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Is there a simplicial set classifying subobjects of groupoids?
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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 $textSub_mathbfGpd(-)$ and $textHom_mathbfsSet(-,Omega)$ from $mathbfGpdtomathbfSet$ are naturally isomorphic?
I know that $mathbfGpd$ doesn't have a subobject classifier for the same reason $mathbfGrp$ 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 $mathbfGpd$ 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 $T_1circ T_1Rightarrow T_1$, $F_1circ T_1Rightarrow F_1$, $T_1circ F_1Rightarrow F_1$, $F_1circ F_1Rightarrow F_1$, and $F_1circ F_1Rightarrow T_1$, each representing a possible assignment of inclusion or exclusion in the subgroupoid to a triple of arrows satisfying $g_1circ g_2=g_3$. 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.
category-theory simplicial-stuff higher-category-theory groupoids
asked Aug 1 at 21:25
user581091
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up vote
1
down vote
favorite
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 $textSub_mathbfGpd(-)$ and $textHom_mathbfsSet(-,Omega)$ from $mathbfGpdtomathbfSet$ are naturally isomorphic?
I know that $mathbfGpd$ doesn't have a subobject classifier for the same reason $mathbfGrp$ 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 $mathbfGpd$ 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 $T_1circ T_1Rightarrow T_1$, $F_1circ T_1Rightarrow F_1$, $T_1circ F_1Rightarrow F_1$, $F_1circ F_1Rightarrow F_1$, and $F_1circ F_1Rightarrow T_1$, each representing a possible assignment of inclusion or exclusion in the subgroupoid to a triple of arrows satisfying $g_1circ g_2=g_3$. 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.
category-theory simplicial-stuff higher-category-theory groupoids
asked Aug 1 at 21:25
user581091
61
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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 $textSub_mathbfGpd(-)$ and $textHom_mathbfsSet(-,Omega)$ from $mathbfGpdtomathbfSet$ are naturally isomorphic?
I know that $mathbfGpd$ doesn't have a subobject classifier for the same reason $mathbfGrp$ 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 $mathbfGpd$ 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 $T_1circ T_1Rightarrow T_1$, $F_1circ T_1Rightarrow F_1$, $T_1circ F_1Rightarrow F_1$, $F_1circ F_1Rightarrow F_1$, and $F_1circ F_1Rightarrow T_1$, each representing a possible assignment of inclusion or exclusion in the subgroupoid to a triple of arrows satisfying $g_1circ g_2=g_3$. 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.
category-theory simplicial-stuff higher-category-theory groupoids
asked Aug 1 at 21:25
user581091
61
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 $textSub_mathbfGpd(-)$ and $textHom_mathbfsSet(-,Omega)$ from $mathbfGpdtomathbfSet$ are naturally isomorphic?
I know that $mathbfGpd$ doesn't have a subobject classifier for the same reason $mathbfGrp$ 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 $mathbfGpd$ 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 $T_1circ T_1Rightarrow T_1$, $F_1circ T_1Rightarrow F_1$, $T_1circ F_1Rightarrow F_1$, $F_1circ F_1Rightarrow F_1$, and $F_1circ F_1Rightarrow T_1$, each representing a possible assignment of inclusion or exclusion in the subgroupoid to a triple of arrows satisfying $g_1circ g_2=g_3$. 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.
category-theory simplicial-stuff higher-category-theory groupoids
asked Aug 1 at 21:25
user581091
61
asked Aug 1 at 21:25
user581091
61
asked Aug 1 at 21:25
user581091
61
asked Aug 1 at 21:25
user581091
61
61
add a comment |Â
add a comment |Â
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