It is maybe remarkable that this phenomenon has more in common with the notion of concrete objects in a local topos than just the terminology.
I find myself in the middle trying to understand this (have been talking to Dave Carchedi about related things for half the day).

I’ll restrict your setup to goupoids and $\infty$-groupoids.

For instance let $Core(Set)$ be the maximal groupoid in $Set$ and say a functor $H \to Core(Set)$ makes the groupoid $H$ concrete if it is faithful. Equivalently that means that the functor is 0-truncated (full and faithful would be $(-1)$-truncated and an equivalence would be $(-2)$-truncated).

Next let $Core(Grpd)$ be the maximal 2-groupoid in $Grpd$. A concrete 2-groupoid in your sense above would be given by a 1-truncated functor $H \to Core(Grpd)$, I suppose.

Generally, if we’d continue in this way we would say that a concrete $n$-groupoid is given by an $\infty$-functor $H \to Core((n-1)Grpd)$ which is $(n-1)$-truncated.

In each case this would really be saying something only about the highest degree nontrivial morphisms.

Now the analogous setup: in a cohesive topos, an object $X$ is concrete if $X\to coDisc \Gamma X$ is a monomorphism. In other words: a cohesive set is concrete if the cohesive 0-functor $X \to coDisc \Gamma X$ is $(-1)$-truncated.

Given then a cohesive $(\infty,1)$-topos, let $X$ be an $n$-truncated obect in it. We might be tempted to say that it is concrete if the canonical morphism $X \to coDisc \Gamma X$ is $(n-1)$-truncated.

For instance in $Smooth\infty Grpd$ $= Sh_{(\infty,1)}(CartSp_{smooth})$ for $G$ a Lie group, the object $\mathbf{B}G$ is concrete in this sense, due to the fact that it has a diffeological space of morphisms (and diffeological spaces are precisely the concrete 0-truncated objects).

However, there seems to be a problem with this way of speaking: also $\mathbf{B}G_{conn}$, the coefficient object for $G$-bundles with connection, is concrete in this sense, because the conditon as stated above only pays attention to morphisms in degree 1 and does not notice the non-concrete cohesive structure in the objects (sheaves of differential forms in positive degree are as far from concrete as possible). In other words, $\mathbf{B}G_{conn} \to coDisc \Gamma \mathbf{B}G_{conn}$ is 0-truncated but far from $(-1)$-truncated.

This is a low degree indication of the analogue of the problem that you mention in the above entry: as we keep raising $n$, the condition that $X \to coDisc \Gamma X$ be $(n-1)$-truncated misses an increasing amount of information that intuitively we’d think should matter for concreteness, until it vanishes entirely for untruncated objects.

So what’s the right way to think about such conditons? For concrete objects in a cohesive $\infty$-topos, I am thinking that a good way to say that an $n$-truncated object $X$ is concrete is to say that

$X \to coDisc \Gamma X$ is $(n-1)$-truncated;

and $\tau_{\leq n-1} X \to coDisc \Gamma \tau_{\leq n-1} X$ is $(n-2)$-truncated;

and so on

and finally $\tau_{\leq 0} X \to coDisc \Gamma \tau_{\leq 0} X$ is $(-1)$-truncated.

In other words, that’s close to saying that all its categorical homotopy groups – its homotopy sheaves – are concrete sheaves.

With that definition for instance $\mathbf{B}G$ would be concrete, but $\mathbf{B}G_{conn}$ would not, as it should be.

Now, this tower of conditions going all the way from $n$ to 0 is too strong for the analog of the notion of “concrete category”. But maybe some variant of this would be good to consider. Maybe such a tower for an interval of values?

## Re: Concrete ∞-Categories

Not too dissimilar a conclusion to yours about infinitely categorified Klein geometry.