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The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology:

If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\mathcal U$ is contractible, then $X$ is homotopy equivalent to the nerve $N \mathcal U$.

Many versions of this lemma are summarized in this paper by Heal; note in particular those versions in which the sets in $\mathcal U$ need not be open, such as Borsuk's version, where $\mathcal U$ is a finite regular cover by closed subsets of a compact space $X$. The version that I have found with the weakest hypotheses on $\mathcal U$ is in this paper by Nagórko as Theorems 3.3 and 3.4.

The lemma may also be strengthened by providing an equivariant homotopy equivalence in the case where $X$ is acted upon by some group $G$ and $\mathcal U$ is invariant under the action. Such a lemma should have the form

If $X$ is acted upon by $G$ such that $\mathcal U$ is a covering by not-necessarily-open subsets, then under some hypotheses on $X$, $G$, and $\mathcal U$, there is a $G$-equivariant homotopy equivalence $X \to N\mathcal U$.

Such lemmas appear as Theorem 4.6 in this paper by González & González, Lemma 2.5 in this paper by Hess & Hirsch, and Proposition 2.2 in this paper by Paris.

The title question of this post may be understood in three ways, all of which I am interested in:

  1. Is there a equivariant version of Nagórko's nerve lemma?

  2. Is there a nerve lemma with hypotheses on $\mathcal U$ weaker than Nagórko's?

  3. What is the equivariant nerve lemma with the weakest hypotheses on $X$, $G$, and $\mathcal U$?

Improve this question
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    $\begingroup$ There is a spacial nerve version which I believe works for any open cover. Instead of realizing a simplicial set built from the combinatorics of the poset associated to the open cover, one also includes the topology of the open subsets. This paper of Bendersky and Gitler has a short description of it in example 1.1. $\endgroup$
    – Connor Malin
    Commented Jul 11, 2021 at 19:53
  • $\begingroup$ This can be thought to generalize the discrete version because the condition of the intersections being contractible implies that the spaces of simplices in the above nerve are homotopy discrete. This implies the quotient to the path components is a weak equivalence of simplicial spaces which with suitable cofibrancy implies weakly equivalent realizations. $\endgroup$
    – Connor Malin
    Commented Jul 11, 2021 at 19:54
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    $\begingroup$ The paper by Dugger and Isakson on hypercovers is also relevant; in particular they prove strong versions of the result Connor mentions: link.springer.com/article/10.1007/s00209-003-0607-y. There's also a version of the Nerve Lemma due to Bjorner in which intersections need only be highly connected: sciencedirect.com/science/article/pii/S0097316503000153 $\endgroup$
    – Dan Ramras
    Commented Jul 11, 2021 at 20:12

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