Homotopy theory of enriched Mackey functors : closed multicategories, permutative enrichments, and algebraic foundations for spectral Mackey functors / Niles Johnson, the Ohio State University at Newark, Donald Yau, the Ohio State University at Newark.

Format:

Book

Author/Creator:

Johnson, Niles, author.

Yau, Donald Y. (Donald Ying), 1977- author.

Series:

London Mathematical Society lecture note series ; 492.

London Mathematical Society lecture note series ; 492

Language:

English

Subjects (All):

Functor theory.

Homotopy theory.

Categories (Mathematics).

Physical Description:

1 online resource (xxxix, 483 pages) : digital, PDF file(s).

Edition:

First edition.

Place of Publication:

Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2025.

Summary:

This work develops techniques and basic results concerning the homotopy theory of enriched diagrams and enriched Mackey functors. Presentation of a category of interest as a diagram category has become a standard and powerful technique in a range of applications. Diagrams that carry enriched structures provide deeper and more robust applications. With an eye to such applications, this work provides further development of both the categorical algebra of enriched diagrams, and the homotopy theoretic applications in K-theory spectra. The title refers to certain enriched presheaves, known as Mackey functors, whose homotopy theory classifies that of equivariant spectra. More generally, certain stable model categories are classified as modules - in the form of enriched presheaves - over categories of generating objects. This text contains complete definitions, detailed proofs, and all the background material needed to understand the topic. It will be indispensable for graduate students and researchers alike.

Contents:

Cover

Series page

Title page

Imprints page

Dedication

Contents

List of Main Facts

List of Notations

Preface

1 Motivations from Equivariant Topology

1.1 Equivariant Spaces and Presheaves on the Orbit Category

1.2 The Burnside Category and Abelian G-Mackey Functors

1.3 Equivariant Spectra and Presheaves on the Burnside 2-Category

1.4 Stable Model Categories as Spectral Presheaf Categories

Part 1 Background on Multicategories and K-Theory Functors

2 Categorically Enriched Multicategories

2.1 Multicategories

2.2 Pointed Multicategories

2.3 M1-Modules

2.4 Permutative Categories

3 Infinite Loop Space Machines

3.1 Homotopy Theories via Complete Segal Spaces

3.2 Category of Pointed Diagrams

3.3 Γ-Objects

3.4 G[sub(*)]-Objects

3.5 Segal and Elmendorf-Mandell K-Theory

4 Homotopy Theory of Multicategories

4.1 Free Permutative Categories

4.2 Free Permutative Category as a Left 2-Adjoint

4.3 Componentwise Right Adjoint of the Counit

4.4 Free Permutative Category as a Nonsymmetric Cat-Multifunctor

4.5 Homotopy Equivalences between Multicategories and Permutative Categories

Part 2 Homotopy Theory of Pointed Multicategories, M1-Modules, and Permutative Categories

5 Pointed Multicategories and M1-ModulesModel All Connective Spectra

5.1 Pointed Free Permutative Categories

5.2 Relating Unpointed and Pointed Free Permutative Categories

5.3 Pointed Free Permutative Category as a Left 2-Adjoint

5.4 Free Permutative Categories of M1-Modules

5.5 Examples of Pointed Free Permutative Categories

5.6 Componentwise Right Adjoint of the Pointed Adjunction

5.7 Homotopy Theory of Pointed Multicategories

5.8 Homotopy Theory of M1-Modules

6 Multiplicative Homotopy Theory of Pointed Multicategories and M1-Modules.

6.1 The Strong Multilinear Functor F[sup(n)][sub()]

6.2 Pointed Free Permutative Category as a Nonsymmetric Cat-Multifunctor

6.3 Comparison Transformations

6.4 Multiplicative Homotopy Theory of Pointed Multicategories

6.5 Multiplicative Homotopy Theory of M1-Modules

Part 3 Enrichment of Diagrams and Mackey Functors in Closed Multicategories

7 Multicategorically Enriched Categories

7.1 Enrichment in a Multicategory

7.2 Enrichment in an Endomorphism Multicategory

7.3 Enrichment in the Multicategory of Permutative Categories

7.4 Self-Enrichment of the Multicategory of Permutative Categories

7.5 Bilinear Evaluation for Permutative Categories

7.6 Opposite Enriched Categories

8 Change of Multicategorical Enrichment

8.1 Change of Enrichment along a Multifunctor

8.2 Preservation of Opposite Enriched Categories

8.3 Change of Enrichment along a Monoidal Functor

8.4 Composition of Change-of-Enrichment 2-Functors

8.5 2-Functoriality of Change of Enrichment

9 The Closed Multicategory of Permutative Categories

9.1 Closed Multicategories

9.2 Internal Hom Permutative Categories

9.3 Multicategorical Evaluation for Permutative Categories

9.4 The Closed Multicategory Structure

9.5 Closed Multicategories of Lax and Strong Multilinear Functors

10 Self-Enrichment and Standard Enrichment of Closed Multicategories

10.1 Self-Enrichment of Closed Multicategories

10.2 Standard Enrichment of a Multifunctor

10.3 Compositionality of Standard Enrichment

10.4 Factorization of K-Theory Standard Enrichment

11 Enriched Diagrams and Mackey Functors of Closed Multicategories

11.1 Enriched Diagrams and Mackey Functors as Modules

11.2 Change of Enrichment of Enriched Diagrams and Mackey Functors

11.3 Diagram and Mackey Functor Change-of-Enrichment Functors.

11.4 Composition of Diagram Change-of-Enrichment Functors

11.5 Spectral Mackey Functors from K-Theory

11.6 Spectral Mackey Functors from Multicategorical Mackey Functors

Part 4 Homotopy Theory of Enriched Diagrams and Mackey Functors

12 Homotopy Equivalences between Enriched Diagram and Mackey Functor Categories

12.1 Comparing Enriched Diagram and Mackey Functor Categories

12.2 Comparing E[sub(*)]F[sub(*)][sup(ξ)] and the Identity

12.3 Comparing F[sub(*)][sup(ξ)]E[sub(*)] and the Identity

12.4 Homotopy Equivalent Enriched Diagram and Mackey Functor Categories

13 Applications to Multicategories and Permutative Categories

13.1 Homotopy Equivalent Multicategorical and Permutative Enriched Diagrams

13.2 Permutative to Multicategorical Enriched Diagrams

13.3 Multicategorical to Permutative Enriched Diagrams

13.4 Homotopy Equivalent M1-Modules and Permutative Enriched Diagrams

13.5 Explanation of the Equivalences of Homotopy Theories

13.6 Homotopy Equivalent Multicategorical and M1-Modules Enriched Diagrams

13.7 Explanation of the Equivalences of Homotopy Theories

APPENDICES

Appendix A Categories

A.1 Monoidal Categories

A.2 2-Categories

Appendix B Enriched Category Theory

B.1 Enriched Categories

B.2 Enriched Monoidal Categories

B.3 Self-Enriched Symmetric Monoidal Categories

B.4 Change of Enrichment

Appendix C Multicategories

C.1 Enriched Multicategories

C.2 Categorically Enriched Multicategories

C.3 Endomorphism Multicategories

C.4 Pointed Multicategories

Appendix D Open Questions

Bibliography

Index.

Notes:

Title from publisher's bibliographic system (viewed on 10 Jan 2025).

Includes bibliographical references and index.

ISBN:

9781009519540

1009519549

9781009519564

1009519565