arXiv subjects

Brenda Johnson

Publications and source records attributed to Brenda Johnson.

7 recordsLinked to original sources

arXiv2024

Constructing monads from cubical diagrams and homotopy colimits

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.

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arXiv2023

Cofibrantly generated model structures for functor calculus

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 R\"ondigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.

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arXiv2020

Enriched functor categories for functor calculus

In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.

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arXiv2017

Directional derivatives and higher order chain rules for abelian functor calculus

In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional derivatives of functions, we define a higher order directional derivative for functors of abelian categories. We show that our higher order directional derivative is related to the iterated partial directional derivatives of the second author and McCarthy by a Faà di Bruno style formula. We obtain a higher order chain rule for our directional derivatives using a feature of the cartesian differential category structure, and with this provide a formulation for the $n$th layers of the Taylor tower of a composition of functors $F\circ G$ in terms of the derivatives and directional derivatives of $F$ and $G$, reminiscent of similar formulations for functors of spaces or spectra by Arone and Ching. Throughout, we provide explicit chain homotopy equivalences that tighten previously established quasi-isomorphisms for properties of abelian functor calculus.

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arXiv2016

Combinatorial models for Taylor polynomials of functors

Goodwillie's calculus of homotopy functors associates a tower of polynomial approximations, the Taylor tower, to a functor of topological spaces over a fixed space. We define a new tower, the varying center tower, for functors of categories with a fixed initial object, such as algebras under a fixed ring spectrum. We construct this new tower using elements of the Taylor tower constructions of Bauer, Johnson, and McCarthy for functors of simplicial model categories, and show how the varying center tower differs from Taylor towers in terms of the properties of its individual terms and convergence behaviour. We prove that there is a combinatorial model for the varying center tower given as a pro-equivalence between the varying center tower and towers of cosimplicial objects; this generalizes Eldred's cosimplicial models for finite stages of Taylor towers. As an application, we present models for the de Rham complex of rational commutative ring spectra due to Rezk on the one hand, and Goodwillie and Waldhausen on the other, and use our result to conclude that these two models will be equivalent when extended to $E_{\infty}$-ring spectra.

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arXiv2014

Unbased calculus for functors to chain complexes

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring. Much of the construction of the Taylor tower for functors to spectra carries over to this context. However, one of the essential steps in the construction requires proving that a particular functor is part of a cotriple. For this, one needs to prove that certain identities involving homotopy limits hold up to isomorphism, rather than just up to weak equivalence. As the target category of chain complexes is not a simplicial model category, the arguments for functors to spectra need to be adjusted for chain complexes. In this paper, we take advantage of the fact that we can construct an explicit model for iterated fibers, and prove that the functor is a cotriple directly. We use related ideas to provide concrete infinite deloopings of the first terms in the resulting Taylor towers when evaluated at the initial object in the source category.

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arXiv2014

Cross effects and calculus in an unbased setting

We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lane's cross effects functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillie's tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of C_f.

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