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• CommentRowNumber1.
• CommentAuthortomr
• CommentTimeJan 14th 2023
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.
1. 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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 14th 2023
• (edited Jan 14th 2023)

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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 14th 2023
• (edited Jan 14th 2023)

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.