Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit $$ \underset{i \in I}{\mathrm{holim}}X(i), $$ where the maps $X(i) \to X(j)$ are nullhomotopic for $i \leq j$ and $X(i) \to X(j)$ a weak homotopy equivalence whenever $i \cong j$.

Can we conclude that $\underset{i \in I}{\mathrm{holim}}X(i)$ is trivial?

An example

The example I have in mind is whenever $I$ is the poset of non-zero subspaces of $\mathbb{R}^n$ (topologized as a disjoint union of Grassmannians), and $X$ is the sphere functor $\mathbb{S}: V \mapsto S^V$. Then when $\dim V < \dim W$ the map $S^V \to S^W$ is null homotopic and when $\dim V = \dim W$ the map $S^V \to S^W$ is a weak homotopy equivalence.

I'd like to conclude that the homotopy limit, $$ \underset{0 \neq U \subseteq \mathbb{R}^n}{\mathrm{holim}}S^U $$ is weakly contractible.

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    $\begingroup$ No, e.g., the homotopy limit of $*\to X \leftarrow *$. $\endgroup$ – Charles Rezk May 2 at 14:54
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    $\begingroup$ Even simpler, consider a diagram $I$ which has no non-identity maps (i.e. no elements are comparable)... The homotopy limit is then just the product, which has no reason to be contractible. $\endgroup$ – Najib Idrissi May 2 at 14:55
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    $\begingroup$ I guess we could also take the diagram which is just a single point. This happens in your specific example when $n = 1$, and you get something non-contractible. $\endgroup$ – Phil Tosteson May 2 at 15:27
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    $\begingroup$ For what it's worth, the example you're interested in is the space of sections of the fiberwise one-point compactification of the tautological line bundle on $\mathbb{R}P^{n-1}$. (Proof: Right Kan extend along $\mathrm{Sub}_{\ne 0}(\mathbb{R}^n) \to \{1<2<...<n\}$; this latter category has an initial object so just evaluate there. You have a homotopy limit over the space of lines, and you can identify that with the space of sections as stated.) $\endgroup$ – Dylan Wilson May 2 at 18:55
  • $\begingroup$ @DylanWilson This is interesting! Do you happen to know of a reference for this kind of homotopy limit? $\endgroup$ – Charles May 2 at 21:04

For the general question, the answer is no. Let $X$ be a pointed topological space. Consider the diagram used to construct $* \times_X *$, it is a poset and all the transition maps are null-homotopic, but the homotopy limit is $\Omega X$.


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