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Edit: The idea is that the total space $EG$ of the classifying bundle of $G$ is contractible and a cofibrant replacement of the point $1$ on which $G$ acts freely. Thus, the construction $BG = EG/G$ is taking a stack-y quotient $BG = 1//G$.

There is a bit more to this idea than may first appear; let me take a related example (which may appear to have some Eulerian "wishful thinking" in it, but have a little faith here!). One way of taking the reciprocal is to pass to a geometric series, so that one suggestive notation for the free monoid construction

$$\sum_{n \geq 0} X^{\otimes n}$$

(in a suitable monoidal category; see my other comment on categorifying exponentiation) is a categorified reciprocal $1/(1 - X)$. We can apply this idea in group cohomology for a group $G$ as follows: think of $\mathbb{Z}$ as being an abelianized point, and consider a standard $G$-free resolution of $\mathbb{Z}$ such as the normalized homogeneous bar resolution, which we can think of as an abelianized $EG$. In one way of constructing this bar resolution (see e.g. Hilton-Stammbach p. 217), the degree $n$ component of $EG$ is

$$\mathbb{Z}G \otimes IG^{\otimes n}$$

where $IG$ is the augmentation ideal, i.e., the kernel of the augmentation map $\varepsilon: \mathbb{Z}G \to \mathbb{Z}$. As a bare module (or seen in degree 0), $IG$ can be seen as an abelianized "$G - 1$". However, in the differential-graded world, it is better to think of it as in degree 1, and this degree 1 shift $\Sigma IG$ can be seen as a categorified "$-IG = 1 - G$" (this may make more sense in the "super-world"; see for example my old notes on the Lie operad when I was doing some work with Saunders Mac Lane, or consider for example the occurrence of signs in the Euler characteristic). So now the total space of the bar resolution $EG$ is the sum of the degree $n$ components

$$\mathbb{Z}G \otimes \sum_{n \geq 0} (\Sigma IG)^{\otimes n}$$

which is an abelianized categorified form of $g \cdot \sum_{n \geq 0} (1 - g)^n$ which is formally $1$ by the geometric series. Very similar types of categorified geometric series constructions occur in Joyal's theory of species (see especially his article on virtual species in Springer LNM 1234), which constructs the Lie operad by categorified constructions [if you read between the lines!], and in the bar resolution for operads as discussed by Ginzburg-Kapranov; I tried to amplify this in my notes on the Lie operad.

Just to put one final gloss on this: consider the Schubert cell decomposition of projective space as a finite geometric series. For a field $k$ we have

$$\mathbb{P}^{n-1}(k) = \frac{k^n - 1}{k - 1} = 1 + k + k^2 + \ldots + k^{n-1}$$

(the '$1$' in the numerator is a zero vector, and the denominator is nonzero scalars $k^\ast$). We can pass to a limit and get infinite-dimensional projective space. Keeping in mind that degree shifts introduce some sign changes in the geometric series, the infinite-dimensional projective space $\mathbb{RP}^\infty$ would be a model of the homotopy quotient $1//\mathbb{R}^* \simeq 1//\mathbb{Z}_2$.