nLab
algebraic category

Algebraic categories

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 CompHausTopComp Haus Top. 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.

Definitions

Let AA be a concrete category; that is, AA is equipped with a forgetful functor U:ASetU\colon A \to Set to the category of sets. For some authors, such a category is called ‘concrete’ only if UU is representable, but that follows in all the cases considered below; in particular, if AA has free objects (that is, if UU has a left adjoint FF), then UU is representable by F(1)F(1), where 11 is a singleton.

Definition (based on AHS 23.38)

The concrete category AA is algebraic if the following conditions hold:

Definition (based on AHS …)

The concrete category AA is monadic if the following conditions hold:

Definition (based on AHS 24.11)

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

Definition (based on AHS 24.4)

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

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

Properties

Every monadic category is algebraic; an algebraic category is monadic if and only if the forgetful functor UU 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 UFU \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 UFU \circ F may be thought of as mapping a set xx to the set of words with alphabet taken from xx 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 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 xx to the set of ultrafilters on xx. (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

Our definitions are taken from

Actually, AHS discusses the more general concept of algebraic (etc) functors, generalising from U:ASetU\colon A \to Set to arbitrary functors (not necessarily faithful, not necessarily to SetSet). We actually take our definitions from AHS's characterisation theorems in the case of faithful functors to SetSet. 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.

Revised on March 26, 2014 03:36:37 by Urs Schreiber (185.37.147.12)