Deriving calculus with cotriples

Johnson, Brenda; McCarthy, Randy

doi:10.1090/s0002-9947-03-03318-x

Cited by 32 publications

(96 citation statements)

References 21 publications

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“…Le résultat élémentaire suivant est classique, il suit par exemple de la démonstration de la proposition 1.6 de [24]. Lemme 6.8.…”

Section: Postcomposition Par Un Foncteur Polynomial Strictunclassified

Homological finiteness of functors and applications

Djament¹,

Touzé²

2021

Preprint

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Nous donnons des conditions suffisantes pour qu'un foncteur de longueur finie d'une catégorie additive vers des espaces vectoriels de dimensions finies possède une résolution projective dont les termes sont de type fini. Pour les foncteurs polynomiaux, nous étudions également une propriété de finitude homologique plus faible, qui s'applique à la stabilité homologique à coefficients tordus des monoïdes de matrices. Ces résultats s'inspirent de travaux de Schwartz et Betley-Pirashvili, qu'ils généralisent, et utilisent les décompositions à la Steinberg sur une catégorie additive que nous avons récemment obtenues avec Vespa. Nous montrons également, en guise d'application, une propriété de finitude pour l'homologie stable de groupes linéaires sur des anneaux appropriés.

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“…Le résultat élémentaire suivant est classique, il suit par exemple de la démonstration de la proposition 1.6 de [24]. Lemme 6.8.…”

Section: Postcomposition Par Un Foncteur Polynomial Strictunclassified

Homological finiteness of functors and applications

Djament¹,

Touzé²

2021

Preprint

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“…In this section we give a brief summary of main constructions of Goodwillie Calculus as developed by B.Johnson and R.McCarthy. For details we refer to [9], which is an application of [6], [7] and [8].…”

Section: Goodwillie Calculusmentioning

confidence: 99%

“…Since then Goodwillie Calculus has been used extensively and new methods were developed to provide alternative descriptions in different categories. In particular, in [9], B.Johnson and R.McCarthy introduce a construction using cotriples that produces a Taylor tower of functors from any pointed category with coproducts to the category of chain complexes Ch(K) over a commutative ring K.…”

Section: Introductionmentioning

confidence: 99%

“…In [9], the authors observed that exponential functors are defined as a solution to a rather elementary functional equation, which allowed them to compute the cross effects and consequently, the layers (or fibers) of Taylor towers of these functors.…”

Section: Introductionmentioning

confidence: 99%

“…This leads to some interesting applications of our computations. For example in [13], we follow the approach suggested in [9] and use the Taylor towers of exponential functors to derive the topological analogue (in framework of [3]) of the commutative to associative spectral sequence, also known as the fundamental spectral sequence for T HH.…”

Section: Introductionmentioning

confidence: 99%

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Functional equations and their related operads

Minasian

2005

Trans. Amer. Math. Soc.

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Abstract. Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little n-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of n-Poisson algebras in terms of the homology of configuration spaces.

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A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus

Osborne

Tebbe

2018

Association for Women in Mathematics Series

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One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when n = 1. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when n = 2. We describe how to obtain the second higher order directional derivative chain rule for abelian functors. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors.

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Deriving calculus with cotriples

Cited by 32 publications

References 21 publications

Homological finiteness of functors and applications

Homological finiteness of functors and applications

Functional equations and their related operads

A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus

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