# Topological realisation of a stack (explicit description)

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?

• There is a homotopy class of simplicial topological spaces associated to $X$, constructed in roughly the way that you describe. – Jason Starr Mar 26 '19 at 9:10

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.