nLab restriction (double category theory)

Idea

The notion of restrictions in a double category abstracts that of the classical operation of restriction of scalars of bimodules. Indeed, they are usually available in framed bicategories, which are formal analogoues to double categories of bimodules; and indeed restrictions in a double category of bimodules correspond to restrictions of scalars.

The dual notion is that of extension.

Definition

In a double category $\mathbb{D}$, the restriction of a loose arrow $p:A \nrightarrow B$ along tight morphisms $f:X\to A$, $g:Y \to B$ is a universal filling of the niche: That is, a 2-cell: such that every other 2-cell as below left factors as below right:

Properties

Admitting all restrictions is sufficient for a double category to be a framed bicategory. In fact, restrictions are given by cartesian maps for the functor $(L,R):\mahtbb{D}_1 \to \mathbb{D}_0 \times \mathbb{D}_0$, thus admitting all restrictions means to admit all cartesian lifts, making $(L,R)$ into a fibration.

As proven in Shulman ‘08, restrictions can be constructed out of companions and conjoints alone. Indeed, notice first that a companion of $f$ is equivalently given as the restriction $A(f,1)$, where $A$ denotes the unit loose arrow at $A$; and dually a conjoint of $f$ is the restriction $B(1,f)$. Moreover, it’s easy to see that (denoting by $\odot$ looseward composition):

$$(p \odot q)(f,g) = p(f,1) \odot q(1,g)$$

Thus we conclude

$$p(f,g) = (A \odot p \odot B)(f,g) = A(f,1) \odot p \odot B(1,g).$$

This can be used to see that admitting all restrictions is equivalent to admitting all extensions, since extensions can also be obtained by composing with companions and conjoints, and companions and conjoints are also special instances of extensions.

See also

• framed bicategory
• extension (double category theory)

References

• Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories 20 18 (2008) 650–738 [tac:2018, arXiv:0706.1286]