## Why is the base manifold of a Lie groupoid required to be second-countable?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

I wonder why one requires that the base manifold of a Lie groupoid is second-countable?

-

1
Isn't second-countability in the definition of manifold? en.wikipedia.org/wiki/… – Zev Chonoles May 15 2011 at 20:41

2
Usually manifolds tout court are required to be second countable. – Mariano Suárez-Alvarez May 15 2011 at 20:42

2
Sorry, I should state this question more carefully. Of course, Zev Chonoles and Mariano Suarez-Alvarez are right: the usual definition of a manifold requires second-countability and Hausdorff and locally euclidean. My question should merely be: At which point in the theory of Lie groupoids does one really need that the base manifold is second-countable? When constructing a Lie groupoid from a foliation one actually has to be a bit careful at this point. If one takes uncountably many charts the base manifold of the Lie groupoid won't be second-countable. – Dave Lewis May 15 2011 at 20:57

2
@Dave Lewis: Can I request that you edit your question to include your comments above? (Mark the edit as an edit, so that @Zev and @Mariano 's comments still make sense.) It sounds like you have a more specific direction that you're thinking about, and in any case clearly recognize that "When constructing a Lie groupoid from a foliation one actually has to be a bit careful at this point", for example. I do not know of a good reason to have questions on MO that are only one sentence long, and there are many good reasons for including a few paragraphs. – Theo Johnson-Freyd May 15 2011 at 22:50

2
@Zev, Mariano, and Dave: If you require manifolds to be second countable, then a disjoint union of manifolds is not always a manifold. Replacing second countability by paracompactness allows you to keep all good properties of second countable manifolds and makes the category of manifolds closed under coproducts, which seems like a good property to have. – Dmitri Pavlov May 16 2011 at 4:19
show 2 more comments

## 1 Answer

Answer #1:
There is no real reason for imposing that the base manifold of a groupoid be second countable.

Answer #2:
You lose some desirable properties if you don't impose second countability: For example, without it, the homotopy type of the geometric realisation of the nerve will no longer be an invariant of the Morita equivalence class of the groupoid.

-

Re Answer #2: Weird!  I would have expected that the homotopy type of the nerve was a well-defined invariant for any topological groupoid, and that the construction should factor through forgetting from Manifolds to Homotopy Types.  Could you either explain more, or include a reference? – Theo Johnson-Freyd May 15 2011 at 22:52
@Theo: I take my favourite non-second countable manifold: the long line $L$, and I look at the cover consisting of all of its bounded connected open subsets. The corresponding Cech groupoid is Morita equivalent to $L$. There is an obvious projection from the geometric realization of the Cech groupoid back to $L$. But there is no section of that map: that's because the cover does not admit partitions of unity. More generally, you can show that the projection does not admit a homotopy inverse. – André Henriques May 15 2011 at 23:12
Ah, Andre - that is why you redefine Morita equivalence not to use 'local sections', but 'local sections wrt a numerable cover'. This class of weak equivalences of topological/Lie groupoids (take your pick) is closer to what people think of when they restrict to paracompact spaces. – David Roberts May 16 2011 at 0:35
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9230931401252747, "perplexity_flag": "middle"}