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Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem

Jaehyeok Lee

TL;DR

This work generalizes the Eilenberg–Watts theorem to the enriched setting inside a Bénabou cosmos $\mathcal{V}$ and connects it to six-functor formalisms by proving an equivalence between $\text{Comm}_{b\otimes b'}$ and the coreflective subcategory of cocontinuous lax monoidal $\mathcal{V}$-enriched functors $\textit{Mod}_b\to\textit{Mod}_{b'}$. It develops an enriched Eilenberg–Watts theorem, introduces six functors for morphisms $f:b\to b'$, and establishes a main adjoint equivalence that specializes to the classical bimodule description when $\mathcal{C}=\textit{Mod}_{b'}$. An alternative approach via Day convolution shows the same equivalence as commutative monoids in the convolution category, reinforcing the tensor-enriched perspective. The results unify algebraic, geometric, and enriched-categorical viewpoints, enabling tensorial and six-functor style formalisms to be studied in a broad, internal setting beyond ordinary schemes. This paves the way for tensor-enriched geometric frameworks over arbitrary cosmoi $\mathcal{V}$ and deepens the role of enrichment in bridging algebra and geometry.

Abstract

Let $b$, $b'$ be commutative monoids in a Bénabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over $b\otimes b'$ is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over $b$, $b'$.

Tensor Enriched Categorical Generalization of the Eilenberg-Watts Theorem

TL;DR

This work generalizes the Eilenberg–Watts theorem to the enriched setting inside a Bénabou cosmos and connects it to six-functor formalisms by proving an equivalence between and the coreflective subcategory of cocontinuous lax monoidal -enriched functors . It develops an enriched Eilenberg–Watts theorem, introduces six functors for morphisms , and establishes a main adjoint equivalence that specializes to the classical bimodule description when . An alternative approach via Day convolution shows the same equivalence as commutative monoids in the convolution category, reinforcing the tensor-enriched perspective. The results unify algebraic, geometric, and enriched-categorical viewpoints, enabling tensorial and six-functor style formalisms to be studied in a broad, internal setting beyond ordinary schemes. This paves the way for tensor-enriched geometric frameworks over arbitrary cosmoi and deepens the role of enrichment in bridging algebra and geometry.

Abstract

Let , be commutative monoids in a Bénabou cosmos. Motivated by six-functor formalisms in algebraic geometry, we prove that the category of commutative monoids over is equivalent to the category of cocontinuous lax monoidal enriched functors between the monoidal enriched categories of right modules over , .
Paper Structure (12 sections, 13 theorems, 81 equations)

This paper contains 12 sections, 13 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eilenberg-Watts theorem in the enriched setting
  4. The six functors
  5. Main result
  6. An approach via Day convolution
  7. Appendix
  8. Proposition \ref{['prop SixFunctors MbtensorCcat']}
  9. Proposition \ref{['prop SixFunctors laxtensorCadj']}
  10. Lemma \ref{['lem MainResult MainThm rightadj']}
  11. Proposition \ref{['prop MainResult lambdaF tensornat']}
  12. Proposition \ref{['prop DayConvolution monoidalproduct']}

Key Result

Lemma 2.1

Let $\mathcal{C}$, $\mathcal{D}$ be monoidal enriched categories and let be an enriched adjunction with unit, counit enriched natural transformations $\eta$, $\varepsilon$. Suppose the right enriched adjoint $\mathcal{G}:\mathcal{D}\to\mathcal{C}$ has a structure of a lax monoidal enriched functor.

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Proposition 4.1
  • ...and 15 more