# 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

$A:T\to \mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$A : T \to Set \,.

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

$T\mathrm{Alg}:=\left[T,\mathrm{Set}{\right]}_{×}\subset \left[T,\mathrm{Set}\right]\phantom{\rule{thinmathspace}{0ex}}.$T Alg := [T,Set]_\times \subset [T,Set] \,.

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

## Properties

###### Proposition

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

###### Proposition

$T\mathrm{Alg}$ is a reflective subcategory of $\left[T,\mathrm{Set}\right]$:

$T\mathrm{Alg}\stackrel{←}{↪}\left[T,\mathrm{Set}\right]\phantom{\rule{thinmathspace}{0ex}}.$T Alg \stackrel{\leftarrow}{\hookrightarrow} [T,Set] \,.
###### Proof

With the above this follows using the adjoint functor theorem.

###### Corollary

The category $T\mathrm{Alg}$ has all colimits.

for more see Lawvere theory for the moment

## Examples

Revised on November 9, 2010 10:29:42 by David Corfield (129.12.18.222)