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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.
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.
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.
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