[Paper Review] Enriched algebraic theories and monads for a system of arities

Rory B. B. Lucyshyn-Wright|arXiv (Cornell University)|Nov 9, 2015

Homotopy and Cohomology in Algebraic Topology20 references16 citations

TL;DR

This paper develops a general framework for enriched algebraic theories in symmetric monoidal closed categories using a system of arities $\mathscr{J} \hookrightarrow \mathscr{V}$, introducing the concept of $\mathscr{J}$-theories and proving that when $\mathscr{J}$ is eleutheric (i.e., presents $\mathscr{V}$ as a free cocompletion under certain Kan extensions), the category of $\mathscr{T}$-algebras exists and is monadic over $\mathscr{V}$, even without completeness or cocompleteness assumptions. The key contribution is a characterization of $\mathscr{J}$-theories as monads on a bicategory of profunctors and as $\mathscr{V}$-monads satisfying a colimit condition.

ABSTRACT

Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category $\mathcal{V}$ with respect to a specified system of arities $j:\mathcal{J} \hookrightarrow \mathcal{V}$. Lawvere's notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted $\mathcal{V}$-enriched $\mathcal{J}$-cotensor theory, or $\mathcal{J}$-theory for short. For suitable choices of $\mathcal{V}$ and $\mathcal{J}$, such $\mathcal{J}$-theories include the enriched algebraic theories of Borceux and Day, the enriched Lawvere theories of Power, the equational theories of Linton's 1965 work, and the $\mathcal{V}$-theories of Dubuc, which are recovered by taking $\mathcal{J} = \mathcal{V}$ and correspond to arbitrary $\mathcal{V}$-monads on $\mathcal{V}$. We identify a modest condition on $j$ that entails that the $\mathcal{V}$-category of $\mathcal{T}$-algebras exists and is monadic over $\mathcal{V}$ for every $\mathcal{J}$-theory $\mathcal{T}$, even when $\mathcal{T}$ is not small and $\mathcal{V}$ is neither complete nor cocomplete. We show that $j$ satisfies this condition if and only if $j$ presents $\mathcal{V}$ as a free cocompletion of $\mathcal{J}$ with respect to the weights for left Kan extensions along $j$, and so we call such systems of arities eleutheric. We show that $\mathcal{J}$-theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on $\mathcal{J}$, and (ii) $\mathcal{V}$-monads on $\mathcal{V}$ satisfying a certain condition. We prove a characterization theorem for the categories of algebras of $\mathcal{J}$-theories, considered as $\mathcal{V}$-categories $\mathcal{A}$ equipped with a specified $\mathcal{V}$-functor $\mathcal{A} ightarrow \mathcal{V}$.

Motivation & Objective

• To generalize Lawvere’s algebraic theories to enriched settings using a system of arities $\mathscr{J} \hookrightarrow \mathscr{V}$ in a symmetric monoidal closed category $\mathscr{V}$.
• To identify conditions under which the category of $\mathscr{T}$-algebras exists and is monadic over $\mathscr{V}$, even when $\mathscr{V}$ lacks completeness or cocompleteness.
• To define and characterize 'eleutheric' systems of arities, where $\mathscr{V}$ is a free cocompletion of $\mathscr{J}$ under weights for left Kan extensions.
• To show that $\mathscr{J}$-theories are equivalent to monads in a bicategory of profunctors on $\mathscr{J}$, and to $\mathscr{V}$-monads satisfying a colimit condition.
• To provide a characterization of the $\mathscr{V}$-categories of $\mathscr{T}$-algebras as those equipped with a specified $\mathscr{V}$-functor to $\mathscr{V}$.

Proposed method

• Introduces the notion of a $\mathscr{J}$-theory as a $\mathscr{V}$-category with $J$-cotensors of a single object $S$, generalizing Lawvere and enriched Lawvere theories.
• Defines $\mathscr{T}$-algebras as $\mathscr{V}$-functors preserving $J$-cotensors for $J \in \mathscr{J}$, generalizing the notion of algebras over algebraic theories.
• Identifies a condition on $j: \mathscr{J} \hookrightarrow \mathscr{V}$ called 'eleutheric'—where $\mathscr{V}$ is the free $\Phi^+$-cocompletion of $\mathscr{J}$ under left Kan extensions—ensuring existence and monadicity of $\mathscr{T}$-algebras.
• Establishes that $\mathscr{J}$-theories correspond to monads in a one-object bicategory of $\mathscr{V}$-profunctors on $\mathscr{J}$, via a bicategorical correspondence.
• Proves that $\mathscr{J}$-theories are equivalent to $\mathscr{V}$-monads that preserve $\Phi^+$-colimits, linking them to monad-theoretic formulations.
• Uses the theory of $\Phi$-presentable objects and $\Phi^+$-colimits to show that $\mathscr{V}_{\Phi}$ is closed under $\Phi^{+-}$-colimits, $\Phi$-colimits, and retracts, enabling the construction of the eleutheric condition.

Experimental results

Research questions

• RQ1 Under what conditions on a system of arities $\mathscr{J} \hookrightarrow \mathscr{V}$ does the category of $\mathscr{T}$-algebras exist and form a monadic category over $\mathscr{V}$, even when $\mathscr{V}$ is not complete or cocomplete?
• RQ2 When is a system of arities $\mathscr{J} \hookrightarrow \mathscr{V}$ eleutheric, meaning $\mathscr{V}$ is the free $\Phi^+$-cocompletion of $\mathscr{J}$ under left Kan extensions?
• RQ3 How can $\mathscr{J}$-theories be equivalently described as monads in a bicategory of profunctors on $\mathscr{J}$?
• RQ4 What is the precise relationship between $\mathscr{J}$-theories and $\mathscr{V}$-monads that preserve $\Phi^+$-colimits?
• RQ5 How can the category of $\mathscr{T}$-algebras be characterized as a $\mathscr{V}$-category equipped with a specified $\mathscr{V}$-functor to $\mathscr{V}$?

Key findings

• The category of $\mathscr{T}$-algebras exists and is monadic over $\mathscr{V}$ for every $\mathscr{J}$-theory $\mathscr{T}$ if and only if the system of arities $j: \mathscr{J} \hookrightarrow \mathscr{V}$ is eleutheric.
• A system of arities $j$ is eleutheric if and only if $\mathscr{V}$ is the free $\Phi^+$-cocompletion of $\mathscr{J}$ under weights for left Kan extensions along $j$, where $\Phi^+$ is the class of $\Phi$-flat weights.
• Every $\mathscr{J}$-theory corresponds to a monad in a one-object bicategory of $\mathscr{V}$-profunctors on $\mathscr{J}$, establishing a bicategorical equivalence.
• Every $\mathscr{J}$-theory also corresponds to a $\mathscr{V}$-monad on $\mathscr{V}$ that preserves $\Phi^+$-colimits, providing a monadic characterization.
• The $\mathscr{V}$-category of $\mathscr{T}$-algebras is equivalent to the full sub-$\mathscr{V}$-category of $\mathscr{V}$-categories $\mathscr{A}$ equipped with a $\mathscr{V}$-functor $\mathscr{A} \to \mathscr{V}$ satisfying a certain universal property.
• The full sub-$\mathscr{V}$-category $\mathscr{V}_{\Phi}$ of $\Phi$-presentable objects in $\underline{\mathscr{V}}$ is closed under $\Phi^{+-}$-colimits, $\Phi$-colimits, and retracts, which is essential for verifying the eleutheric condition.