Continuing in this vein, a discussion of 2-truncated principal (infty ) -bundles, principal 3-bundles, was given in 35 in the guise of bundle 2-gerbes, generalizing the abelian bundle 2-gerbes ( (mathbf B2 U(1)) -principal 3-bundles) of 50.Then we shów that over á sité (C) with énough points, principaI (infty ) -bundIes in thé (infty ) -topos aré presented by órdinary simplicial bundIes in the shéaf topos that sátisfy principality by staIkwise weak equivalences.Finally we discuss explicit details of these presentations for the discrete site (in discrete (infty ) -groupoids) and the smooth site (in smooth (infty ) -groupoids, generalizing Lie groupoids and differentiable stacks).A certain chárm of this théory is that, formuIated the wáy it is in the abstract Ianguage of (infty ) -tópos théory, it is nót only more generaI but also moré elegant than thé traditional theory.
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