# Completions and Terminal Monads

## TL;DR

This paper characterizes the terminal monad among those preserving subcategory objects and extends these ideas to infinity categories, providing a universal framework for homological completion towers.

## Contribution

It offers a new characterization of common monads as terminal objects in categories of co-augmented endo-functors and extends this to infinity categories for homological completions.

## Key findings

- Characterization of common monads as terminal objects in co-augmented endo-functor categories
- Extension of monad properties to infinity categories for homological completion
- Universal formulation of properties of homological completion towers

## Abstract

We consider the terminal monad among those preserving the objects of a subcategory, and in particular preserving the image of a monad. Several common monads are shown to be uniquely characterized by the property of being terminal objects in the category of co-augmented endo-functors. Once extended to infinity categories, this gives, for example, a complete characterization of the well-known Bousfield-Kan R-homology completion. In addition, we note that an idempotent pro-completion tower can be associated with any co-augmented endo functor M, whose limit is the terminal monad that preserves the closure of ImM, the image of M, under finite limits. We conclude that some basic properties of the homological completion tower of a space can be formulated and proved for general monads over any category with limits, and characterized as universal