Monoidal Grothendieck construction

Abstract

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this 'global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a 'fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems. Contents 1. Introduction 1 2. Preliminaries 3 3. The Monoidal Grothendieck Construction 13 4. (Co)cartesian case: fibrewise and global monoidal structures 23 5. Examples

Structured versus Decorated Cospans

“…Proof. This was shown by Moeller and the third author [37,Theorems 3.13 & 4.2]. In summary, for a cocartesian base A we have correspondences…”

“…where the composite of the bottom two arrows is the coproduct (20). The left-hand isomorphism comes from pseudonaturality of φ as in (37), whereas the right-hand isomorphism follows from universal properties and pseudofunctoriality of F. Thus, we have a globular 2-isomorphism…”

“…In [37], the classical Grothendieck construction is generalized to the monoidal setting: given a (symmetric) monoidal category A, there is a bijection between (symmetric) monoidal opindexed categories, which are (symmetric) lax monoidal pseudofunctors F : (A, ⊗ A ) → (Cat, ×), and (symmetric) monoidal opfibrations, which are opfibrations U : X → A where X, A are (symmetric) monoidal, U is (symmetric) strict monoidal, and ⊗ X preserves cocartesian liftings. This bijection sends a monoidal opindexed category F : (A, ⊗ A ) → (Cat, ×) to the monoidal opfibration U : ∫ F → A.…”

“…In that case, opfibrations (X, +) → (A, +) that strictly preserve coproducts and initial object bijectively correspond to pseudofunctors into the 2category of cocartesian categories. For more details, see [37,Corollary 4.7] and the related discussion.…”

Graphical Regular Logic

“…Note that Poset is a subcategory of Cat. This allows us to take the monoidal Grothendieck construction P of P : FRg(T) → Poset, [MV18]. A P -graphical term is an object in the comma category P ↓FRg(T).…”