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Free monad sequences and extension operations

Christian Sattler

TL;DR

This work addresses extending maps in algebraic weak factorization systems by first developing a parametrized theory of free algebras, free monads, and free monoids in non-cocomplete settings through well-pointed and pointed endofunctors. It then shows how extension operations along a generating category $oldsymbol{I}$ can be lifted to the coalgebras of the left class via the algebraic small object argument, establishing a robust framework for algebraic-WFS extensions that leverages van Kampen colimits and pushout-stability. The results yield a constructive, modular method for transferring free-structure data and performing extensions in categories with pullbacks, including Grothendieck topoi, and have implications for model-categorical constructions where extension along trivial cofibrations or fibrations is desired. Overall, the paper provides a principled approach to functorially extending fibrations and related structures in non-cocomplete contexts, with explicit control over colimits via a parametrizing class ${ rak{M}}$.

Abstract

In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for free monad and monoid sequences. In the second part, we deal with the problem of functorially extending via pullback squares a category of maps along the category of coalgebras of an algebraic weak factorization system. This generalizes the classical problem of extending a class of maps along the left class of a weak factorization system in the sense of pullback squares where the vertical maps are in the chosen class and the bottom map is in the left class. Such situations arise in the context of model structures where one might wish to extend fibrations along trivial cofibrations. We derive suitable conditions for the algebraic analogue of weak saturation of the extension problem, using the results of the first part to reduce the technical burden.

Free monad sequences and extension operations

TL;DR

This work addresses extending maps in algebraic weak factorization systems by first developing a parametrized theory of free algebras, free monads, and free monoids in non-cocomplete settings through well-pointed and pointed endofunctors. It then shows how extension operations along a generating category can be lifted to the coalgebras of the left class via the algebraic small object argument, establishing a robust framework for algebraic-WFS extensions that leverages van Kampen colimits and pushout-stability. The results yield a constructive, modular method for transferring free-structure data and performing extensions in categories with pullbacks, including Grothendieck topoi, and have implications for model-categorical constructions where extension along trivial cofibrations or fibrations is desired. Overall, the paper provides a principled approach to functorially extending fibrations and related structures in non-cocomplete contexts, with explicit control over colimits via a parametrizing class .

Abstract

In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for free monad and monoid sequences. In the second part, we deal with the problem of functorially extending via pullback squares a category of maps along the category of coalgebras of an algebraic weak factorization system. This generalizes the classical problem of extending a class of maps along the left class of a weak factorization system in the sense of pullback squares where the vertical maps are in the chosen class and the bottom map is in the left class. Such situations arise in the context of model structures where one might wish to extend fibrations along trivial cofibrations. We derive suitable conditions for the algebraic analogue of weak saturation of the extension problem, using the results of the first part to reduce the technical burden.

Paper Structure

This paper contains 14 sections, 19 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Van Kampen colimits
  4. Categories with a class of maps
  5. Strong categories of functors, adjunctions, monads
  6. Functoriality of free algebras, monads, and monoids
  7. Wellpointed endofunctors
  8. Statements for pointed endofunctors
  9. Pointed objects in a monoidal category
  10. Proofs for pointed endofunctors
  11. Reviewing the algebraic small object argument
  12. Extension operations
  13. Extension operations
  14. Lifting extension operations

Key Result

proposition 2.2

Let $\Xi$ be a pullback-stable class of van Kampen colimiting cocones in ${\mathcal{E}}$. Then $\operatorname{cod} \colon {\mathcal{E}}^\to_{{\mathsf{cart}}} \to {\mathcal{E}}$ lifts colimits in $\Xi$, and ${\mathcal{E}}^\to_{{\mathsf{cart}}} \to {\mathcal{E}}^\to$ preserves these lifted colimits.

Theorems & Definitions (62)

  • definition 2.1
  • proposition 2.2
  • proof
  • definition 2.3
  • definition 2.4
  • definition 2.5
  • definition 2.6
  • corollary 2.7
  • proof
  • definition 2.8
  • ...and 52 more