# nLab enriched Lawvere theory

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of enriched Lawvere theory is a generalization of Lawvere theories to the setting of enriched categories.

## Definitions and elementary properties

###### Definition

An enriched Lawvere theory over a locally finitely presentable? $V$-enriched category $A$ is a small $V$-enriched category $L$ together with a $V$-enriched functor $J\colon A_f^{op}\to L$ that induces an identity map on the set of objects. Here $A_f$ denotes the full subcategory of finitely presentable objects in $A$. The category $V$ is a locally finitely presentable closed symmetric monoidal category.

###### Definition

An algebra over an enriched Lawvere theory is an object $X$ in $A$ equipped with a functor $M\colon L\to V$ whose precomposition with $J$ is isomorphic to the representable functor of $X$.

###### Proposition

For the case $A=Set$ the above definition recovers the usual notion of a Lawvere theory and an algebra over a Lawvere theory.

###### Proposition

The forgetful functor from algebras to $A$ is a finitary enriched monadic functor? (over $V$).

###### Proposition

The category of enriched Lawvere theories over $A$ is equivalent to the category of finitary enriched monads? over $A$. The corresponding $V$-enriched categories of algebras are also equivalent.

## References

Last revised on January 21, 2021 at 13:43:59. See the history of this page for a list of all contributions to it.