# Pre-orders induced by subcategories

Let $\mathcal{C}$ denote a category. Then for every wide subcategory $\mathcal{D}$ of $\mathcal{C}$, there is an associated pre-order $\leq_\mathcal{D}$ on the objects of $\mathcal{C}$ defined by asserting that $X \leq_\mathcal{D} Y$ iff there exists $f \in \mathrm{Arr}(\mathcal{D})$ such that $f : X \rightarrow Y$.

Does this observation go anywhere?

I'm especially interested in the case where we're given a category $\mathcal{C}$, and we let $\mathcal{M}$ and $\mathcal{E}$ denote the wide subcategories whose arrows are the monomorphisms and epimorphisms of $\mathcal{C}$ respectively. Note that if $\mathcal{C}$ is $\mathrm{Set}$ and $\mathcal{M}$ and $\mathcal{E}$ are defined in this way, then for all objects $X$ and $Y$, it holds that $X \leq_\mathcal{M} Y$ iff $Y \leq_\mathcal{E} X$.

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Where would you want it to go? – Qiaochu Yuan Apr 23 '13 at 3:16
@QiaochuYuan, Well I'm trying to understand these new ideas in category theory using ideas I'm familiar with, like pre-orders. So for instance, if a terminal object is precisely a maximum element in an induced pre-order, this would clarify things for me. – goblin Apr 23 '13 at 3:19
I think you should think of statements about categories as specializing to statements about preorders rather than reducing to statements about preorders (so the notion of a terminal object specializes to, but is more general than, the notion of a maximum element). Some categories just don't look much like a preorder, e.g. in the category of abelian groups there's a morphism between any pair of objects. – Qiaochu Yuan Apr 23 '13 at 4:01
@QiaochuYuan, thank you for your advice. I still wait and see if I receive answers to this question though, because I like seeing the connections between things, and then may be more going on than just a specialization/generalization relationship. – goblin Apr 23 '13 at 5:02

If you like preorders and monomorphisms, you'll like subobjects. For any object $c$ of a category $C$, we can consider the category of all monomorphisms $b \to c$ into $c$, and this forms a preorder in which all of the morphisms themselves come from monomorphisms in $C$. When $c$ is an object in a familiar category this usually ends up being equivalent to the poset of subobjects of $c$ in the usual sense (e.g. when $c$ is a set or a group).