nLab
algebra over a Lawvere theory

Contents

Definition

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

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

A:TSet. A : T \to Set \,.

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

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

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

Properties

Proposition

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

Proposition

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

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

With the above this follows using the adjoint functor theorem.

Corollary

The category TAlgT 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)