# Lifting as a KZ-doctrine

## Abstract

In a cartesian closed category with an initial object and a dominance that classifies it, an *intensional* notion of approximation between maps —the *path relation* (c.f. link relation)— is defined. It is shown that if such a category admits *strict/upper-closed factorisations* then it preorderenriches (as a cartesian closed category) with respect to the path relation. By imposing further axioms we can, on the one hand, endow maps and proofs of their approximations (viz. *paths*) with the 2-dimensional algebraic structure of a sesqui-category and, on the other, characterise lifting as a preorder-enriched lax colimit. As a consequence of the latter the *lifting* (or partial map classifier) *monad* becomes a *KZ-doctrine*.

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