Algebraic constructs (A,U), such as Vec, Grp, Mon, and Lat, can be fully described by the following data, called the monad associated with (A,U):
1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associated free functor,
2. the natural transformation η : idSet → T formed by universal arrows, and
3. the natural transformation μ : T ◦ T → T given by the unique homomorphism μX : T(TX) → TX that extends idTX.
In these cases, there is a canonical concrete isomorphism K between (A,U) and the full concrete subcategory of Alg(T) consisting of those T-algebras TX →x X that satisfy the equations x ◦ ηX = idX and x ◦ Tx = x ◦ μX. The latter subcategory is called the Eilenberg-Moore category of the monad (T, η, μ). The above observation makes it possible, in the following four steps, to express the “degree of algebraic character” of arbitrary concrete categories that have free objects:
Step 1: With every concrete category (A,U) over X that has free objects (or, more generally, with every adjoint functor A →U X) one can associate, in an essentially unique way, an adjoint situation (η, ε) : F -|U : A → X.
Step 2: With every adjoint situation (η, ε) : F -|U : A → X one can associate a monad T = (T, η, μ) on X, where T = U ◦ F : X → X.
Step 3: With every monad T = (T, η, μ) on X one can associate a concrete subcategory of Alg(T) denoted by (XT, UT) and called the category of T-algebras.
Step 4: With every concrete category (A,U) over X that has free objects one can associate a distinguished concrete functor (A,U) →K (XT , UT) into the associated category of T-algebras called the comparison functor for (A, U).
Concrete categories that are concretely isomorphic to a category of T-algebras for some monad T have a distinct “algebraic flavor”. Such categories (A,U) and their forgetful functors U are called monadic. It turns out that a concrete category (A, U ) is monadic iff it has free objects and its associated comparison functor (A,U) →K (XT , UT) is an isomorphism. Thus, for concrete categories (A,U) that have free objects, the associated comparison functor can be considered as a means of measuring the “algebraic character” of (A,U); and the associated category of T-algebras can be considered to be the “algebraic part” of (A,U). In particular,
(a) every finitary variety is monadic,
(b) the category TopGrp, considered as a concrete category
- over Top, is monadic,
- over Set, is not monadic; the associated comparison functor is the forgetful functor TopGrp → Grp, so that the construct Grp may be considered as the “algebraic part” of the construct TopGrp,
(c) the construct Top is not monadic; the associated comparison functor is the forgetful functor Top → Set itself, so that the construct Set may be considered as the “algebraic part” of the construct Top; hence the construct Top may be considered as having a trivial “algebraic part”.
Among constructs, monadicity captures the idea of “algebraicness” rather well. Unfortunately, however, the behavior of monadic categories in general is far from satisfactory. Monadic functors can fail badly to reflect properties of the base category (e.g., the existence of colimits or of suitable factorization structures), and they are not closed under composition.