Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the object $\textbf{n}$.

**Question:** Suppose $X: \Lambda^{op} \to Top$ is a cyclic space. What is a description of the homotopy colimit of this functor?

Just to put this in a bit of context, if $Y: \Delta^{op} \to Top$ is a simplicial space then it has a geometric realisation $|Y|$. One can also take the homotopy colimit of $Y$, and under some reasonable hypotheses there will be an equivalence $\mathrm{hocolim}_{\Delta^{op}} Y \simeq |Y|$.

There is an inclusion $\Delta \to \Lambda$, so a cyclic space $X$ can be considered as a simplicial space and one can thus make a geometric realisation $|X|$. This space is supposed to have a circle action. I suppose my question should be:

How is the hocolim of $X$ over $\Delta^{op}$ related to the hocolim of $X$ over $\Lambda^{op}$.