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 Post subject: Category Theory for beginners (1)Posted: Sat Jan 16, 2016 11:10 pm
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 313
Prove the following:

1. If $A$ is an initial or a final object in a category $\mathcal{C}$, then $A$ is unique up to isomorphism.
2. An object $0$ of a category $\mathcal{C}$ is a zero object of $\mathcal{C}$ if and only if it is an initial and a final object of $\mathcal{C}$.

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 Post subject: Re: Category Theory for beginners (1)Posted: Sat Jan 16, 2016 11:13 pm
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Joined: Mon Nov 09, 2015 1:36 am
Posts: 443
Location: Ioannina, Greece
Hi Nikos,

1. Suppose that ${\rm{I}}_1$,${\rm{I}}_2$ are initial object in a category $\mathcal{C}$. Because ${\rm{I}}_1$ is initial, exists unique morfism ${\rm{I}}_1\xrightarrow{f_1}{\rm{I}}_2$ and similarly, because ${\rm{I}}_2$ is initial, exists unique morfism ${\rm{I}}_2\xrightarrow{f_2}{\rm{I}}_1$.
But then the morphisms ${\rm{I}}_1\xrightarrow{f_2\circ f_1}{\rm{I}}_1$ , ${\rm{I}}_2\xrightarrow{f_1\circ f_2}{\rm{I}}_2$ must be unique also. So $f_2\circ f_1=id_{{\rm{I}}_1}$, $f_1\circ f_2=id_{{\rm{I}}_2}$ and ${\rm{I}}_1$, ${\rm{I}}_2$ are equal up to isomorphism.

2. For the second, I assume that you are asking to prove the uniqueness of the zero object (when exists one). Is this correct? What do you define as zero object?

_________________
Grigorios Kostakos

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 Post subject: Re: Category Theory for beginners (1)Posted: Sat Jan 16, 2016 11:15 pm
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Joined: Tue Nov 10, 2015 8:25 pm
Posts: 313
The definition I read was the following: "An object 0 in a category $\mathcal{C}$ is a zero object of $\mathcal{C}$ if_f both the sets Mor$(0, A)$ and Mor$(A, 0)$ contain a single morphism for each object of $A$ of $\mathcal{C}$. So, (2) is self-evident, and due to it another definition of THE zero object is that "it is simultaneously an initial and a final object in a category $\mathcal{C}$."

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