Consider the category C of pointed simplicial sets. The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C models the suspension and loop functors on the underlying ∞-category of C.
There is another model for this adjunction, namely, the Kan suspension and loop functors. (Very) roughly speaking, the idea is to “shift” simplices in degree by 1 either in the positive direction for the suspension functor or in the negative direction for the loop functor. One reference for the Kan suspension functor is Section III.5 in Goerss and Jardine's “Simplicial homotopy theory”.
The resulting models for the suspension and loop functors are noticeably smaller than the usual ones, which is sometimes beneficial when doing concrete computations.
Of course, the suspension and loop functors are one of the easiest examples of homotopy colimits and limits, namely the homotopy pushout of *←X→* and the homotopy pullback of *→Y←*, which makes me wonder if Kan's constructions can be extended to other types of homotopy limits and colimits.
A related question is whether it is possible to do similar constructions for unpointed simplicial sets.
Is there a model for homotopy colimits of pointed simplicial sets that reduces to the Kan suspension functor in the case of the homotopy pushout of the diagram *←X→*? What about unpointed simplicial sets and/or homotopy limits?