# Terminology for a partition of unity for an étale groupoid

I would like to ask about terminology for a partition of unity for an étale groupoid.

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.

1) The source map $s$ and the target map $t$ are both étale (i.e., induce isomorphisms on tangent spaces).

2) The map $(s,t):X_1 \to X \times X$ is proper and unramified, with finite fibers (unramified means injective on tangent spaces).

For such a Lie groupoid, a partition of unity is defined as follows.

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 $t_!s^* \rho \equiv 1$.

a) What does "$\rho$ has proper support with respect to $t:X_1 \to X_0$" means?

b) How do we define the operator $t_!$?

I would be most grateful if you could tell me references on these terminology.

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For a), I guess it means that the inclusion of the support (i.e. closure of...) of $s^*\rho$ into $X_1$ composed with $t$ is a proper map (i.e. the inverse image of a compact set is compact). For b), $t_!$ means "integration along fibers": as a function on $X_0,\ t_!s^*\rho$ takes $x\in X_0$ to $\int_{t^{-1}(x)}s^*\rho,$ and this function is required to be the constant function 1. –  shenghao Oct 28 '11 at 10:16
@shenghao Thank you very much for your comment. Your definition for a) is reasonable, and then integration along the fibre is well defined. –  H. Shindoh Oct 28 '11 at 11:04
The analogy is that for a partition of unity for an open cover $\coprod U_a \to [0,1]$, the function is locally finite - this is part a) - and the sum is identically 1 - which makes sense because of the first point. –  David Roberts Oct 29 '11 at 22:46