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Cofibrantly generated model structures for functor calculus

Lauren Bandklayder, Julia E. Bergner, Rhiannon Griffiths, Brenda Johnson, Rekha Santhanam

TL;DR

This work develops a general criterion for when functor-calculus–driven model structures on categories of simplicial functors are cofibrant generation localizations. By employing a Bousfield endofunctor $Q$ and a flexible collection of test morphisms, the authors provide a unified route to construct and certify cofibrantly generated localizations, including the hf (homotopy functor), $n$-excisive, and discrete degree-$n$ calculi. They recover known structures of Goodwillie calculus and discrete calculus, and supply a systematic cofibrant-generation framework that extends to new calculi. The results facilitate comparisons across different functor calculi and pave the way for applying these techniques to additional towers built from comonads or similar endofunctors. Overall, the paper strengthens the methodological foundation for constructing and analyzing functor-calculus model structures in a broad, unified setting.

Abstract

Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this approach are cofibrantly generated. Our examples recover the homotopy functor and $n$-excisive model structures of Biedermann and Röndigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.

Cofibrantly generated model structures for functor calculus

TL;DR

This work develops a general criterion for when functor-calculus–driven model structures on categories of simplicial functors are cofibrant generation localizations. By employing a Bousfield endofunctor and a flexible collection of test morphisms, the authors provide a unified route to construct and certify cofibrantly generated localizations, including the hf (homotopy functor), -excisive, and discrete degree- calculi. They recover known structures of Goodwillie calculus and discrete calculus, and supply a systematic cofibrant-generation framework that extends to new calculi. The results facilitate comparisons across different functor calculi and pave the way for applying these techniques to additional towers built from comonads or similar endofunctors. Overall, the paper strengthens the methodological foundation for constructing and analyzing functor-calculus model structures in a broad, unified setting.

Abstract

Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this approach are cofibrantly generated. Our examples recover the homotopy functor and -excisive model structures of Biedermann and Röndigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.
Paper Structure (15 sections, 41 theorems, 45 equations)

This paper contains 15 sections, 41 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgements
  4. Background on simplicial model categories
  5. Background on functor calculus
  6. Goodwillie calculus
  7. Discrete functor calculus
  8. Polynomial approximations for simplicial functors
  9. Bousfield's $Q$-Theorem
  10. The homotopy functor model structure
  11. The $n$-excisive model structure
  12. The degree $n$ model structure
  13. Cofibrant generation for Bousfield $Q$-model structures
  14. Cofibrant generation and the $n$-excisive model structure
  15. Cofibrant generation and discrete functor calculus

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a right proper model category together with an endofunctor $Q \colon \mathcal{M} \rightarrow \mathcal{M}$ and a natural transformation $\eta \colon \operatorname{id} \Rightarrow Q$ satisfying the following axioms: Then there exists a right proper model structure, denoted by $\mathcal{M}_Q$, on the same underlying category as $\mathcal{M}$ with the same cofibrations as $\mathc

Theorems & Definitions (97)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Proposition 2.7
  • Theorem 2.9
  • ...and 87 more