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More examples:

  • the functor Δ → Δ sending a totally ordered set S to S ∐ S, where the elements in the left copy are all less than the elements in the right copy. Restriction along this functor is the "edgewise subdivision", e.g., it sends Δ2 to a complex with four nondegenerate 2-simplices whose geometric realization is homeomorphic to |Δ2|.

  • the functor Δ → Δ sending S to Sop ∐ S. Restriction along this functor sends the nerve of a category C to the nerve of the twisted arrow category of C.

I guess all the examples I know can be built out of objects of Δ, op, and the join Δ × Δ → Δ.
show/hide this revision's text 1

More examples:

  • the functor Δ → Δ sending a totally ordered set S to S ∐ S, where the elements in the left copy are all less than the elements in the right copy. Restriction along this functor is the "edgewise subdivision", e.g., it sends Δ2 to a complex with four nondegenerate 2-simplices whose geometric realization is homeomorphic to |Δ2|.

  • the functor Δ → Δ sending S to Sop ∐ S. Restriction along this functor sends the nerve of a category C to the nerve of the twisted arrow category of C.