nLab algebra over a Lawvere theory

1. 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.

2. 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

3. Examples

• groups

• rings

• $k$-associative algebras

• smooth algebras

4. 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