Any ordinary bicategory can be regarded as a linear bicategory with .
If is a linear bicategory (such as the delooping of a linearly distributive category) such that has local coproducts and has local products, then the bicategory (whose objects are families of objects of and whose morphisms are matrices of 1-cells in ) is again a linear bicategory. For instance, if is the Boolean algebra of truth values, then .
A linear bicategory in which every 1-cell has both a linear right adjoint and a linear left adjoint is a horizontal categorification of a non-symmetric star-autonomous category?. But in Cockett-Koslowski-Seely this is called a “closed linear bicategory”, with the term “-linear bicategory” reserved for something stronger analogous to a cyclic star-autonomous category?.
References
Cockett and Koslowski and Seely, Introduction to linear bicategories, Mathematical Structures in Computer Science, 10 (2), 2000 (165 - 203)
Last revised on October 20, 2017 at 11:04:34.
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