Terminology for a partition of unity for an étale groupoid - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:41:21Z http://mathoverflow.net/feeds/question/79372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79372/terminology-for-a-partition-of-unity-for-an-etale-groupoid Terminology for a partition of unity for an étale groupoid H. Shindoh 2011-10-28T09:20:40Z 2011-10-28T09:20:40Z <p>I would like to ask about terminology for a partition of unity for an étale groupoid.</p> <p>I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in Definition 22. Let $X_1 \rightrightarrows X_0$ be a Lie groupoid satisfying the following conditions.</p> <p>1) The source map $s$ and the target map $t$ are both étale (i.e., induce isomorphisms on tangent spaces).</p> <p>2) The map $(s,t):X_1 \to X \times X$ is proper and unramified, with finite fibers (unramified means injective on tangent spaces).</p> <p>For such a Lie groupoid, a partition of unity is defined as follows.</p> <blockquote> <p>A partition of unity for the groupoid $X_1 \rightrightarrows X_0$ is an $\mathbb{R}$-valued $C^\infty$-function $\rho$ on $X_0$ with the property that $s^*\rho$ has proper support with respect to $t:X_1 \to X_0$ and $t_!s^* \rho \equiv 1$.</p> </blockquote> <p>a) What does "$\rho$ has proper support with respect to $t:X_1 \to X_0$" means?</p> <p>b) How do we define the operator $t_!$?</p> <p>I would be most grateful if you could tell me references on these terminology.</p>