Let A be a concrete category; that is, A is equipped with a forgetful functor U:A→Set to the category of sets. For some authors, such a category is called ‘concrete’ only if U is representable, but that follows in all the cases considered below; in particular, if A has free objects (that is, if U has a left adjoint F), then U is representable by F(1), where 1 is a singleton.

Definition (based on AHS 23.38). The concrete category A is algebraic if the following conditions hold:

- A has free objects.
- The category A has all binary coequalizers.
- The forgetful functor U preserves and reflects extremal epimorphisms.

Definition (based on AHS …). The concrete category A is monadic if the following conditions hold:

- A has free objects.
- The adjunction FU is monadic.

Definition (based on AHS 24.11). An algebraic (or monadic) category is bounded if the following condition holds:

- For some cardinal number κ and every κ-directed colimit in A, the universal cocone is jointly surjective in Set.

Definition (based on AHS 24.4). An algebraic (or monadic) category is finitary if the following condition holds:

- For every finitely directed colimit in A, the universal cocone is jointly surjective in Set.

Note that this a weakening of the condition that the forgetful functor U is fintary (that is, that U preserves directed colimits); every universal cocone in Set is jointly surjective, but not conversely.

### Definitions

An algebraic theory is: a small category 'T' with finite products.

An algebra for the theory 'T' is a functor A: T -> Set preserving the finite products.

Alg T is the category of the algebras of T.

### Example - Sets

algebraic theory T_{1} for sets |