#Abstract Algebra help

22 messages · Page 1 of 1 (latest)

turbid sky
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I’m using Algebra Chapter 0 written by Paolo Aluffi to study AA. In exercise 2 of the first chapter, there’s this question which asks the reader to formulate a definition of epimorphisms and then prove a result with them, analogous to how a function which is a monomorphism is also injective.

My question is: what in tarnation is an epimorphism and how does a function that is an epimorphism can be shown to be surjective

heavy raptorBOT
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hidden sphinx
turbid sky
hidden sphinx
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a' and a" are functions from some object Z to A, and your function f A->B holds f◦a' = f◦a".
Intuitively, if f is injective, it gives different outputs for different inputs. So if we throw results from a' and a" into it, and it's exactly the same, it must mean a' and a" are exactly the same.

Now surjectivity, in a similar way. Surjectivity must mean the entire codomain is generated by f. So maybe instead of using a' and a" before applying f, you apply a' and a" after applying f

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This basically gives the answer already, but I think to formulate it yourself and really prove it implies surjectivity you still need to understand and think about it

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Look at it this way, you're probably used to defining surjectivity as

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$\forall y\in B,; \exists x\in A : f(x)=y$

obtuse crownBOT
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Jehare

1049706877480616037.png
hidden sphinx
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But now, instead of using ∀, you just use a function, a'. And because you're using an entire function instead of a ∀y in B, you can just compare the functions, instead of checking every element in codomain individually

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Just like you did with monomorphisms. To check injectivity you usually go like \
$\forall a,b\in A,; f(a)=f(b)\implies a=b$ \
But now you don't have to use $\forall$ but you use functions $\alpha'$ and $\alpha''$ and if the functions are equal, you basically did the same as checking if the implication holds for every pair of elements, thus monomorphism and injectivity

obtuse crownBOT
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Jehare

1049706877480616037.png
magic mossBOT
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@turbid sky

:HelpIcon:| Help Reminder

Hello inverse_cupid, this is a friendly reminder that your help request has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command. This thread will be automatically closed in 3 days if it remains inactive.

magic mossBOT
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@hidden sphinx The user still needs help with this help request.

turbid sky
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Ƒυcκ sorry

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+close

magic mossBOT
# turbid sky +close
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magic mossBOT
# magic moss

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cerulean crest
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@turbid sky close it too

turbid sky
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+close

magic mossBOT
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