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I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ...

I need some further help in order to fully understand some aspects of the definition of the category Rel ... ...

The definition of the category Rel ... reads as follows:View attachment 8387

I am having trouble fully understanding the definition of the identity arrow ...I will use an example to illustrate my problems ...Let the category \(\displaystyle \text{Rel}_1\) consist of two sets \(\displaystyle A, B\) where

\(\displaystyle A = \{ 1, 2, 3 \}\)

\(\displaystyle B = \{ 3, 4 \} \)

so ... two arrows, for example, may be \(\displaystyle f_1 : A \to B\) where \(\displaystyle f_1 = \{ (1, 3), (1, 4), (3, 4) \}\)

and \(\displaystyle f_2 : A \to B\) where \(\displaystyle f_2 = \{ (1, 3) \}\)

Now .. consider \(\displaystyle f_3 : A \to A\) where \(\displaystyle f_3 = \{ (1, 1) \}\)

and \(\displaystyle f_4 : A \to A\) where \(\displaystyle f_4 = \{ (2, 2) \}\)

and \(\displaystyle f_5 : A \to A\) where \(\displaystyle f_5 = \{ (3, 3) \}\) ... BUT ...... according to Awodey's definition these arrows are all equal to the identity arrow of A ...... the identity arrow is meant to be unique ... ?Does this mean \(\displaystyle f_3 = f_4 = f_5\) ... ? ... but why and how are they equal ...

Can someone please clarify the above ... ?

Peter

I need some further help in order to fully understand some aspects of the definition of the category Rel ... ...

The definition of the category Rel ... reads as follows:View attachment 8387

I am having trouble fully understanding the definition of the identity arrow ...I will use an example to illustrate my problems ...Let the category \(\displaystyle \text{Rel}_1\) consist of two sets \(\displaystyle A, B\) where

\(\displaystyle A = \{ 1, 2, 3 \}\)

\(\displaystyle B = \{ 3, 4 \} \)

so ... two arrows, for example, may be \(\displaystyle f_1 : A \to B\) where \(\displaystyle f_1 = \{ (1, 3), (1, 4), (3, 4) \}\)

and \(\displaystyle f_2 : A \to B\) where \(\displaystyle f_2 = \{ (1, 3) \}\)

Now .. consider \(\displaystyle f_3 : A \to A\) where \(\displaystyle f_3 = \{ (1, 1) \}\)

and \(\displaystyle f_4 : A \to A\) where \(\displaystyle f_4 = \{ (2, 2) \}\)

and \(\displaystyle f_5 : A \to A\) where \(\displaystyle f_5 = \{ (3, 3) \}\) ... BUT ...... according to Awodey's definition these arrows are all equal to the identity arrow of A ...... the identity arrow is meant to be unique ... ?Does this mean \(\displaystyle f_3 = f_4 = f_5\) ... ? ... but why and how are they equal ...

Can someone please clarify the above ... ?

Peter

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