A lattice in an inner product space is called integral if the inner product of pairs of lattice vectors takes values in the integer numbers.
Integral lattices may encode higher structures, such as braided 2-groups (Baez 2015) and multiplicative bundle gerbes (Ganter 2014).
Lecture notes:
Discussion of related higher structures:
Nora Ganter. Categorical Tori, SIGMA 14 014 (2018) [arXiv:1406.7046, doi:10.3842/SIGMA.2018.014]
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