arXiv subjects

Max Demirdilek

Publications and source records attributed to Max Demirdilek.

3 recordsLinked to original sources

arXiv2026

Linearly distributive coherence in the absence of units

Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories carry two tensor products, with structural morphisms given by associators and distributors relating the two tensor products. In several examples, including Grothendieck--Verdier categories, also known as $\ast$-autonomous categories, these distributors need not be invertible. We give a self-contained proof that linearly distributive categories without units are coherent, while units may obstruct coherence. With the same techniques, we also establish an analogous coherence result for Frobenius linearly distributive functors without units. These results admit a reformulation in terms of directed paths in associahedra and multiplihedra.

math.CO↗

arXiv2026

Grothendieck-Verdier functors

We introduce Grothendieck-Verdier functors between Grothendieck-Verdier, or $\ast$-autonomous, categories. Such functors are lax monoidal functors equipped with a morphism expressing compatibility with Grothendieck-Verdier duality. We show that the resulting $2$-category is $2$-equivalent to that of linearly distributive categories with negation and Frobenius linearly distributive functors. We further extend this $2$-equivalence to the braided setting. We then establish a lifting theorem for Grothendieck-Verdier functors: given a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, we identify additional structure such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$. This structure turns the functor into a Grothendieck-Verdier functor. As applications, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras.

math.CT↗

arXiv2025

Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories

Grothendieck-Verdier categories (also known as $\ast$-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras. In this paper, we develop a surface-diagrammatic calculus for Grothendieck-Verdier categories, extending the string-diagrammatic calculus of Joyal and Street for rigid monoidal categories into a third dimension. This extension naturally arises from the non-invertibility of coherence data in Grothendieck-Verdier categories. We show that key properties of Frobenius algebras in rigid monoidal categories carry over to the Grothendieck-Verdier setting. Moreover, we introduce higher Frobenius-Schur indicators for suitably finite $k$-linear pivotal Grothendieck-Verdier categories and prove their invariance under pivotal Frobenius linearly distributive equivalences. The proofs are carried out using the surface-diagrammatic calculus. To facilitate the verification of some of our results, we provide auxiliary files for the graphical proof assistant homotopy$.$io.

math.CT↗