Is the geometric realization of simplicial functors interesting?

Question

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.

Answer 1

If I understand your question correctly, you may find some interesting results of a similar nature in the paper by Vogt (Homotopy limits and colimits, Math. Z., 134, (1973), 11 – 52.) which was followed up by Jean-Marc Cordier and myself in some papers (Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65 – 90) and (Maps between homotopy coherent diagrams,Top. and its Appls., 28, (1988), 255 – 275) in particular. The results there are particular cases of results proved later by Joyal and by Lurie in a much more general context, but in the specific case that you mention in the question, i.e. functors with values in the category of simplicial sets, it is perhaps easier to unpick the geometric interpretation of the results and thus their application to cases of interest to yourself.

My own feeling is that the simplicial set valued functors often involve combinatorial structures that would be obscured if you pass to the geometric realisation. In the case of presheaves on spaces, you seem to be heading off into the realm of stacks, gerbes, etc. and for the geometric interpretation of these, a good point of entry is Larry Breen's notes (Notes on 1-and 2-gerbes, in J. Baez and J. May, eds., Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, 193–235, 2009).

As to how one regards simplicial sets or more generally simplicial objects in some nice category, I have found the simplicial approcah an extremely useful means of encoding combinatorial relationships (e.g. in homotopy coherence). If one gets used to working with the face and degeneracy relations and with, e.g. products of simplicial sets via shuffles and similar devices, then one can obtain explicit formulae for relationships, whereas if one uses the corresponding spatial machinery one has to work quite hard to get out as much detail.

There are more recent expositions of all these ideas e.g. in Riehl and Verity's book (Elements of ∞-Category Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press), in Jacob Lurie's online Kerodon, and, of course, quite a few other sources.

I hope this helps.