nLab algebraic category

Idea

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 $Comp Haus Top$. However, all of these notions are related, and we will discuss them here.

The definitions in AHS also includes a requirement violating the principle of equivalence, which we omit: that of unique strict lifts of isomorphisms, which serves to fix algebraic categories up to isomorphism (instead of mere equivalence).

Definitions

Let $A$ be a concrete category; that is, $A$ is equipped with a forgetful functor $U\colon A \to Set$ 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 $F \dashv U$ is monadic.

Definition (based on AHS 24.11)

An algebraic (or monadic) category is bounded if the following condition holds:

• For some cardinal number $\kappa$ and every $\kappa$-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 is a weakening of the condition that the forgetful functor $U$ is finitary (that is, that $U$ preserves directed colimits); every universal cocone in $Set$ is jointly surjective, but not conversely.

Properties

Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor $U$ 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 $U \circ F$. (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:

• A concrete category is bounded monadic if and only if it is equationally presentable (presented by a variety) with a small set of operations (and hence equations).
• A concrete category is bounded algebraic if and only if it is presented by a quasivariety with a small set of operations.
• A concrete category is finitary monadic if and only if it is the category of algebras for some finitary variety; that is, we have only a small set of finitary operations.
• A concrete category is finitary algebraic if and only if it is the category of algebras for some finitary quasivariety.

Also, every algebraic category whose forgetful functor preserves filtered colimits is the category of models for some first-order theory. The converse is false.

Examples

The typical categories studied in algebra, such as Grp, Ring, Vect, etc, are all finitary monadic categories. The monad $U \circ F$ may be thought of as mapping a set $x$ to the set of words with alphabet taken from $x$ 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.

The category of cancellative monoids is finitary algebraic but not monadic. The category Field of fields is not even algebraic.

Assuming the ultrafilter principle, the category of compact Hausdorff spaces is monadic, but not bounded algebraic. The monad in question takes a set $x$ to the set of ultrafilters on $x$. (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.

References

The original definitions can be found in

• F. William Lawvere, Algebraic theories, algebraic categories, and algebraic functors. 1965 Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), 413–418. North-Holland, Amsterdam.

Our definitions are taken from

• AHS: Jiří Adámek, Horst Herrlich, George Strecker; Abstract and Concrete Categories: The Joy of Cats, Sections 23 & 24; web.

Actually, AHS discusses the more general concept of algebraic (etc) functors, generalising from $U\colon A \to Set$ to arbitrary functors (not necessarily faithful, not necessarily to $Set$). We actually take our definitions from AHS's characterisation theorems in the case of faithful functors to $Set$. We probably should discuss the more general concept, perhaps at algebraic functor?; we already have monadic functor.

• Peter Johnstone; Stone Spaces, Section 3.8

For Johnstone, a concrete category is ‘algebraic’ if and only if it is monadic. However, Johnstone also discusses equationally presentable categories.

Another modern reference is

• Jiří Adámek, Jiří Rosický, Enrico M. Vitale,

  Algebraic theories, Cambridge Tracts in Mathematics 184 (2011), Cambridge University Press. doi:10.1017/cbo9780511760754.