Constructing monads from cubical diagrams and homotopy colimits
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929263929720832 |
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| author | Bauer, Kristine Brooks, Robyn Hess, Kathryn Johnson, Brenda Rasmusen, Julie Schreiner, Bridget |
| author_facet | Bauer, Kristine Brooks, Robyn Hess, Kathryn Johnson, Brenda Rasmusen, Julie Schreiner, Bridget |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing monads from cubical diagrams and homotopy colimits Bauer, Kristine Brooks, Robyn Hess, Kathryn Johnson, Brenda Rasmusen, Julie Schreiner, Bridget Algebraic Topology Category Theory 55P65, 18C15, 16D90 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. |
| title | Constructing monads from cubical diagrams and homotopy colimits |
| topic | Algebraic Topology Category Theory 55P65, 18C15, 16D90 |
| url | https://arxiv.org/abs/2403.01724 |