**Definitions**

**Transfors between 2-categories**

**Morphisms in 2-categories**

**Structures in 2-categories**

**Limits in 2-categories**

**Structures on 2-categories**

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**.

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:

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 β08, 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, itβs 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.

- Mike Shulman,
*Framed bicategories and monoidal fibrations*, Theory and Applications of Categories**20**18 (2008) 650β738 [tac:2018, arXiv:0706.1286]

Last revised on August 14, 2024 at 12:02:17. See the history of this page for a list of all contributions to it.