Tate's central extension originates from 1968 and has since found many
applications to curves. In the 80s Beilinson found an n-dimensional
generalization: cubically decomposed algebras, based on ideals of bounded and
discrete operators in ind-pro limits of vector spaces. Kato and Beilinson
independently defined '(n-)Tate categories' whose objects are formal iterated
ind-pro limits in general exact categories. We show that the endomorphism
algebras of such objects often carry a cubically decomposed structure, and thus
a (higher) Tate central extension. Even better, under very strong assumptions
on the base category, the n-Tate category turns out to be just a category of
projective modules over this type of algebra
