Introduction

Basic concepts and definitions

• Class of algebras
  Definition 0.1   A class of algebras is defined in a precise sense as an algebraic object in the groupoid category.
• Monad on a category $\mathcal{C}$ , and a T-algebra in $\mathcal{C}$
  Definition 0.2   Let us consider a category $\mathcal{C}$ , two functors: $T: \mathcal{C} \to \mathcal{C}$ (called the monad functor) and $T^2: \mathcal{C} \to \mathcal{C} = T \circ T$ , and two natural transformations: $\eta: 1_ \mathcal{C} \to T$ and $\mu: T^2 \to T$ . The triplet $(\mathcal{C},\eta,\mu)$ is called a monad on the category $\mathcal{C}$ . Then, a T-algebra $(Y,h)$ is defined as an object $Y$ of a category $\mathcal{C}$ together with an arrow $h: TY \to Y $ called the structure map in $\mathcal{C}$ such that:

  1. $$Th: T^2 \to TY,$$
  2. $$h \circ Th = h \circ \mu_Y,$$ where: $\mu_Y: T^2 Y \to TY;$ and
  3. $$ h \circ \eta_Y = 1_Y.$$
• Category of Eilenberg-Moore algebras of a monad $T$

  An important definition related to abstract classes of algebras and universal algebras is that of the category of Eilenberg-Moore algebras of a monad $T$ :

  Definition 0.3   The category $\mathcal{C}^T$ of $T$ -algebras and their morphisms is called the Eilenberg-Moore category or category of Eilenberg-Moore algebras of the monad T.

Remarks

• a. Algebraic category definition
  Remark 0.1   With the above definition, one can also define a category of classes of algebras and their associated groupoid homomorphisms which is then an algebraic category.

  Another example of algebraic category is that of the category of C*-algebras.

  Generally, a category $\mathcal{A}_C$ is called algebraic if it is monadic over the category of sets and set-theoretical mappings, $Set$ ; thus, a functor $G: \mathcal{D} \to \mathcal{C}$ is called monadic if it has a left adjoint $F: \mathcal{C}\to \mathcal{D}$ forming a monadic adjunction $(F,G,\eta,\epsilon)$ with $G$ and $\eta, \epsilon$ being, respectively, the unit and counit; such a monadic adjunction between categories $\mathcal{C}$ and $\mathcal{D}$ is defined by the condition that category $\mathcal{D}$ is equivalent to the to the Eilenberg-Moore category $\mathcal{C} ^T$ for the monad $$T = GF.$$

• b. Equivalence classes
  Remark 0.2   Although all classes can be regarded as equivalence, weak equivalence, etc., classes of algebras (either specific or general ones), do not define identical, or even isomorphic structures, as the notion of `equivalence' can have more than one meaning even in the algebraic case.