A $1$-groupoid can be thought of as a Kan complex in the usual way. Is there a simplicial set $\Omega$ such that the contravariant functors $\text{Sub}_{\mathbf{Gpd}}(-)$ and $\text{Hom}_{\mathbf{sSet}}(-,\Omega)$ from $\mathbf{Gpd}\to\mathbf{Set}$ are naturally isomorphic?
I know that $\mathbf{Gpd}$ doesn't have a subobject classifier for the same reason $\mathbf{Grp}$ doesn't; not every subgroup is a kernel. It seems to me that the relaxed composition in simplicial sets prevents this argument from going through.
Monics in $\mathbf{Gpd}$ are morphisms that are injective on objects and arrows, so I would imagine that $\Omega$, should it exist, would have two objects $F_0$ and $T_0$ (representing exclusion or inclusion of objects), with hom-sets into and out of $F_0$ being singletons, and two $1$-cells $F_1$ and $T_1$ from $T_0$ to itself (representing exclusion or inclusion of arrows). There would have to be $2$-cells representing compositions $T_1\circ T_1\Rightarrow T_1$, $F_1\circ T_1\Rightarrow F_1$, $T_1\circ F_1\Rightarrow F_1$, $F_1\circ F_1\Rightarrow F_1$, and $F_1\circ F_1\Rightarrow T_1$, each representing a possible assignment of inclusion or exclusion in the subgroupoid to a triple of arrows satisfying $g_1\circ g_2=g_3$. I'm not sure how to construct the rest of the cells, or even if it's possible.
More importantly, is the above reasoning nonsense? I'm new to the area.