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Category Theory for beginners (1)

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Tsakanikas Nickos
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Category Theory for beginners (1)

#1

Post by Tsakanikas Nickos » Sat Jan 16, 2016 11:10 pm

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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Grigorios Kostakos
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Re: Category Theory for beginners (1)

#2

Post by Grigorios Kostakos » Sat Jan 16, 2016 11:13 pm

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?
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Tsakanikas Nickos
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Re: Category Theory for beginners (1)

#3

Post by Tsakanikas Nickos » Sat Jan 16, 2016 11:15 pm

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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