I'm learning about the category of objects over an object $X$. The definition is simple enough, for a category $\mathcal{C}$ and an object $X$ in it define the category $\mathcal{C}/X$ whose objects consist of the morphisms $Y\rightarrow X$ in $\mathcal{C}$ with the obvious morphisms between objects.
The notation seems to suggest that this should be a sort of "quotient". I wanted to check my intuition against this:
- If $I$ is an ideal of $R$ then the ideals of $R/I$ are in bijection with the ideals of $R$ containing $I$.
So if we only care about ideals the quotient $R/I$ consists of all ideals over $I$, so I imagine one could define an arrow $J\rightarrow I$ if and only if $I\subseteq J$. This however seems a bit artificial because I am taking the arrows to be what I need.
Hopefully my question won't be too vague - Here it goes: Is there a nice (general) way to interpret the category of objects over $X$ as a quotient in some category? I am particularly interested on how it would work with quotient topological spaces!
No. The similarity to quotient notation is essentially entirely coincidental here. The motivation is that you are just considering objects "over" $X$ and so you write the category by putting $\mathcal{C}$ "over" $X$. The use of the $/$ symbol is kind of a pun here, conflating this notion of "over" with the entirely unrelated notion of "over" as in division.