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Completions and Terminal Monads

Emmanuel Dror Farjoun, Sergei O. Ivanov

TL;DR

The paper develops a universal framework in which completions and localizations are characterized as terminal monads: the terminal object among co-augmented endofunctors that preserve a designated subcategory or the image of a given monad. It provides explicit constructions, notably the equalizer formula $T_M \cong Eq(M \rightrightarrows M^2)$, and expresses terminal monads via structured double-duals and operadic hom-objects, enabling concrete computations. The authors extend these ideas to the $\infty$-categorical setting, connect to the Bousfield–Kan $R_\u2080$-completion, and develop a pro-idempotent completion tower that mirrors the classical Bousfield–Kan tower while generalizing to arbitrary categories with limits. The work unifies known completions (profinite, ultrafilter/Stone–Čech, nilpotent and $I$-adic, etc.) under a universal property, and introduces an operadic/monadic perspective that clarifies how completions interact with additional algebraic structures, with potential applications to homotopy theory, algebra, and beyond.

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

Completions and Terminal Monads

TL;DR

The paper develops a universal framework in which completions and localizations are characterized as terminal monads: the terminal object among co-augmented endofunctors that preserve a designated subcategory or the image of a given monad. It provides explicit constructions, notably the equalizer formula , and expresses terminal monads via structured double-duals and operadic hom-objects, enabling concrete computations. The authors extend these ideas to the -categorical setting, connect to the Bousfield–Kan -completion, and develop a pro-idempotent completion tower that mirrors the classical Bousfield–Kan tower while generalizing to arbitrary categories with limits. The work unifies known completions (profinite, ultrafilter/Stone–Čech, nilpotent and -adic, etc.) under a universal property, and introduces an operadic/monadic perspective that clarifies how completions interact with additional algebraic structures, with potential applications to homotopy theory, algebra, and beyond.

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
Paper Structure (20 sections, 26 theorems, 110 equations)

This paper contains 20 sections, 26 theorems, 110 equations.

Table of Contents

  1. Introduction and main results
  2. Examples:
  3. Acknowlegements
  4. A sample of results
  5. Outline of the rest of the paper
  6. -Completions
  7. Terminal monads associated with a functor
  8. in terms of
  9. The equalizer as the terminal monad.
  10. Example
  11. Explicit expressions for
  12. Remark.
  13. Examples in the category of sets and groups
  14. Variations on the set of ultrafilters.
  15. Examples: Groups and modules
  16. ...and 5 more sections

Key Result

Proposition 1.1

Let $\mathpzc{D}\subseteq \mathpzc{C}$ be a full subcategory of a category with limits. The co-density functor, or the $\mathpzc{D}$-completion, $T_\mathpzc{D}:\mathpzc{C}\to \mathpzc{C}$ is the terminal object in the category of co-augmented functors $Id\to F\in \mathpzc{C}^{\mathpzc{C}},$ such tha

Theorems & Definitions (53)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 43 more