An algebraic theory, also called a Lawvere theory, is a category that describes a set of equational axioms such as group axioms. Any algebraic theory has a presentation with generators and relations where generators here are function (constant) symbols such as *, e, and relations are considered as equational axioms like x * e = x.
In this talk, I will give an introduction to algebraic theories and their homology groups.
We will see that, like Morse’s inequalities, given an algebraic theory, the difference of numbers of generators and relations is below by the difference of 1st and 2nd Betti numbers of the theory. As an instance of this inequality, we can give another proof that we need at least 2 equational axioms to present a theory of groups if we fix function symbols (*, ^-1, e), which was proved first by Alfred Tarski.