# Homotopy limit over a diagram of nullhomotopic maps

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.

• No, e.g., the homotopy limit of $*\to X \leftarrow *$. – Charles Rezk May 2 at 14:54
• 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. – Najib Idrissi May 2 at 14:55
• 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. – Phil Tosteson May 2 at 15:27
• 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.) – Dylan Wilson May 2 at 18:55
• @DylanWilson This is interesting! Do you happen to know of a reference for this kind of homotopy limit? – 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 $$\Omega X$$.