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In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

I answered a similar question to this a while back.

Specifically the question was asking about the result that the category covering spaces on BC:=|N(C)| is canonically isomorphic to presheaves on C.

In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

I answered a similar question to this a while back.

Specifically the question was asking about the result that the category covering spaces on BC:=|N(C)| is canonically isomorphic to presheaves on C.

In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

I answered a similar question to this a while back.

In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

I answered a similar question to this a while back.

Specifically the question was asking about the result that the category covering spaces on BC:=|N(C)| is canonically isomorphic to presheaves on C.

In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

In general, this is not well-defined (you could look at the completion of the category by homotopy colimits, I guess, but for some reason I feel like this isn't very useful). A sheaf on the geometric realization is equivalent to a covering space. The geometric realization throws out too much data for this sort of thing to be very useful. You should really read HTT by Lurie, because this is what the book is about ([generalizations of] sheaf toposes on infinty-categories (which are special simplicial sets). You usually want to look at sheaves of Kan complexes, which are a higher-categorical equivalnt of categories fibered in groupoids.

Technically you can view any simplicial set as the $\omega$-nerve of a strict $\omega$-category, so in that sense, you could look at strict $\omega$-functors, but these are of very little interest in general.

I answered a similar question to this a while back.