TITLE: Is there a category-theoretic definition of induced subquivers (subgraphs, subposets etc.)?
QUESTION [0 upvotes]: Let $Q=(V,E)$ denote a quiver and suppose $V' \subseteq V.$ Then (definition:) $V'$ gives rise to an "induced subquiver" of $Q$, call it $Q'$ simply by defining that $Q' = (V',E')$ where $E'$ denotes the maximum subset of $E$ such that every element of $E'$ has both its source and its target in $V'$.
Indeed, the same basic idea works for induced subgraphs and induced subposets, too.
Question. Suppose $Q$ is an object of a category $\mathcal{C}$ and $U : \mathcal{C} \rightarrow \mathrm{Set}$ is a functor, and furthermore assume that $f : V' \rightarrow U(Q)$ denotes an injective function. Is there a way to define "the subobject of $Q$ induced by $f$ with respect to $U$"? I want it to give the correct answer for the case where $\mathcal{C}$ is the category of quivers (graphs, posets etc.) and $U$ is the forgetful functor.

REPLY [1 votes]: Let $Q \in \mathcal C$ and $f \colon V' \to U(Q)$ an injective function (i.e. a monomorphism in $\mathbf {Set}$.
Then we can consider the following subcategory $U^{-1}(f)$ in which:

objects all those monomorphisms $p \colon X \to Q$ in $\mathcal C$ such that $U(p)=f$;
for every pair $p \colon X \to Q$ and $p \colon Y \to Q$ in $U^{-1}(f)$ a morphism $g \colon p \to q$ is just a morphism $g \colon X \to Y$ in $\mathcal C$ such that $q \circ g=p$ and $U(g)=1_V$;
composition and identities are the obvious ones.

In this category every morphism $g \in U^{-1}(f)(p,q)$ is a monomorphism, since $q \circ g=p$ in $\mathcal C$ and $p$ is a monomorphism of $\mathcal C$.
The terminal object in $U^{-1}(f)$ should be a good candidate to be the subobject generated by $f$.
This works fine you categories, meaning that in these cases the terminal object is (up to isomorphism) the subobject generated by $f$ (that's because in the categories you've considered subobjects are monomorphisms, up to isomorphism, they admit an epi-mono factorization).
A further generalization could be made by considering a sub-category of inclusions in $\mathcal C$ and requiring that objects and morphisms of $U^{-1}(f)$ are inclusions, that is objecs and morphisms of the subcategory of $\mathcal C$. Of course the construction above become a particular case when you consider a subcategory formed by the monomorphisms of $\mathcal C$. 
I suppose that the subcategory of inclusion should verify some properties, but I'm not so sure about which ones.
{"set_name": "stack_exchange", "score": 0, "question_id": 633715}