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Constructing monads from cubical diagrams and homotopy colimits

Kristine Bauer, Robyn Brooks, Kathryn Hess, Brenda Johnson, Julie Rasmusen, Bridget Schreiner

Abstract

This paper is the first step in a general program for defining cocalculus towers of functors via sequences of compatible monads. Goodwillie's calculus of homotopy functors inspired many new functor calculi in a wide range of contexts in algebra, homotopy theory and geometric topology. Recently, the third and fourth authors have developed a general program for constructing generalized calculi from sequences of compatible comonads. In this paper, we dualize the first step of the Hess-Johnson program, focusing on monads rather than comonads. We consider categories equipped with an action of the poset category $\mathcal{P}(n)$, called $\mathcal{P}(n)$-modules. We exhibit a functor from $\mathcal{P}(n)$-modules to the category of monads. The resulting monads act on categories of functors whose codomain is equipped with a suitable notion of homotopy colimits. In the final section of the paper, we demonstrate the monads used to construct McCarthy's dual calculus as an example of a monad arising from a $\mathcal{P}(n)$-module. This confirms that our dualization of the Hess-Johnson program generalizes McCarthy's dual calculus, and serves as a proof of concept for further development of this program.

Constructing monads from cubical diagrams and homotopy colimits

Abstract

This paper is the first step in a general program for defining cocalculus towers of functors via sequences of compatible monads. Goodwillie's calculus of homotopy functors inspired many new functor calculi in a wide range of contexts in algebra, homotopy theory and geometric topology. Recently, the third and fourth authors have developed a general program for constructing generalized calculi from sequences of compatible comonads. In this paper, we dualize the first step of the Hess-Johnson program, focusing on monads rather than comonads. We consider categories equipped with an action of the poset category , called -modules. We exhibit a functor from -modules to the category of monads. The resulting monads act on categories of functors whose codomain is equipped with a suitable notion of homotopy colimits. In the final section of the paper, we demonstrate the monads used to construct McCarthy's dual calculus as an example of a monad arising from a -module. This confirms that our dualization of the Hess-Johnson program generalizes McCarthy's dual calculus, and serves as a proof of concept for further development of this program.
Paper Structure (10 sections, 10 theorems, 82 equations)

This paper contains 10 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Monads and Comonads
  5. $(\mathcal{P}(n),\cap,\underline{n})$-modules
  6. Systems of homotopy colimits
  7. Main theorem
  8. A model for iterated homotopy cofibers
  9. Obtaining monads from modules
  10. Recovering dual calculus

Key Result

Lemma 2.7

For any adjunction $L\dashv R:\mathscr{A}\to\mathscr{B}$ and any monad $\mathbb{T}=(T,\mu,\eta)$ on $\mathscr{B}$, the composite $R T L$ underlies a monad on $\mathscr{A}$, which we denote $R\mathbb{T} L$.

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Lemma 2.9
  • ...and 29 more