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Dmitri Pavlov
Dmitri Pavlov
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I have seen the notion forof a delooping groupoid defined as a 1-groupoid having a single object * and the automorphisms onof * to be the group G, with the composition of 1-morphisms to bebeing the usual group operation. In this sense, a group is a pointed 0-connected groupoid. In higher cases too, an n-group is a pointed connected n-groupoid. But

But I have seen this for a discrete group G. What about a continuous group? Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or a 2-groupoid? I have seen claims that BU(1) to be theis a 2-group. I guess it follows from the fact that the homotopy type of U(1) is same asequivalent to B$\mathbb{Z}$,then U making BU(1) is like B$^{2}\mathbb{Z}$. 

Could someone briefly explain how to see this as a 2-group? Do we have to always bring this to something relating to discrete groups? Thanks.

I have seen the notion for delooping groupoid as a 1-groupoid having a single object * and the automorphisms on * to be the group G, the composition of 1-morphisms to be the usual group operation. In this sense, a group is a pointed 0-connected groupoid. In higher cases too, an n-group is a pointed connected n-groupoid. But I have seen this for discrete group G. What about a continuous group? Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or 2-groupoid? I have seen BU(1) to be the 2-group. I guess it follows from the fact that the homotopy type of U(1) is same as B$\mathbb{Z}$,then U(1) is like B$^{2}\mathbb{Z}$. Could someone briefly explain how to see this a 2-group? Do we have to always bring this to something relating to discrete groups? Thanks.

I have seen the notion of a delooping groupoid defined as a 1-groupoid having a single object * and the automorphisms of * to be the group G, with the composition of 1-morphisms being the usual group operation. In this sense, a group is a pointed 0-connected groupoid. In higher cases too, an n-group is a pointed connected n-groupoid.

But I have seen this for a discrete group G. What about a continuous group? Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or a 2-groupoid? I have seen claims that BU(1) is a 2-group. I guess it follows from the fact that the homotopy type of U(1) is equivalent to B$\mathbb{Z}$, making BU(1) like B$^{2}\mathbb{Z}$. 

Could someone briefly explain how to see this as a 2-group? Do we have to always bring this to something relating to discrete groups? Thanks.

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Pinak Banerjee
Pinak Banerjee
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Delooping groupoid

I have seen the notion for delooping groupoid as a 1-groupoid having a single object * and the automorphisms on * to be the group G, the composition of 1-morphisms to be the usual group operation. In this sense, a group is a pointed 0-connected groupoid. In higher cases too, an n-group is a pointed connected n-groupoid. But I have seen this for discrete group G. What about a continuous group? Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or 2-groupoid? I have seen BU(1) to be the 2-group. I guess it follows from the fact that the homotopy type of U(1) is same as B$\mathbb{Z}$,then U(1) is like B$^{2}\mathbb{Z}$. Could someone briefly explain how to see this a 2-group? Do we have to always bring this to something relating to discrete groups? Thanks.