Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the action of $G$ is always strict. A $G$-space $X$ is given by a group homomorphism $G\to \text{Aut}(X)$, where $\text{Aut}(-)$ denotes the group of continuous automorphisms.


If instead I want to allow $\sigma:G\times X\to X$ to solve $\sigma(e)\sim \text{id}\_X$ and
$$\sigma\circ(\text{id}\_G\times \sigma)\sim \sigma\circ (\mu\_G\times \text{id}\_X)\,, $$
only up to homotopies ($\mu\_G$ here is the multiplication on $G$) and possibly have higher homotopies I need to think about $\infty$-groupoids, as [this paper](https://www.jstor.org/stable/1997628?seq=1) shows that there is an obstruction to strictifying homotopy group actions.


1. Is there a well defined notion of equivariant cohomology theories in this setting?
2. For an $\infty$-groupoid, one can take its homotopy quotient (the colimit). Can one define the equivariant cohomology as the cohomology of this quotient?



Homotopy group action and equivariant cohomology theories
From modern perspective this is much more straightforward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity categories. $G$-spectra are just functors $BG\to \mathrm{Sp}$. You can think of a $G$-spectrum as a functor on $G$-spaces by
$E \mapsto (X\mapsto \mathrm{Map}\_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E))$ where $\mathbb{S}[-]= \Sigma^{\infty}$ is the stabilization functor, applied pointwise to functors from $BG$. Hence after accepting some notions like functors and stabilization in infinity category theory you immediately get a theory of equivariant stuff of this "up to homotopy" flavour. In particular, if $E$ has trivial $G$-action then by trivial-colimit adjunction and colimits preservation of the stabilization we get $$\mathrm{Map}\_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E)\simeq \mathrm{Map}\_{\mathrm{Sp}}(\mathbb{S}[X]\_{hG},E)\simeq \mathrm{Map}\_{\mathrm{Sp}}(\mathbb{S}[X\_{hG}],E)$$ and you indeed get the cohomology of the homotopy quotient.


In some sense, the surprising thing from this modern perspective is the existence of the "strict" version, which is slightly harder to define internally to modern homotopy theory, even though it is doable.


