Math 210 A : Algebra , Homework 7
نویسنده
چکیده
We claim that monomorphisms in Sets and Groups are the usual injective maps and homomorphisms. First, suppose that f : A → B is a set injection and that fg = gh. Then for all c ∈ C, we have f(g(c)) = f(h(c)) implies g(c) = h(c) by injectivitiy. Hence g = h. Conversely, suppose f : A → B is not an injection. Then let f(a) = f(a′) for some a 6= a′ ∈ A. Then let g(c) = a and h(c) = a′ for all c ∈ C. Therefore f(g(c)) = f(a) = f(a′) = f(h(c)) for all c ∈ C, hence fg = fh. But since a 6= a′, g 6= h. Therefore f cannot be a monomorphism. Therefore monomorphisms in Sets are exactly injective maps.
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