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Thread: Subobjects in a category

  1. #1
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    Subobjects in a category

    In a category $\displaystyle \mathcal{C}$ a subobject of an object $\displaystyle A$ is an object $\displaystyle B$ such that there exists a monic $\displaystyle \phi : A \longrightarrow B$. Simple enough.

    But I need to show that if $\displaystyle A$ is a subobject of $\displaystyle B$ and $\displaystyle B$ is a subobject of $\displaystyle A$, then $\displaystyle A = B$.

    It looks obvious but I think I am missing something.
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  2. #2
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    Ah, If $\displaystyle \theta : B \longrightarrow A$ is the other monic then their composition is monic, but their compositions are identity morphisms for A and B; $\displaystyle \theta \circ \phi = 1_A$ and $\displaystyle \phi \circ \theta = 1_B$. Hence they are isomorphic! Or is there a strict equality? help
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