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?
How to express inclusion with arrows?
0
$\begingroup$
category-theory
-
1In the category of sets, yes, provided you label morphisms $1 \to X$ by their values. Otherwise false in general. – 2012-11-15
-
1Thank you. What if we replace the 1 with "all separator objects" of $\mathcal{C}$? – 2012-11-15
-
0Please formulate exactly what you mean: as it stands your claim does not generalise. – 2012-11-15
-
0What about (perhaps regular/split) *monomorphisms*? For what purpose you need it? – 2012-11-17
-
0Sorry, now this questions sounds a bit pointless and confusing to me. I am now happy with the axioms for sub-object. – 2012-11-17