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Multiplicative structures on comodules in higher categories

Takeshi Torii

TL;DR

This work extends multiplicative structures to comodules over bialgebras in the setting of $\infty$-categories by developing duoidal and mixed-duoidal frameworks, and by constructing generalized bimodule and right module objects that lift $\mathcal{O}$-monoidal structures to categories of (co)modules. The central result shows that for an $(\mathcal{O},\mathbf{Ass})$-bialgebra $\Gamma$ in ${}_A\mathrm{BMod}_A(\mathcal{M})$, where $\mathcal{M}$ is an $(\mathcal{O}\times\mathbf{Ass})$-monoidal $\infty$-category and $A$ an $(\mathcal{O}\times\mathbf{Ass})$-algebra, the right $\Gamma$-comodules $\mathrm{Rcomod}_{(A,\Gamma)}(\mathcal{M})$ form an $\mathcal{O}$-monoidal $\infty$-category with a strong $\mathcal{O}$-monoidal forgetful functor. The paper also develops parallel theories for bicomodules and right modules, and analogously for right comodules via the RM operad, providing a coherent, higher-algebraic framework for multiplicative structures on a broad class of (co)modules. These results lay foundational groundwork for applying higher category techniques to multiplicative phenomena in homotopy theory, representation theory, and related fields.

Abstract

In this paper we study multiplicative structures on comodules over bialgebras in the setting of $\infty$-categories. We show that the $\infty$-category of comodules over an $(\mathcal{O},\mathbf{Ass})$-bialgebra in a mixed $(\mathcal{O},\mathbf{Ass})$-duoidal $\infty$-category has the structure of an $\mathcal{O}$-monoidal $\infty$-category for any $\infty$-operad $\mathcal{O}$.

Multiplicative structures on comodules in higher categories

TL;DR

This work extends multiplicative structures to comodules over bialgebras in the setting of -categories by developing duoidal and mixed-duoidal frameworks, and by constructing generalized bimodule and right module objects that lift -monoidal structures to categories of (co)modules. The central result shows that for an -bialgebra in , where is an -monoidal -category and an -algebra, the right -comodules form an -monoidal -category with a strong -monoidal forgetful functor. The paper also develops parallel theories for bicomodules and right modules, and analogously for right comodules via the RM operad, providing a coherent, higher-algebraic framework for multiplicative structures on a broad class of (co)modules. These results lay foundational groundwork for applying higher category techniques to multiplicative phenomena in homotopy theory, representation theory, and related fields.

Abstract

In this paper we study multiplicative structures on comodules over bialgebras in the setting of -categories. We show that the -category of comodules over an -bialgebra in a mixed -duoidal -category has the structure of an -monoidal -category for any -operad .

Paper Structure

This paper contains 19 sections, 27 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. $\infty$-operads
  3. $\infty$-operads over perfect operator categories
  4. Monoidal $\infty$-categories
  5. Sections of two-variable fibrations
  6. Algebra objects over products of $\infty$-operads
  7. Duoidal $\infty$-categories and bialgebras
  8. Mixed duoidal $\infty$-categories
  9. Bialgebras in mixed duoidal $\infty$-categories
  10. Multiplicative structures on bicomodules
  11. The nonsymmetric generalized $\infty$-operad $\mathrm{BM}$
  12. Looping construction of bimodules
  13. Multiplicative structures on bi(co)modules
  14. Bicomodules in $(\mathcal{O}\times\mathbf{Ass})$-monoidal $\infty$-categories
  15. Multiplicative structures on right comodules
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}^{\otimes}$ be an $\infty$-operad over a perfect operator category. Suppose that $\mathcal{M}$ is an $(\mathcal{O}\times\mathbf{Ass})$-monoidal $\infty$-category in which $\mathcal{M}$ has geometric realizations and the tensor products preserve them separately in each variable, and t

Theorems & Definitions (50)

  • Theorem 1.1: cf. Theorem \ref{['thm:right-comodule-main-theorem']}
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: cf. CH
  • Lemma 2.5
  • Proposition 2.6
  • Definition 3.1: cf. Torii1
  • Remark 3.2
  • Definition 3.3: cf. Torii1 and Torii2
  • ...and 40 more