Algebraic weighted colimits

Abstract

In this thesis weighted colimits in 2-categories equipped with promorphisms are studied. Such colimits include most universal constructions with counits, like ordinary colimits in categories, weighted colimits in enriched categories, and left Kan extensions. In the first chapter we recall the notion of 2-categories equipped with promorphisms (also simply called equipments), that provide a coherent way of adding bimodule-like morphisms to a 2-category. In the second chapter we recall two ways of defining weighted colimits in equipments. Most important to us is their original definition, introduced by Wood, in equipments that are endowed with a closed structure. The second notion, of what we call pointwise weighted colimits, was introduced by Grandis and Par\'e. It requires no extra structure and generalises Street's notion of pointwise left Kan extensions in 2-categories. The main result of the second chapter gives a condition, on closed equipments, under which these two notions coincide. In the third chapter we consider monads on equipments. The main idea of this thesis, given in the fourth chapter, generalises the notions of lax and colax morphisms, of algebras over a 2-monad, to notions of lax and colax promorphisms, of algebras over a monad on an equipment. One of these, that of right colax promorphisms, is well suited to the construction of weighted colimits. In particular, given a monad T on an equipment K, we will show that T-algebras, colax T-morphisms and right colax T-promorphisms form a double category T-rcProm. Although weaker than equipments, double categories still allow definition of weighted colimits, and our main result states that the forgetful functor T-rcProm -> K lifts all weighted colimits whenever K is closed, under some mild conditions on T.