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1. Green 2-functors

Author

Dell'Ambrogio, Ivo, Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Applied Mathematics, General Mathematics, 20J05, 20C99, 18B40, 18N10, 19A22, 55P91, K-Theory and Homology (math.KT), Mathematics - Category Theory, Group Theory (math.GR), Mathematics::Algebraic Topology, Mathematics::K-Theory and Homology, Mathematics::Category Theory, Mathematics - K-Theory and Homology, FOS: Mathematics, Category Theory (math.CT), Representation Theory (math.RT), [MATH]Mathematics [math], Mathematics - Group Theory, Mathematics - Representation Theory
Abstract

We extend the theory of Mackey 2-functors introduced in arXiv:1808.04902 by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical Green functors familiar from representation theory and topology arise by decategorification, in various ways, of some Green 2-functor occurring in Nature., 46 pages. A few remaining typos corrected in journal proofs. Final version, to appear in Trans. AMS

Published
2022
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2. An introduction to Mackey and Green 2-functors

Author

Dell'Ambrogio, Ivo, Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Mathematics - K-Theory and Homology, FOS: Mathematics, K-Theory and Homology (math.KT), Group Theory (math.GR), Representation Theory (math.RT), [MATH]Mathematics [math], 20J05, 18B40, 18N10, 19A22, Mathematics - Group Theory, Mathematics - Representation Theory
Abstract

For half a century, Mackey and Green functors have been successfully used to model the induction and restriction maps which are ubiquitous in the representation theory of finite groups. In the examples, the latter maps are typically distilled in some way from induction and restriction functors between additive categories. This naturally leads to the notions of Mackey and Green 2-functors and to axioms which better capture this underlying layer of information. Such structures have been used in algebra, geometry and topology for a long time. We survey examples and applications of this young (yet arguably overdue) theory., 23 pages

Published
2023

3. Properadic Homotopical Calculus

Author

Bruno Vallette, Johan Leray, Eric Hoffbeck, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut universitaire de France, DIM Math Innov – Région Île de France, and ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016)

Subjects
General Mathematics, Mathematics::Algebraic Topology, 01 natural sciences, Mathematics::K-Theory and Homology, Mathematics::Quantum Algebra, Mathematics::Category Theory, Mathematics - Quantum Algebra, 0103 physical sciences, FOS: Mathematics, Calculus, medicine, Quantum Algebra (math.QA), Category Theory (math.CT), [MATH]Mathematics [math], 0101 mathematics, Calculus (medicine), 18D50, 18G55, 16T10, 17B62, [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT], Mathematics, Series (mathematics), Homotopy, 010102 general mathematics, Mathematics - Category Theory, medicine.disease, Transfer (group theory), [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], 010307 mathematical physics
Abstract

In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a series, we generalise the seminal notion of infini-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak--Fukaya--Latschev and we produce new explicit formulas., Submitted version

Published
2020
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4. Homology of infinity-operads

Author

Hoffbeck, Eric, Moerdijk, Ieke, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Utrecht University [Utrecht], and ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016)

Subjects
Mathematics::K-Theory and Homology, Mathematics::Category Theory, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Category Theory, Category Theory (math.CT), Mathematics - Algebraic Topology, 18N70, 55N35, 18M70, Mathematics::Algebraic Topology, [MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
Abstract

In a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define an adjoint bar-cobar (or "Koszul") duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of trees defining infinity-operads. We emphasize that our methods are completely elementary and explicit., Comment: 25 pages

Published
2021

5. On Krull-Schmidt bicategories

Author

Dell'Ambrogio, Ivo, Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Mathematics::Category Theory, Mathematics::Rings and Algebras, FOS: Mathematics, Mathematics - Category Theory, Category Theory (math.CT), [MATH]Mathematics [math], Representation Theory (math.RT), 20J05, 18B40, 18N10, 18N25, Mathematics - Representation Theory
Abstract

We study the existence and uniqueness of direct sum decompositions in additive bicategories. We find a simple definition of Krull-Schmidt bicategories, for which we prove the uniqueness of decompositions into indecomposable objects as well as a characterization in terms of splitting of idempotents and properties of 2-cell endomorphism rings. Examples of Krull-Schmidt bicategories abound, with many arising from the various flavors of 2-dimensional linear representation theory., Comment: 21 pages, final version. Several examples added

Published
2021
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6. Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis

Author

Bérénice Delcroix-Oger, Joan Bellier-Millès, Eric Hoffbeck, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, ANR-13-BS02-0005,CATHRE,Catégories, Homotopie et Réécriture(2013), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Université Sorbonne Paris Cité (USPC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Rewriting, General Mathematics, 01 natural sciences, Mathematics::Algebraic Topology, symbols.namesake, Mathematics::K-Theory and Homology, Converse, Partition poset, FOS: Mathematics, Algebraic Topology (math.AT), Partition (number theory), Operad, 0101 mathematics, 18D50, 06A11, 16S15, Link (knot theory), Mathematics, Mathematics::Commutative Algebra, Applied Mathematics, 010102 general mathematics, Basis (universal algebra), [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Poincaré conjecture, symbols, 2010 MSC: 18D50, 06A11, 16S15, Counterexample
Abstract

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré-Birkhoff-Witt basis. Furthermore, we give counter-examples to the converse., In this version, we have removed the part on a potential EL-labelling on the poset associated to the operad Perm because of an error reported to us by Rafael Gonz\'alez, Josh Hallam and Yeison Quiceno

Published
2021
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7. Cohomological Mackey 2-functors

Author

Paul Balmer, Ivo Dell’Ambrogio, Department of Mathematics [UCLA], University of California [Los Angeles] (UCLA), University of California (UC)-University of California (UC), Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Mathematics::K-Theory and Homology, General Mathematics, Mathematics::Category Theory, FOS: Mathematics, Mathematics - Category Theory, Category Theory (math.CT), Group Theory (math.GR), [MATH]Mathematics [math], Representation Theory (math.RT), Mathematics - Group Theory, Mathematics::Algebraic Topology, Mathematics - Representation Theory
Abstract

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in arXiv:1808.04902, obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory., Comment: A few remaining typos corrected in journal proofs. 28 pages, final version, to appear in J. Inst. Math. Jussieu

Published
2021
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8. Mackey 2-Functors and Mackey 2-Motives

Author

Paul Balmer, Ivo Dell'Ambrogio, Department of Mathematics [UCLA], University of California [Los Angeles] (UCLA), University of California (UC)-University of California (UC), Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
010102 general mathematics, 0103 physical sciences, 010307 mathematical physics, 0101 mathematics, [MATH]Mathematics [math], 01 natural sciences
Abstract

International audience

Published
2020
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9. On the comparison of spans and bisets

Author

Dell'Ambrogio, Ivo, Huglo, James, Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Mathematics::K-Theory and Homology, Mathematics::Category Theory, FOS: Mathematics, Category Theory (math.CT), Mathematics - Category Theory, Group Theory (math.GR), [MATH]Mathematics [math], Representation Theory (math.RT), Mathematics::Algebraic Topology, Mathematics - Group Theory, 18A25, 18B40, 18C15, 18D05, 18D10, 20J05, 16B50, Mathematics - Representation Theory
Abstract

We compare the bicategory of spans with that of bisets (a.k.a. bimodules, distributors, profunctors) in the context of finite groupoids. We construct in particular a well-behaved pseudo-functor from spans to bisets. This yields an application to the axiomatic representation theory of finite groups, namely a new proof and a strengthening of Ganter and Nakaoka's identification of Bouc's category of biset functors as a reflective subcategory of global Mackey functors. To this end, we also prove a tensor-monadicity result for linear functor categories., 28 pages. Final version, to appear in Cah. Topol. G\'eom. Differ. Cat\'eg

Published
2020

10. Green equivalences in equivariant mathematics

Author

Paul Balmer, Ivo Dell'Ambrogio, Department of Mathematics [UCLA], University of California [Los Angeles] (UCLA), University of California (UC)-University of California (UC), Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)

Subjects
Pure mathematics, General Mathematics, 01 natural sciences, Representation theory, Mathematics::Algebraic Topology, Mathematics - Algebraic Geometry, Mathematics::K-Theory and Homology, Mathematics::Category Theory, 0103 physical sciences, FOS: Mathematics, Algebraic Topology (math.AT), Category Theory (math.CT), Mathematics - Algebraic Topology, 0101 mathematics, Representation Theory (math.RT), [MATH]Mathematics [math], Equivalence (measure theory), Algebraic Geometry (math.AG), Topology (chemistry), Mathematics, 010102 general mathematics, Mathematics - Category Theory, Stable homotopy theory, Equivariant map, 010307 mathematical physics, Mathematics - Representation Theory
Abstract

We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry., 25 pages. References updated, final version. To appear in Math. Ann

Published
2020
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11. Protoperads II: Koszul duality

Author

Johan Leray, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016)

Subjects
Pure mathematics, Koszul duality, General Mathematics, Homotopy, 010102 general mathematics, Diagonal, 0102 computer and information sciences, Type (model theory), Adjunction, 01 natural sciences, Mathematics::Algebraic Topology, 010201 computation theory & mathematics, Mathematics::K-Theory and Homology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Mathematics::Category Theory, Lie algebra, Homogeneous space, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, 0101 mathematics, Commutative property, Mathematics
Abstract

In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad DLie. As a corollary, we deduce that the properad DPois which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in non commutative geometry.

Published
2019

12. Protoperads I: Combinatorics and definitions

Author

Johan Leray, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016)

Subjects
Mathematics::Category Theory, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology
Abstract

This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced S-modules. We study the associated monoids, called protoperads, which are a generalization of operads. As operads encode algebraic operations with several inputs and one outputs, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and show that there exists a notion of free protoperad. We also show that the monoidal product introduced here is related to Vallette's one on the category of S-bimodules, via the induction functor.

Published
2019

13. Multiplicative persistent distances

Author

Ginot, Grégory, Leray, Johan, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016)

Subjects
Mathematics::K-Theory and Homology, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], Mathematics::Algebraic Topology
Abstract

We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In particular, we prove that there exists a persistent A-infinity-structure associated to data sets and and we define the associated distance. We prove the stability of these new distances for Cech or Vietoris Rips complexes with respect to the Gromov-Hausdorff distance, and we compare these new distances with each other and the classical one, building some examples which prove that they are not equal in general and refine effectively the classical bottleneck distance.

Published
2019
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14. Goodwillie's calculus of functors and higher topos theory

Author

Anel, Mathieu, Biedermann, Georg, Finster, Eric, Joyal, André, Carnegie Mellon University [Pittsburgh] (CMU), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Gallinette : vers une nouvelle génération d'assistant à la preuve (GALLINETTE), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des Sciences du Numérique de Nantes (LS2N), IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique Bretagne-Pays de la Loire (IMT Atlantique), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS), Design, study and implementation of languages for proofs and programs (PI.R2), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université du Québec à Montréal = University of Québec in Montréal (UQAM), European Project: 637339,H2020 ERC,ERC-2014-STG,CoqHoTT(2015), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Gallinette : vers une nouvelle génération d'assistant à la preuve (LS2N - équipe GALLINETTE), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-École Centrale de Nantes (ECN)-Centre National de la Recherche Scientifique (CNRS)-IMT Atlantique (IMT Atlantique), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), ANR-16-CE40-0003,ChroK,Homotopie chromatique et K-théorie(2016), Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Cité (USPC)-Université Paris 13 (UP13)-Institut Galilée-Université Paris 8 Vincennes-Saint-Denis (UP8), Université de Nantes - Faculté des Sciences et des Techniques, Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Nantes - Faculté des Sciences et des Techniques, and Université du Québec à Montréal (UQAM)

Subjects
Mathematics::K-Theory and Homology, 55P65, 18F05, Mathematics::Category Theory, [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], FOS: Mathematics, Algebraic Topology (math.AT), Category Theory (math.CT), Mathematics - Category Theory, Mathematics - Algebraic Topology, Mathematics::Algebraic Topology, ComputingMilieux_MISCELLANEOUS
Abstract

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors., 40 pages, (a slightly modified version of) this paper is accepted for publication by the Journal of Topology

Published
2018
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