Timeline for Lie functor preserves "surjections" in synthetic differential geometry?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 3, 2017 at 0:08 | vote | accept | ಠ_ಠ | ||
| Jan 21, 2016 at 9:29 | comment | added | André Henriques | It's true that if you define a Lie group to be a second countable manifold equipped with a group structure, then a surjective homomorphism of Lie groups is mapped to a surjective homomorphism of Lie algebras. But this ceases to be true as soon as you allow manifolds with uncountably many connected components (think about the identity map from $\mathbb R$ with the discrete topology to $\mathbb R$ with its usual topology). So you see: it's fragile statement. And fragile statements often don't generalize very well. | |
| Jan 21, 2016 at 3:05 | comment | added | ಠ_ಠ | @ André As far as I know, classically a surjective Lie group homomorphism is mapped to a surjective Lie algebra homomorphism by the Lie functor. | |
| Jan 21, 2016 at 1:43 | comment | added | André Henriques | By the way, the inclusion $\mathbb Q\to \mathbb R$ is not an epimorphism in the category of rings. It is the inclusion $\mathbb Z\to \mathbb Q$ which is an epimorphism in the category of rings. | |
| Jan 21, 2016 at 1:34 | comment | added | Peter Samuelson | Ah, right - sorry for the stupid comment. | |
| Jan 21, 2016 at 1:24 | comment | added | André Henriques | Here, $\mathbb Q$ is endowed with the discrete topology (it is a zero-dimensional Lie group). Te map $\mathbb Q\to \mathbb R$ is an epimorphism because if $f:\mathbb R\to G$ is a homomorphism of Lie groups, then $f$ is completely determined by $f|_{\mathbb Q}$ (by continuity). | |
| Jan 21, 2016 at 1:00 | comment | added | Peter Samuelson | What topology are you putting on $\mathbb Q$ so that it is a Lie group? And why is it an epimorphism? I know the inclusion is an epimorphism in the category of rings, but is it an epimorphism in the category of abelian groups? | |
| Jan 21, 2016 at 0:47 | comment | added | André Henriques | I'm not sure what you mean by "the Lie functor preserves surjections in classical differential geometry". Can you try to make that statement more precise? | |
| Jan 20, 2016 at 22:09 | comment | added | ಠ_ಠ | Ah, I hadn't realized that epis in the category of Lie groups were not precisely surjective smooth homomorphisms. I believe the Lie functor does preserve surjections in classical differential geometry, so I have edited my question. | |
| Jan 20, 2016 at 21:18 | history | answered | André Henriques | CC BY-SA 3.0 |