Indeed, I forgot to take the opposite category when I considered this possibility.

]]>The category $\mathbf{\Delta}^{op}$ has an initial object, so its homotopy limit is natural equivalent to the value of the diagram at that vertex.

]]>Homotopy colimits of simplicial diagrams and homotopy limits of cosimplicial diagrams have their own special names: realization and totalization.

Is there a special name for homotopy limits of simplicial diagrams? In general, are there any examples in the literature where such homotopy limits are computed?

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