Equipments are categorical structures of dimension $2$ having two separate types of $1$-arrows -vertical and horizontal- and supporting restriction and extension of horizontal arrows along vertical ones. Equipments are $2$-functors satisfying certain conditions, but they can also be understood as double categories satisfying a fibrancy condition. In the zoo of $2$-dimensional categorical structures, they fit nicely between $2$-categories and double categories, and are generally considered as the $2$-dimensional categorical structures where synthetic category theory is done, and in reasonable cases, where monoidal bicategories are more naturally defined.

In this talk I discuss the problem of lifting a $2$-category into an equipment along a given category of vertical arrows, and how this problem allows us to define a notion of length on general double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct the double category from a bicategory along its set of vertical arrows. I will review the length of double categories, and I will discuss the crossed product of a decorated bicategory construction -a method for constructing different double categories from a given bicategory is presented-. One of the main ingredients of the construction are so-called canonical squares. I will show that in certain classes of equipments -fully faithful and absolutely dense- every square that can be canonical is indeed canonical. I will explain how from this, it can be concluded that fully faithful and absolutely dense equipments are of length 1, and so they can be 'easily' reconstructed from their horizontal bicategories.