nLab
linear bicategory

Linear bicategories

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 bicategories with one object coincide with the deloopings of (non-symmetric) linearly distributive categories.
  • Any allegory whose hom-sets are Boolean algebras is a linear bicategory, with \otimes the usual composition and with XZ=¬(¬X ¬Y ) X \parr Z = \neg (\neg X^\circ \otimes \neg Y^\circ)^\circ. In particular, this includes the bicategory of relations in any Boolean category, such as Set (assuming classical logic).
  • Any ordinary bicategory can be regarded as a linear bicategory with =\otimes = \parr.
  • If BB is a linear bicategory (such as the delooping of a linearly distributive category) such that (B,,)(B,\otimes,\top) has local coproducts and (B,,)(B,\parr,\bot) has local products, then the bicategory Mat(B)Mat(B) (whose objects are families of objects of BB and whose morphisms are matrices of 1-cells in BB) is again a linear bicategory. For instance, if BB is the Boolean algebra 2\mathbf{2} of truth values, then Mat(2)Rel(Set)Mat(\mathbf{2}) \cong Rel(Set).

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

  • 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 07:04:34. See the history of this page for a list of all contributions to it.