An algebraic category is a concrete category which behaves very much like the categories familiar from algebra, such as Grp, Ring, and Vect, but characterised in category-theoretic terms. But many other categories are also algebraic, most famously CompHausTop; one can describe these in purely algebraic terms, but only using infinitary (perhaps even largely many) operations.
There are several definitions of ‘algebraic’ in the literature. Here, we will follow AHS (see references) in using a generous interpretation, but other authors follow Johnstone in using ‘algebraic’ to mean monadic (a stricter requirement), while some authors add finiteness conditions that remove examples such as . However, all of these notions are related, and we will discuss them here.
The definitions in AHS also include an evil requirement of unique strict lifts of isomorphisms, which serves to fix algebraic categories up to isomorphism (instead of mere equivalence), which we omit.
Let be a concrete category; that is, is equipped with a forgetful functor to the category of sets. For some authors, such a category is called ‘concrete’ only if is representable, but that follows in all the cases considered below; in particular, if has free objects (that is, if has a left adjoint ), then is representable by , where is a singleton.
The concrete category is algebraic if the following conditions hold:
The concrete category is monadic if the following conditions hold:
An algebraic (or monadic) category is bounded if the following condition holds:
An algebraic (or monadic) category is finitary if the following condition holds:
Note that this a weakening of the condition that the forgetful functor is finitary (that is, that preserves directed colimits); every universal cocone in is jointly surjective, but not conversely.
Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor preserves congruences. (AHS 23.41)
A category is algebraic if and only if it is a reflective subcategory of a monadic category with regular epic reflector; given an algebraic category, this monadic category is the Eilenberg–Moore category of the monad . (AHS 24.3)
Every monadic category is the category of algebras for some variety of algebras, although we must allow potentially a proper class of infinitary axioms; that is, every monadic category is equationally presentable. Similarly, every algebraic category is the category of algebras for some quasivariety of algebras; that is, we allow conditional statements of equations among the axioms. (AHS 24.11)
As special cases of the last item:
The typical categories studied in algebra, such as Grp, Ring, Vect, etc, are all finitary monadic categories. The monad may be thought of as mapping a set to the set of words with alphabet taken from and the connections between letters taken from the appropriate algebraic operations, with two words identified if they can be proved equal by the appropriate algebraic axioms.
Assuming the ultrafilter principle, the category of compact Hausdorff spaces is monadic, but not bounded algebraic. The monad in question takes a set to the set of ultrafilters on . (Without the ultrafilter principle, this monad still exists, but it may be quite small, possibly even the identity monad; passing to locales does not help.)
Similarly, the category of Stone spaces is algebraic, but not monadic or bounded algebraic.
Our definitions are taken from
Actually, AHS discusses the more general concept of algebraic (etc) functors, generalising from to arbitrary functors (not necessarily faithful, not necessarily to ). We actually take our definitions from AHS's characterisation theorems in the case of faithful functors to . We probably should discuss the more general concept, perhaps at algebraic functor?; we already have monadic functor.
For Johnstone, a concrete category is ‘algebraic’ if and only if it is monadic. However, Johnstone also discusses equationally presentable categories.