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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.

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Is the notion of a topological concrete category what you want? – Martin Brandenburg Jan 10 '14 at 13:40
@MartinBrandenburg, perhaps. It will take me a while to digest the definition. – goblin Jan 10 '14 at 14:23
Are you sure you're interested in concrete categories? Because quivers shouldn't be concrete in nlab sense. – Giorgio Mossa Jan 11 '14 at 15:57
@GiorgioMossa, why wouldn't they be concrete? The forgetful functor $\mathrm{Pos} \rightarrow \mathrm{Set}$ is certainly faithful. – goblin Jan 11 '14 at 15:59
@user18921 damned hurry in digitation, I meant quivers :P My bad. – Giorgio Mossa Jan 11 '14 at 16:01
up vote 1 down vote accepted

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.

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@user18921 is this what you were looking for? – Giorgio Mossa Jan 11 '14 at 16:39
It looks about right. Strange that it doesn't work for quivers though. I'm going to leave the answer unaccepted for a while just to see if someone can iron this final issue out. – goblin Jan 11 '14 at 18:40
By the way, I think better notation than $U^{-1}(f)$ would be $U^{-1}(f,Q)$ or some such, since $Q$ is very much part of the definition of this category. – goblin Jan 11 '14 at 18:45
Or perhaps $U^{-1}_{\rightarrow Q}(f).$ – goblin Jan 11 '14 at 18:49
@user18921 I don't know which version you've seen, but the construction in the last one works for quivers too. – Giorgio Mossa Jan 11 '14 at 19:44

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