A category is equationally presentable if it may be described, beginning with the category Set, as the category of sets equipped with certain operations satisfying certain equations involving these operations as objects, and functions that preserve these operations as morphisms. A category is equationally presented if it is equipped with such a description.
We can generalise to categories given by laws which may be implications between equations rather than simply equations themselves; call such a category quasi-equationally presentable or quasi-equationally presented.
In general, we allow largely many operations and equations (that is a proper class of them, too many be parametrised by an object of itself), although each individual operation must have small arity and each equation must be well-founded.
Equationally presentable categories capture precisely the categories of varieties of algebras studied in universal algebra. However, to study them with category theory, it is somewhat more manageable to use monadic categories instead. We have that an algebraic category is monadic if and only if it has free objects; that is, the forgetful functor from to has a left adjoint.
Let a (quasi)-equationally presentable category be bounded if it may be described using only a small set of operations (and hence a small set of equations); in this case it follows that is monadic (if it is equationally presentable) or at least algebraic (if it is only quasi-equationally presentable), although the converse does not hold. Similarly, let be finitary if it may be described using only a small set of operations, each of which has finite arity (in which case it must be bounded). We can characterise boundedness and finite arity nicely in category-theoretic terms; thus one speaks of bounded monadic categories, finitary algebraic categories, etc. See algebraic category for the definitions; it is a theorem that the definitions there are equivalent to the definitions here.
Is it true that a quasi-equationally presentable category is algebraic if it has free objects? Both Johnstone and AHS stop short of this case.
Finitary monadic categories are particularly nice; they are given by Lawvere theories.
Warning: The meaning of ‘algebraic category’ varies in the literature; Johnstone in Stone Spaces defines it to mean monadic, while we follow (here and at algebraic category) AHS in The Joy of Cats . The term ‘equationally presentable’ is from Johnstone, and the claim that this is equivalent to being given by an algebraic theory requires using Johnstone's definition of the latter from the Elephant. The term ‘quasi-equationally presentable’ is my own, following a pattern set down by AHS.
Same as algebraic category.