Constructing monoidal structures on fibered categories via factorizations

TL;DR

The paper develops a method to construct monoidal structures on fibered categories over a small category by using factorizations through subcategories generated by specific morphisms, with applications to perverse sheaves.

Contribution

It introduces a factorization approach to determine monoidal structures on fibered categories from their restrictions to subcategories, extending to morphisms and perverse sheaves.

Findings

Monoidal structures are uniquely determined by restrictions to subcategories.

The method applies to morphisms of fibered categories and their monoidality.

A variant of the factorization method is adapted for perverse sheaves.

Abstract

Let $\mathcal{S}$ be a small category, and suppose that we are given two (non-full) subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$ that generate all morphisms of $\mathcal{S}$ under composition in the same way as morphisms of quasi-projective algebraic varieties are generated by smooth morphisms and closed immersions. We show that a monoidal structure on a given $\mathcal{S}$-fibered category is completely determined by its restrictions to $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$; in fact, any such pair of monoidal structures satisfying a natural coherence condition uniquely determines a monoidal structure over $\mathcal{S}$. The same principle applies to morphisms of $\mathcal{S}$-fibered categories and monoidality thereof. Under further assumptions on the subcategories $\mathcal{S}^{sm}$ and $\mathcal{S}^{cl}$, and with suitable restrictions on the $\mathcal{S}$-fibered categories and morphisms involved, we provide a variant of the above factorization method in which inverse images under closed immersion are partially replaced by the corresponding direct images: the latter variant is more adapted to the setting of perverse sheaves.