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1. • CommentRowNumber1.
   • CommentAuthortomr
   • CommentTimeJan 14th 2023
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   Author: tomr
   Format: TextI am trying to read https://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos (in my own path) and I see bold-B in many places (e.g. BH, where H is infinity-topos), but I have not managed to find name or explanation for it. There is place where B (though with indice) is called as de Rham complex, but I can not connect the dots. I am seeking just name for bold-B. No explanation is needed here - I will be able then found it myself as much as I need.

   I am trying to read https://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos (in my own path) and I see bold-B in many places (e.g. BH, where H is infinity-topos), but I have not managed to find name or explanation for it. There is place where B (though with indice) is called as de Rham complex, but I can not connect the dots.
   
   I am seeking just name for bold-B. No explanation is needed here - I will be able then found it myself as much as I need.
2. • CommentRowNumber2.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeJan 14th 2023
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   Author: Madeleine Birchfield
   Format: MarkdownItexI think the bold B stands for the [[delooping]] of an object. So for the infinity-topos H, BH would be the delooping of said infinity-topos.

   I think the bold B stands for the delooping of an object. So for the infinity-topos H, BH would be the delooping of said infinity-topos.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2023
   • (edited Jan 14th 2023)
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   Author: Urs
   Format: MarkdownItexThe boldface $\mathbf{B}$ is for *delooping* of infinity-group objects in an infinity-topos $\mathbf{H}$: $$ Groups(\mathbf{H}) \underoverset {\underset{\;\;\mathbf{B}\;\;}{\longrightarrow}} {\overset{\Omega}{\longleftarrow}} {\simeq} \mathbf{H}^{\ast/}_{\geq 1} $$ according to [this Prop.](https://ncatlab.org/nlab/show/May+recognition+theorem#RecognitionForGroupsInAnInfinityTopos)

   The boldface $\mathbf{B}$ is for delooping of infinity-group objects in an infinity-topos $\mathbf{H}$:

   $$Groups(\mathbf{H}) \underoverset {\underset{\;\;\mathbf{B}\;\;}{\longrightarrow}} {\overset{\Omega}{\longleftarrow}} {\simeq} \mathbf{H}^{\ast/}_{\geq 1}$$

   according to this Prop.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2023
   • (edited Jan 14th 2023)
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   Author: Urs
   Format: MarkdownItexI write it in boldface to indicate that this delooping is a "geometrically enriched" version of the classical classifying space construction which is traditionally denoted "$B G$": If $\mathbf{H} = SmoothGrpd_\infty$ and $G$ a compact Lie group, then the underlying shape of $\mathbf{B}G$ (the [[delooping groupoid]] [[Lie groupoid]] of $G$) is the classifying space: $$ ʃ \mathbf{B} G \;\simeq\; B G $$ This is Prop. 4.1.12 in *[[schreiber:Equivariant principal infinity-bundles]]*. See also around Prop. 0.2.1 there.

   I write it in boldface to indicate that this delooping is a “geometrically enriched” version of the classical classifying space construction which is traditionally denoted “$B G$”:

   If $\mathbf{H} = SmoothGrpd_\infty$ and $G$ a compact Lie group, then the underlying shape of $\mathbf{B}G$ (the delooping groupoid Lie groupoid of $G$) is the classifying space:

   $$ʃ \mathbf{B} G \;\simeq\; B G$$

   This is Prop. 4.1.12 in Equivariant principal infinity-bundles (schreiber). See also around Prop. 0.2.1 there.