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.
1. Motivations from equivariant topology; Part I. Background on Multicategories and K-Theory Functors: 2. Categorically enriched multicategories; 3. Infinite loop space machines; 4. Homotopy theory of multicategories; Part II. Homotopy Theory of Pointed Multicategories, M1-Modules, and Permutative Categories: 5. Pointed multicategories and M1-modules model all connective spectra; 6. Multiplicative homotopy theory of pointed multicategories and M1-modules; Part III. Enrichment of Diagrams and Mackey Functors in Closed Multicategories: 7. Multicategorically enriched categories; 8. Change of multicategorical enrichment; 9. The closed multicategory of permutative categories; 10. Self-enrichment and standard enrichment of closed multicategories; 11. Enriched diagrams and Mackey functors of closed multicategories; Part IV. Homotopy Theory of Enriched Diagrams and Mackey Functors: 12. Homotopy equivalences between enriched diagram and Mackey functor categories; 13. Applications to multicategories and permutative categories; Appendices: A. Categories; B. Enriched category theory; C. Multicategories; D. Open questions; Bibliography; Index.
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