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The answer to Question 1 is, as expected, "No". Let $(A\rightrightarrows X)=(X\times X\rightrightarrows X)$. If our groupoid is an action groupoid, i.e., of the form $G\ltimes X$, then the $\Gamma$-s …

Let $F\colon (A\rightrightarrows X)\to (B\rightrightarrows Y)$ be a morphism of groupoids (a functor). We say that $F$ is an equivalence of groupoids if it is an equivalence of categories. … By a weak equivalence of $\Gamma$-groupoids we mean a $\Gamma$-functor $F\colon (A\rightrightarrows X)\to (B\rightrightarrows Y)$ that is a weak equivalence of groupoids. …

Conversely, any connected groupoid is isomorphic to an action groupoid, see the answers to my question Connected groupoids and action groupoids. … We say that two $\Gamma$-groupoids are weakly $\Gamma$-equivalent if they can be connected by a chain of quasi-isomorphisms of $\Gamma$-groupoids. …

Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid. Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$, and we have 5 maps: $s,t\colon A\to X$ (the source and the target, surjective), $m\co …

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to an action groupoid $G\ltimes X$ coming from a transitive action o …

I am looking for Albrecht Fröhlich's unpublished text `Groupoids, groupoid spaces and cohomology' (1965). …

No, in general $G=G(\mathbf{Q})$ can be strictly smaller that ${\rm Aut}(\mathbf{Q})$. Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Set $P=G$, $\ Q=G/H$, and define the maps in the obv …