From *-Autonomous to Linearly Distributive via Enrichments and Strengths
Abstract
It is part of the folklore of category theory that every *-autonomous category is a linearly distributive category, but a full proof, with all coherence conditions verified, is not available in the literature. We give such a proof. The key observation is that the linear distributions, and the coherences they must satisfy, are strengths and strength coherences for the internal-hom functors of a biclosed monoidal category, arising from the correspondence between enrichments and strengths. Working with an arbitrary biclosed monoidal category V, we construct a canonical cotensor product on V , show that the two candidate linear distributions are composed strengths satisfying the interchange and dual interchange coherences, and that (V, \par, 0, α , λ , ρ ) is a skew monoidal category in the sense of Szlachányi and of Uustalu, Veltri and Zeilberger. When V is *-autonomous, the skew monoidal structure upgrades to a genuine monoidal structure, and we obtain that every *-autonomous category is a planar linearly distributive category, with no symmetry of ⊗ or cyclicity of the dualizing object required.
Related
- A continuous $3$-distributive frame that is not ω-distributive
- Posets of Hyper (b,t)-ary Partitions are Distributive Lattices
- Proper varieties of distributive l-pregroups are periodic
- Sequential and distributive dual futile cycle: Hopf bifurcation can occur under parameter-rich kinetics but cannot occur under mass action kinetics
- Rank identifications in join-meet ideals of thin distributive lattices
BibTeX
@misc{chichery2026from,
title = {{From *-Autonomous to Linearly Distributive via Enrichments and Strengths}},
author = {Gregory Chichery and Luigi Santocanale},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05770015v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - From *-Autonomous to Linearly Distributive via Enrichments and Strengths
AU - Gregory Chichery
AU - Luigi Santocanale
PY - 2026
DA - 2026/09/29
PB - HAL
UR - https://hal.science/hal-05770015v1
AB - It is part of the folklore of category theory that every *-autonomous category is a linearly distributive category, but a full proof, with all coherence conditions verified, is not available in the literature. We give such a proof. The key observation is that the linear distributions, and the coherences they must satisfy, are strengths and strength coherences for the internal-hom functors of a biclosed monoidal category, arising from the correspondence between enrichments and strengths. Working with an arbitrary biclosed monoidal category V, we construct a canonical cotensor product on V , show that the two candidate linear distributions are composed strengths satisfying the interchange and dual interchange coherences, and that (V, \par, 0, α , λ , ρ ) is a skew monoidal category in the sense of Szlachányi and of Uustalu, Veltri and Zeilberger. When V is *-autonomous, the skew monoidal structure upgrades to a genuine monoidal structure, and we obtain that every *-autonomous category is a planar linearly distributive category, with no symmetry of ⊗ or cyclicity of the dualizing object required.
ER -