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Objective Mackey and Tambara functors via parametrized categories

Ross Street

TL;DR

This work develops a comprehensive, categorified framework for objective Mackey and Tambara functors via parametrized categories built from protocalibrations. By constructing spans $\mathrm{Spn}_{\mathscr R}{\mathscr C}$ and polynomials $\mathrm{Ply}{\mathscr E}$, and establishing a distributive law under compatibility, it shows how objective Mackey functors correspond to functors on spans with $\mathscr R$-cocompleteness, and how Tambara functors arise as structured, biproduct-preserving functors with a Day-convolution-like monoidal structure. The theory yields monoidal refinements to Green functors and provides concrete instances (e.g., Burnside functors, Mackey functors of Mackey functors) while connecting to classical Tambara theory for $G$-sets. Overall, the paper generalizes Lindner’s span approach to a rich bicategorical setting, enabling enriched, parametrized, and distributive-law–driven treatments of equivariant algebra in homotopy-theoretic contexts.

Abstract

The first word in the title is intended in a sense suggested by Lawvere and Schanuel whereby finite sets are objective natural numbers. At the objective level, the axioms defining abstract Mackey and Tambara functors are categorically familiar. The first step was taken by Harald Lindner in 1976 when he recognized that Mackey functors, defined as pairs of functors, were single functors with domain a category of spans. We define objective Mackey and objective Tambara functors as parametrized categories which have local finite products and satisfy some parametrized completeness and cocompleteness restriction. However, we can replace the original parametrizing base for objective Mackey functors by a bicategory of spans while the replacement for objective Tambara functors is a bicategory obtained by iterating the span construction; these iterated spans are polynomials. There is an objective Mackey functor of ordinary Mackey functors. We show that there is a distributive law relating objective Mackey functors to objective Tambara functors analogous to the distributive law relating abelian groups to commutative rings. We remark on hom enrichment matters involving the 2-category $\mathrm{Cat}_{+}$ of categories admitting finite coproducts and functors preserving them, both as a closed base and as a skew-closed base.

Objective Mackey and Tambara functors via parametrized categories

TL;DR

This work develops a comprehensive, categorified framework for objective Mackey and Tambara functors via parametrized categories built from protocalibrations. By constructing spans and polynomials , and establishing a distributive law under compatibility, it shows how objective Mackey functors correspond to functors on spans with -cocompleteness, and how Tambara functors arise as structured, biproduct-preserving functors with a Day-convolution-like monoidal structure. The theory yields monoidal refinements to Green functors and provides concrete instances (e.g., Burnside functors, Mackey functors of Mackey functors) while connecting to classical Tambara theory for -sets. Overall, the paper generalizes Lindner’s span approach to a rich bicategorical setting, enabling enriched, parametrized, and distributive-law–driven treatments of equivariant algebra in homotopy-theoretic contexts.

Abstract

The first word in the title is intended in a sense suggested by Lawvere and Schanuel whereby finite sets are objective natural numbers. At the objective level, the axioms defining abstract Mackey and Tambara functors are categorically familiar. The first step was taken by Harald Lindner in 1976 when he recognized that Mackey functors, defined as pairs of functors, were single functors with domain a category of spans. We define objective Mackey and objective Tambara functors as parametrized categories which have local finite products and satisfy some parametrized completeness and cocompleteness restriction. However, we can replace the original parametrizing base for objective Mackey functors by a bicategory of spans while the replacement for objective Tambara functors is a bicategory obtained by iterating the span construction; these iterated spans are polynomials. There is an objective Mackey functor of ordinary Mackey functors. We show that there is a distributive law relating objective Mackey functors to objective Tambara functors analogous to the distributive law relating abelian groups to commutative rings. We remark on hom enrichment matters involving the 2-category of categories admitting finite coproducts and functors preserving them, both as a closed base and as a skew-closed base.

Paper Structure

This paper contains 10 sections, 30 theorems, 87 equations.

Table of Contents

  1. Parametrized categories and protocalibrations
  2. ${\mathscr R}$-cocompletion
  3. ${\mathscr R}$-Spans and ${\mathscr R}$-cocompleteness
  4. A distributive law from compatibility
  5. Polynomials as spans of spans
  6. Finite coproducts for spans and polynomials
  7. Objective Mackey functors
  8. A monoidal structure on $\mathrm{OMky}_{{\mathscr R}}{\mathscr E}$
  9. An objective Mackey functor of Mackey functors
  10. Objective Tambara functors

Key Result

Proposition 1.1

In any finitely complete bicategory, suppose $p : E\to B$ is a groupoid opfibration. In diagram pseudoptwise, the 2-morphism $\sigma$ exhibits $k$ as a pointwise left extension of $f$ along $p$ if and only if, for all morphisms $b$ with the square a bicategorical pullback (= briefly, bipullback), th

Theorems & Definitions (67)

  • Proposition 1.1
  • Definition 1.2
  • Example 1.3
  • Proposition 1.4: Compare 13 Proposition 9.12, 17 Proposition 7.12
  • Remark 1.5
  • Lemma 1.6
  • proof
  • Corollary 1.7
  • proof
  • Proposition 1.8
  • ...and 57 more