Lawvere et al, algebraic theories, Cambridge University Press 2010

From the two categorical approaches to algebra: algebraic theories (aka. Lawvere theories) and monads, the book -with the exception of Appendix A and Appendix C- entirely concentrates on algebraic theories. Also the book only treats sets and “many sorted sets”. Sets with structure such as topological groups are not treated.

Appendix A

Finitary monads for $K=Set$ yield precisely one-sorted algebraic theories.

Finitary monads for $K=Set^S$ yield precisely $S$-sorted algebraic theories.

Proposition: For every one-sorted algebraic theory $(T, t)$, the concrete category $(Alg T, Alg t)$ is pseudomonadic.

Theorem: The following conditions on a concrete category $(A,U)$ over $Set$ are equivalent:

(1) $(A,U)$ is pseudo-one-sorted algebraic.

(2) $A$ is cocomplete, and $U$ is a conservative right adjoint preserving sifted colimits.

Corollary: Pseudo-one-sorted algebraic categories are up to pseudoconcrete equivalence precisely the categories $Set^M$ of Eilenberg-Moore algebras for finitary monads $M$ on $C$.

Theorem (One-sorted algebraic duality): The category $Alg_u^1$ of uniquely transportable on-sorted algebraic categories is equivalent to the dual of the category $Th^1$ of one-sorted algebraic theories. In fact, the $2$-functor $Alg^1:(Ps Th^1)^{op}\to Ps Alg^1$ is a biequivalence.