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MacLane and Moerdijk's Sheaves in Geometry and Logic has a section on Continuous Group Actions (Sec. III.9). On page 152, there is an isomorphism displayed:

$$Hom_G(G/U, X) \cong X^U$$

In their set-up, $G$ is a topological group, $U$ is an open subgroup, $X$ is a set (space with discrete topology) with an action of $G$. The Hom-set is that of right G-actions. $X^U$ is the set of $U$-fixed points in $X$. (This result probably holds for ordinary groups instead of topological groups. I can't say.) The commentary following the display says "as usual" by which, they presumably mean it is widely known. But I can't find anything like it in the standard text books.

Can somebody figure out what this isomorphism is? Or, tell me where to look to find out?

Uday Reddy
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2 Answers2

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$G/U$ has a distinguished element, namely the coset of the identity. A $G$-morphism $G/U \to X$ is completely determined by where it sends this coset, and the possible points in $X$ it can be sent to are precisely points fixed by the action of $U$. (This should make sense on the point-set level, and then one only has to check that the topological details work out.)

One way of describing this result is that $G$ is the free $G$-set on a point and $G/U$ is the free $G$-set on a $U$-fixed point.

Qiaochu Yuan
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  • Fantastic! Thanks very much. I was probably too exhausted to understand it last night. But, I had chat with an 'expert' this morning, who made it all clear to me :-) – Uday Reddy Apr 01 '13 at 04:45
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Are you familiar with $\hom_G(G, X) \cong X$? Composing with the map $\hom_G(G/U, X) \to \hom_G(G, X)$ (which is monic!) gives you the injection $\hom_G(G/U, X) \to X$ you seek.