Topological realisation of a stack (explicit description)

Question

Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription.

My first guess would be: take a smooth cover $U\to X$ ($U$ is a scheme), then consider the simplicial space $$\cdots \substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow} U_\text{top}\times_XU_\text{top} \ \rightrightarrows U_\text{top}$$ and take its geometric realisation. I think this gives the right answer when $X$ is a scheme or $X=BG$.

Is this guess correct? If not, what goes wrong and what is the correct answer?

Answer 1

Let $\mathrm{Ét}_\mathbb{C}$ be the étale $\infty$-topos of schemes over $\mathbb{C}$, that is the $\infty$-categories of étale sheaves of $\infty$-groupoids over $\mathbb{C}$. This contains necessarily as a subcategory the sheaves of 1-groupoids, that is the étale stacks over $\mathbb{C}$ and so in particular the Artin stacks.

I claim that there is a (homotopy) colimit-preserving functor $\mathrm{Ét}_\mathbb{C}\to \mathrm{Space}$, where the target is the $\infty$-category of spaces, sending every scheme $X$ to the homotopy type of its complex points $X(\mathbb{C})$. This implies the construction you described since every Artin stack is the (homotopy) colimit of the Čech nerve of an atlas, and the (homotopy) colimit of a simplicial space is just its geometric realization.

The claim follows because, by proposition 6.2.4.6 in Lurie's Higher Topos Theory colimit-preserving functors out of $\mathrm{Ét}_\mathbb{C}$ are the same thing as functors from the category of finitely presented $\mathbb{C}$-schemes to spaces that send any étale cover to a family of maps of spaces jointly surjective on $\pi_0$. But it is immediate to see that the functor sending a $\mathbb{C}$-scheme $U$ to the homotopy type of $U(\mathbb{C})$ satisfies the above property.