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Consider the simplicial indexing category $\Delta$. Now, let's denote the subcategory consisting of injections as $\Delta_{inj}$. When we're dealing with a cosimplicial space, which is essentially a functor $C: \Delta \to \text{Spaces}$, we get a Bousfield-Kan spectral sequence as detailed in Simplicial Homotopy theory. This spectral sequence is associated with the Tot tower and converges to the homotopy limit of the cosimplicial space $C$.

Question: Is there a similar spectral sequence when we replace $\Delta$ with the subcategory $\Delta_{inj, \leq n}$, where $\Delta_{inj, \leq n}$ is obtained by truncating $\Delta_{inj}$ at level $n$. Any suggestions or references would be great! Thank you in advance.

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  • $\begingroup$ There should be a spectral sequence from $\Delta^{inj}$. I've been told that If you have a cosimplicial space and you forget the codegeneracies, the resulting $\Delta^{inj}$ spectral sequence should agree with the $\Delta$-spectral sequence from the $E_2$ page on. I don't know why or where to finda reference. $\endgroup$
    – Tim Campion
    Commented Oct 17, 2023 at 19:45
  • $\begingroup$ @TimCampion Are you referring to the fact that $\Delta^{inj} \hookrightarrow \Delta$ is cofinal? $\endgroup$
    – happymath
    Commented Oct 17, 2023 at 20:14

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