[Paper Review] General Operads and Multicategories

Tom Leinster|ArXiv.org|Oct 8, 1998

Advanced Topics in Algebra11 references19 citations

TL;DR

This paper generalizes the theory of monoidal categories by introducing (S, ∗)-structured categories as category objects in the category of algebras for a monad ∗ on a cartesian category S. It establishes a monadic adjunction between (S, ∗)-multicategories and (S, ∗)-structured categories, extending the classical adjunction between multicategories and monoidal categories, and proves that this adjunction is monadic, providing a categorical framework for structured categories via universal algebra.

ABSTRACT

Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to certain conditions on S and *. Different choices of S and * give some of the existing notions of operad and multicategory. We then describe the `algebras' for an (S,*)-multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan's opetopes and Batanin's operads; both of these are included.

Motivation & Objective

• To generalize monoidal categories beyond the classical case (Sets, free monoid) to arbitrary cartesian categories (S, ∗).
• To define (S, ∗)-structured categories as category objects in S()∗, the category of algebras for the monad ∗ on S.
• To establish a monadic adjunction between (S, ∗)-multicategories and (S, ∗)-structured categories.
• To show that the forgetful functor from (S, ∗)-structured categories to (S, ∗)-multicategories is part of a monadic adjunction.
• To provide a categorical framework for structured categories using universal algebra and multicategory theory.

Proposed method

• Define (S, ∗)-structured categories as (S()∗, id)-multicategories, i.e., category objects in the category of ∗-algebras.
• Use the monad ∗ on S to induce a monad on S-Cat, defining (S, ∗)-structured categories as algebras for this monad.
• Construct a free functor F and forgetful functor U between (S, ∗)-multicategories and (S, ∗)-structured categories.
• Establish an adjunction F ⊣ U using canonical natural transformations ϕ and ψ, with P ⊣ Q as plain functors.
• Verify that the adjunction satisfies the conditions for being monadic, using the monadicity theorem.
• Apply the construction to the case (S, ∗) = (Sets, free monoid), recovering the classical adjunction between multicategories and monoidal categories.

Experimental results

Research questions

• RQ1 How can monoidal categories be generalized beyond the case of (Sets, free monoid) to arbitrary cartesian categories (S, ∗)?
• RQ2 What is the correct categorical definition of an (S, ∗)-structured category that generalizes monoidal categories?
• RQ3 Is there a monadic adjunction between (S, ∗)-multicategories and (S, ∗)-structured categories?
• RQ4 How do the free and forgetful functors between these categories behave, and what is their universal property?
• RQ5 What conditions ensure that the adjunction between (S, ∗)-multicategories and (S, ∗)-structured categories is monadic?

Key findings

• The paper constructs a monadic adjunction between (S, ∗)-multicategories and (S, ∗)-structured categories for any cartesian category (S, ∗).
• The adjunction is monadic, meaning that (S, ∗)-structured categories are equivalent to algebras for the monad induced by the free-forgetful adjunction.
• The forgetful functor from (S, ∗)-structured categories to (S, ∗)-multicategories is faithful but not full, indicating a strict categorical distinction between the two structures.
• The construction generalizes the classical adjunction between multicategories and monoidal categories, recovering it when (S, ∗) = (Sets, free monoid).
• The adjunction is defined using canonical natural transformations ϕ and ψ, and the unit and counit commute suitably with these, ensuring coherence.
• The framework provides a uniform categorical treatment of structured categories via universal algebra in the context of multicategories.