nLab
linear bicategory

Linear bicategories

Idea

The notion of linear bicategory (or linearly distributive bicategory for long) is a horizontal categorification of the notion of linearly distributive category, analogous to how bicategories are a horizontal categorification of monoidal categories.

Definition

Definition

A linear bicategory consists of

  1. A set of objects x,y,zx,y,z.
  2. For each x,yx,y a hom-category B(x,y)B(x,y).
  3. Two bicategory structures (,)(\otimes,\top) and (,)(\parr,\bot) on these hom-categories. Thus we have two compositions ,:B(y,z)×B(x,y)B(x,z)\otimes,\parr : B(y,z) \times B(x,y) \rightrightarrows B(x,z) and two units x, xB(x,x)\top_x,\bot_x \in B(x,x), each coherently associative and unital.
  4. Linear distributivities (XY)ZX(YZ)(X \parr Y) \otimes Z \to X \parr (Y\otimes Z) and X(YZ)(XY)ZX \otimes (Y \parr Z) \to (X\otimes Y) \parr Z, satisfying the usual coherence laws for a linearly distributive category.

Examples

Linear adjoints

Definition

A linear adjunction in a linear bicategory consists of 1-cells f:XYf:X\to Y and g:YXg:Y\to X along with a unit η: Xgf\eta : \top_X \to g \parr f and ϵ:fg Y\epsilon : f \otimes g \to \bot_Y satisfying versions of the usual triangle identities that include the linear distributivities, as for dual objects in a linearly distributive category

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 “*\ast-linear bicategory” reserved for something stronger analogous to a cyclic star-autonomous category?.

References