# nLab algebra over a Lawvere theory

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

A Lawvere theory is encoded in its syntactic category $T$, a category with finite products such that all objects are finite products of a given object.

An algebra over a Lawvere theory $T$, or $T$-algebra for short, is a model for this algebraic theory: it is a product-preserving functor

$A : T \to Set \,.$

The category of $T$-algebras is the full subcategory of the functor category on the product-preserving functors

$T Alg := [T,Set]_\times \subset [T,Set] \,.$

For more discussion, properties and examples see for the moment Lawvere theory.

## Properties

###### Proposition

The category $T Alg$ has all limits and these are computed objectwise, hence the embedding $T Alg \to [T,Set]$ preserves these limits.

###### Proposition

$T Alg$ is a reflective subcategory of $[T, Set]$:

$T Alg \stackrel{\leftarrow}{\hookrightarrow} [T,Set] \,.$
###### Proof

With the above this follows using the adjoint functor theorem.

###### Corollary

The category $T Alg$ has all colimits.

for more see Lawvere theory for the moment

## Examples

Revised on February 12, 2014 07:50:50 by Urs Schreiber (88.128.80.11)