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Does$ \forall x \left(1 \stackrel{x}\longrightarrow X\right) \Rightarrow 1 \stackrel{x} \longrightarrow A $ means $A \subset B $? Is there any better way to express this with arrows?

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    Sorry, now this questions sounds a bit pointless and confusing to me. I am now happy with the axioms for sub-object.2012-11-17

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There is a category whose objects are sets and morphism are inclusions.

So if $A$, $B$ are sets then $A \longrightarrow B$ means we have an inclusion $A \subset B$. We have identity morphism since $A \subset A$ and you can compose $A \subset B$ with $B \subset C$ by transitivity to get $A \subset C$.

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    @Hooman, nice point about it being a subcat, I hadn't thought of that. I don't think we can get it as a slice or any other purely categorical way - since category theory generally only classifies things up to isomorphism, we'd have to actually look at the elements inside a set similar to how you were doing.2012-11-15