nLab
algebra over a Lawvere theory

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

$$A:T\to \mathrm{Set}.$$A : T \to Set \,.

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

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

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

Properties

Proposition

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

Proposition

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

$$T\mathrm{Alg}\overleftarrow{\hookrightarrow }[T,\mathrm{Set}].$$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

• groups

• rings

• $k$-associative algebras

• smooth algebras

Related concepts

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

  • model structure on algebras over an operad