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They mention four related 2-groups on pp. 3-4. Where it says ’the T-duality 2-group’ at T-duality 2-group, presumably that’s their $\mathbb{T}\mathbb{D}$. But they’re also speaking of a ’half-geometric T-duality 2-group’.
They say their work on ’crossed interwiners’ is new. Shouldn’t one expect this to be fairly straightforward abstract general work - something that should fall out from a HoTT treatment of infinity-actions? But it seems they want it to capture a certain level of strictness in morphisms of Lie 2-groups (p. 4).
To add:
Calculation of the categorical automorphism group of the T-duality 2-group is in
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