# Homotopy colimit description of stacks

Let $$F$$ be say an Artin stack. If $$p: X \to F$$ is an atlas for $$F$$, can we express $$F$$, in the $$\infty$$-category $$Shv^{\acute{et}}(k)$$ of higher stacks, as a homotopy colimit over the simplicial diagram $$… \rightrightarrows X \times_{F} X \rightrightarrows X$$ in $$Shv ^{\acute{et}}(k)$$? I ask this because we know that if $$U_{\bullet} \to X$$ is a hypercover, for $$X$$ a scheme, this fact is built into the very definition of $$Shv ^{\acute{et}}(k)$$. Indeed, by Dugger, Hollander, Isaksen, (in the language of model categories) one obtains $$Shv ^{\acute{et}}(k)$$ as a Bousfield localization of $$sPre(k)$$ (the category of simplicial presheaves) by inverting morphisms $$hocolim_{\Delta} U_{\bullet} \to X$$

So I’m wondering if we can deduce this fact about a more general higher sheaf or stack $$F$$ from the scheme case.