nLab algebra over a Lawvere theory

Contents

Context

Categorical algebra

Higher algebra

Contents

1. 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.

2. 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

3. Examples

Last revised on March 22, 2021 at 09:18:46. See the history of this page for a list of all contributions to it.