearly doesn't always remain a coproduct of morphisms in any meaningful sense (e.g. push out id+id along !) but is there anything that remains or good special cases?
Spencer Breiner (Jan 30 2023 at 12:31):Pulling back a product along the diagonal gives an equalizer, so dually this should give a quotient
Spencer Breiner (Jan 30 2023 at 12:32):... for this special case
James Deikun (Jan 30 2023 at 13:09):I guess in a way it's the general case, since any map out of a coproduct factors as a coproduct followed by a codiagonal. So first you push out along another coproduct, which I think should give you the coproduct of the two individual pushouts by general nonsense about adjoints, and then you get the coequalizer of the two. Is this right? (If so it's an answer, but maybe not the ideal answer -- I'd like to have one that gives some idea of how much meaningful gluing is going on and how much is just the images missing each other, and how you get a mix of quotientyness and coproductyness.)