While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric realization $|X_c|$ fit into a diagram indexed by $C$ if relations are allowed to hold only up to homotopy coherence (in other words one get a functor from $N(C)$ to the $\infty$-category of spaces).

I do have a relevant application coming from my original problem, but I wanted to see if other nice examples can be given in different contexts. sSet-valued functors seem to be usually called simplicial functors, and the latter are apparently common in derived geometry to give higher analogs of classical concepts.

Let us for example consider a simplicial presheaf over a manifold (or some more general context if that is beneficial, e.g. a Grothendieck site). Is the openwise geometric realization of this presheaf worth to be studied? As an example of what I have in mind, it could be that the simplicial homology of the total sections admit an approximation in terms of simplicial homology over smaller open sets. The latter may be simplified by looking at the geometric realization, as the simplicial set could be a large model of a simpler object, up to homotopy. However, deforming the spaces openwise does not guarantee that the presheaf constraints are still satisfied, and one may have to move to the world of homotopy coherent diagrams.

My impression is that the geometry of these simplicial sets are not studied, and that simplicial sets are regarded as a categorical machinery to extend ordinary notions to higher analogs. However, since I am an outsider, it's worth asking. Thanks!

While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric realization $|X_c|$ fit into a diagram indexed by $C$ if relations are allowed to hold only up to homotopy coherence (in other words one get a functor from $N(C)$ to the $\infty$-category of spaces).

I do have a relevant application coming from my original problem, but I wanted to see if other nice examples can be given in different contexts. sSet-valued functors seem to be usually called simplicial functors, and the latter are apparently common in derived geometry to give higher analogs of classical concepts.

Let us for example consider a simplicial presheaf over a manifold (or some more general context if that is beneficial, e.g. a Grothendieck site). Is the openwise geometric realization of this presheaf worth to be studied? As an example of what I have in mind, it could be that the simplicial homology of the total sections admit an approximation in terms of simplicial homology over smaller open sets. The latter may be simplified by looking at the geometric realization, as the simplicial set could be a large model of a simpler object, up to homotopy. However, deforming the spaces openwise does not guarantee that the presheaf constraints are still satisfied, and one may have to move to the world of homotopy coherent diagrams.

My impression is that the geometry of these simplicial sets are not studied, and that simplicial sets are regarded as a categorical machinery to extend ordinary notions to higher analogs. However, since I am an outsider, it's worth asking. Thanks!

EDIT: my original example is quite simple but occurs naturally in several contexts. If one considers a cosimplicial diagram with values in simplicial sets, there is a (co)homological spectral sequence converging to the (co)homology of the homotopy totalization of the diagram (I think it is called Bousfeld-Kan). If one is ultimately interested in the geometry of such homotopy totalization, it could be convenient to replace the given simplicial sets with homotopy equivalent simpler spaces.

While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric realization $|X_c|$ fit into a diagram indexed by $C$ if relations are allowed to hold only up to homotopy coherence (in other words one get a functor from $N(C)$ to the $\infty$-category of spaces).

I do have a relevant application coming from my original problem, but I wanted to see if other nice examples can be given in different contexts. It seems like sSet-valued functors are usually called simplicial functors, and the latter are somehow trendy in derived geometry to give higher analogs of classical concepts. Forgive my sloppiness as I know very little of the field.

Let us for example consider a simplicial presheaf over a manifold (or some more general context if that is beneficial, e.g. a Grothendieck site). Is the pointwise geometric realization of this presheaf worth to be studied? As an example to what I have in mind, it could be that the simplicial homology of the total sections admit an approximation in terms of simplicial homology over smaller open sets. The latter may be simplified by looking at the geometric realization, as the simplicial set could be a giant model of a very simple object. However, deforming the spaces objectwise does not guarantee that the presheaf constraints are still satisfied, and one may have to move to the world of homotopy coherent diagrams.

My impression is that the geometry of these simplicial sets are not really relevant, and that they are just a categorical machinery to extend ordinary notions to higher analogs. However, since I am an outsider, it's worth asking. Hope this does not sound as "cultural appropriation" to the derived community!! :D

While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric realization $|X_c|$ fit into a diagram indexed by $C$ if relations are allowed to hold only up to homotopy coherence (in other words one get a functor from $N(C)$ to the $\infty$-category of spaces).

I do have a relevant application coming from my original problem, but I wanted to see if other nice examples can be given in different contexts. sSet-valued functors seem to be usually called simplicial functors, and the latter are apparently common in derived geometry to give higher analogs of classical concepts.

Let us for example consider a simplicial presheaf over a manifold (or some more general context if that is beneficial, e.g. a Grothendieck site). Is the openwise geometric realization of this presheaf worth to be studied? As an example of what I have in mind, it could be that the simplicial homology of the total sections admit an approximation in terms of simplicial homology over smaller open sets. The latter may be simplified by looking at the geometric realization, as the simplicial set could be a large model of a simpler object, up to homotopy. However, deforming the spaces openwise does not guarantee that the presheaf constraints are still satisfied, and one may have to move to the world of homotopy coherent diagrams.

My impression is that the geometry of these simplicial sets are not studied, and that simplicial sets are regarded as a categorical machinery to extend ordinary notions to higher analogs. However, since I am an outsider, it's worth asking. Thanks!