Lawvere theory

Summary

A Lawvere theory is a category with finite products whose objects are the natural numbers, where $n$ is the $n$-th power of $1$. Its morphisms $n \to 1$ play the role of the $n$-ary operations of an algebraic theory. They are closed under composition and identified modulo the equations of the theory. Models are finite-product-preserving functors to $\mathbf{Set}$, and the models of the theory of a variety form a category equivalent to the variety. Lawvere theories give a presentation-free description of varieties, so that different choices of operations and axioms for the same variety give the same theory.