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Jonathan Beardsley (Jul 20 2025 at 19:10):If a category has products and coproducts then there's a canonical map X∐Y→X×Y. Does anyone know of conditions on this map so that it's always a monomorphism? I was thinking that the category being "adhesive" might be relevant...but I'm not sure. The example I'm wondering about is n-fold (pointed) simplicial sets, which maybe makes it easier.
Amar Hadzihasanovic (Jul 20 2025 at 19:24):You need some stronger assumption to have such a canonical map; in Set, or any category with a strict initial object , there isn't (if you take then the domain is always isomorphic to and the codomain to ).
Amar Hadzihasanovic (Jul 20 2025 at 19:24):Perhaps you are also assuming that there is a zero object? Then I agree.
Amar Hadzihasanovic (Jul 20 2025 at 19:25):(The fact you are thinking of pointed simplicial sets suggests so.)
Jonathan Beardsley (Jul 20 2025 at 19:48):Ah, yes! I am definitely thinking of pointed things. Sorry, yes, I was just implicitly pointing everything....
Jonathan Beardsley (Jul 20 2025 at 20:11):@Amar Hadzihasanovic what do you mean you agree? you agree that such a map exists, or you agree that adhesive implies that this is a monomorphism?
Amar Hadzihasanovic (Jul 20 2025 at 22:08):Sorry, just that the map exists. I don't immediately see how adhesiveness helps as it relates to properties of pushout/pullback squares and not the universal maps that they induce, do you have some argument in mind?
Jonathan Beardsley (Jul 20 2025 at 22:35):I did. But it's nonsense, lol. So I'm back to zero again.
Mike Shulman (Jul 21 2025 at 01:07):It's easy to prove that it's a monomorphism in pointed sets, and the same proof should work for pointed objects in any topos.
Jonathan Beardsley (Jul 21 2025 at 03:44):ah thanks @Mike Shulman I'll give it a try. :)
Mike Shulman (Jul 21 2025 at 05:48):You could also deduce it for all Grothendieck toposes from the fact for pointed sets by taking presheaves and then sheaves (and for n-simplicial sets you wouldn't even need the last step).