symmetric monoidal (∞,1)-category of spectra
In his 1963 doctoral dissertation, Bill Lawvere introduced a new categorical method for doing universal algebra, alternative to the usual way of presenting an algebraic concept by means of its logical signature (with generating operations satisfying equational axioms). The rough idea is to define an algebraic theory as a category with finite products and possessing a “generic algebra” (e.g., a generic group), and then define a model of that theory (e.g., a group) as a product-preserving functor out of that category. This type of category is what is nowadays called a Lawvere algebraic theory, or just Lawvere theory.
A Lawvere theory or finite-product theory is (equivalently encoded by its syntactic category which is) a category with finite products in which every object is isomorphic to a finite cartesian power of a distinguished object (called the generic object for the theory ).
For a Lawvere theory, we are to think of the hom-set as the set of -ary operations definable in the theory. For instance for the theory of abelian groups, includes operations like , , and . For -ary or nullary operations, we have .
and homomorphism of -algebras is a natural transformation between such functors.
Such a functor picks out a single set and picks for every -ary abstract operation an actual -ary operation on the elements of this set
such that these operations are all compatible with each other in the way governed by the composition rules of morphisms in .
It is common to adopt the (slightly evil) convention that every object of is equal to a chosen power of . Thus, if is the category of finite cardinals and functions between them, then the unique (up to isomorphism) product-preserving functor that takes the 1-element cardinal to is commonly supposed to be surjective on objects (rather than, less evilly, essentially surjective), or even an isomorphism on objects so that each morphism has a well-defined arity .
Some people use ‘finite-product theory’ to mean any (small) category with finite products, reserving ‘Lawvere theory’ to refer to finite product theories with the property that every object is isomorphic to a product of finitely many copies of a given object . Finite-product theories can be regarded as a special case of multisorted Lawvere theories (see below) where the set of sorts is itself. Some, but not all, the above discussion generalizes to this case.
As finite-product theories, Lawvere theories are at one end of a spectrum of theories of differing logical strengths. For example, there are left exact theories, regular theories, geometric theories, and so on, which require for their interpretation categories of differing degrees of strength in their internal logic. See also classifying topos.
If is a category with finite products, then a group (object) in may be defined as a product-preserving functor . For example, a topological group may be identified with a functor , and a Lie group with a product-preserving functor into the category of smooth manifolds. An analogous statement holds for any finitary algebraic theory, when formulated in terms of its Lawvere theory .
A multisorted or multityped Lawvere theory for a given set of sorts is a category with finite products together with a function such that every object of is isomorphic to a finite product of objects of the form . An example is the theory for ring-module pairs, which may be regarded as a two-sorted theory in which one sort is interpreted as a ring and the other as a module over that ring.
An infinitary Lawvere theory allows for infinitary operations. An example is the theory of suplattices, where we have, for every cardinal number , an operation to take the supremum of elements. While Lawvere theories correspond to finitary monads on , infinitary Lawvere theories correspond to arbitrary monads.
A Fermat theory is a Lawvere theory equipped with a notion of differentiation.
A finite-product theory can also be presented without including all the products of the basic types as actual objects. This yields the notion of cartesian multicategory.
The tautological example of a Lawvere theory is the algebraic theory of no operations. This is also called the theory of equality.
This is the initial object in the category of Lawvere theories.
An algebra over this theory is just a bare set:
For any Lawvere theory, there is a canonical morphism . On categories of algebras this induces the functor
This sends each algebra to its underlying set . For more on this see the section Free T-algebras below.
We consider here the theory of groups (defined however you like). To get the corresponding Lawvere theory , let (for any natural number ) be a free group on generators, and define the Lawvere theory to be the category opposite to the category of free groups and group homomorphisms. The generic object of is taken to be .
The category of free groups has finite coproducts since (in other words, the inclusion
creates coproducts in ), so has finite products, and we have in . Any group defines a product-preserving functor
since contravariant hom-functors take coproducts to products. Thus any group gives a model of .
The other direction is more interesting. Let
be a model of , i.e., a product-preserving functor. We will define a group structure on , the underlying set of the group.
To understand this, let’s consider how group multiplication would be defined. The idea is that in is a “generic group”, so we first need to understand how multiplication works there. The idea is that the product in the generic group
corresponds to a homomorphism
which by freeness corresponds to an element , and the element we are after should be the product of the generators of the free group . The generators themselves correspond to the two coproduct inclusions
Then, since is assumed to preserve products, we obtain a map
and this defines the group multiplication on . The group identity and group inversion on are defined by following similar recipes.
It may be checked that the notion of homomorphism of -models (as defined above) coincides with the usual notion of group homomorphism. In summary, the category of groups is equivalent to the category of models of .
In particular, any hom-functor
preserves products, and so defines a group. This group is precisely the free group on generators, and a little thought shows that the generators correspond to the natural transformations
induced by the projection maps .
All of the discussion above for the case of groups generalizes to any finitary algebraic theory (i.e., any single-sorted theory whose signature consists of function symbols of finite arity, subject to universally quantified equational axioms). In summary:
The Lawvere theory is the category opposite to the category of free algebras on finitely many generators,
The category of algebras is in turn equivalent to the category of product-preserving functors , and
The free algebras are retrieved as the representable functors .
Most of the standard structures that are considered in algebra indeed are models of algebraic/Lawvere theories in the precise sense. The following list gives a few familiar examples and a few not so familiar ones, but there are many more. Beware though that there are also some familiar examples that seem to be algebraic but are not, these we discuss below.
If is any theory given by a signature consisting of finitary operations? (but no relations) on a single sort, and a set of axioms all of which are universally quantified equations between terms, then a model of can be described as an algebra of a Lawvere theory.
This includes most cases arising in a typical undergraduate course in modern algebra, as the examples above suggest.
There are also well-known criteria for a category of single-sorted structures , with underlying set-functor , to be the category of algebras of a Lawvere theory.
A concrete category is a category of algebras over a Lawvere theory precisely if
Another characterization is:
Suppose given a language generated by a set of (single-sorted) finitary operations, and a class of structures for . Then is the class of models for a set of universally quantified equations between terms of if and only if
(H) The class is closed under homomorphic images,
(S) The class is closed under subalgebras,
(P) The class is closed under taking products.
Here are some notable examples of mathematical structures that look algebraic, but are not models of an algebraic theory in the present sense:
The class of fields is not the class of algebras of a Lawvere theory.
Neither is the class of integral domains.
This might seem obvious since multiplicative inversion in fields is not a global operation, or otherwise the cancellation law of multiplication in integral domains is not a universally quantified axiom (since we have to make an exception of ). But one should be careful that there isn’t some sneaky alternative axiomatization for these structures which counters these objections!
The second clause in Theorem 1 suggests another type of non-example:
There is a whole class of infinitary sup-operations for sup-lattices (one for every arity = cardinal), but again one may wonder how one rules out any alternative finitary axiomatizations. But this is fairly clear by invoking the second clause and considering the following example: if created filtered colimits, then the countable copower of 2-element sup-lattices (which turns out to be the power set with its usual order) would be the filtered colimit (in fact a union) over finite subsets of finite copowers , hence a countable union of finite sup-lattices, which is clearly impossible.
Let be a Lawvere theory and a -algebra. A congruence on is an equivalence relation on the set such that whenever for all whenever any and are pairwise eqivalent, , then also for every operation the results are equivalent: .
For a -algebra and a congruence on , the relation induced by for each are functions and define a -algebra structure on .
For two morphisms of -algebras, the canonical morphism is the coequalizer of and .
Write for the (syntactic category of the) algebraic theory of sets (described above). Then for any other (syntactic category of a) Lawvere theory, there is a canonical morphism
By precomposition with we obtain a corresponding functor on -algebras, which we write
and call the underlying set functor.
The functor has a left adjoint .
This is a standard example of a free functor, called the free -algebra functor.
For Set, let be the -algebra whose underlying set is the set of formal expressions with the evident composition operation.
The adjunction isomorphism
is in this case just the Yoneda lemma.
Notice that this extends to a functor
which is the composite
More generally, for not necessarily finite, let be the poset of finite subsets of and their inclusions.
Then is the filtered colimit of the representables corresponding to the finite subsets
As discussed below, these filtered colimits of -algebras are computed objectwise.
The following establishes that more generally any morphism of Lawvere theories leads to an adjunction between their categories of algebras.
Let and be Lawvere theories and a morphism. Write for the functor on categories of algebras induced by precomposition with .
The functor has a left adjoint .
Here is an elementary proof:
Since for any set we have it follows that . For the smallest -congruence containing we have that is a -algebra.
This one checks is .
Here is a more high-powered way to obtain this using the monads whose algebras are -algebras:
for a -algebra let the the evident reflexive coequalizer
A famous result by G. Higman in group theory says that a finitely generated group can be embedded in a finitely presented group precisely if it has a presentation whose defining relations are a recursively enumerable set of words. Clearly, this question can be asked for every similar algebraic theory and it has been in fact conjectured by the group theorist W. Boone that the same result holds more generally for every single-sorted algebraic theory.
The origin of the categorical formulation of algebraic theories as Lawvere theories is in
The concept was then streamlined in
Also still worthwhile reading are the following early papers:
Textbook treatments are
Francis Borceux, Handbook of categorical algebra 2 – Categories and structures , Encyclopedia of Mathematics and its Applications, Cambridge UP 1994. (chap. 3)
M. C. Pedicchio, F. Rovatti, Algebraic Categories , pp.269-310 in Pedicchio, Tholen (eds.), Categorical Foundations , Encyclopedia of Mathematics and its Applications 97, Cambridge UP 2004.
A recent monograph is
The concept of an internal algebraic theory in topos theory is dicussed in
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014; sec. 6.4, pp.190-198)
For a comparison with the concept of monads see
Distributive laws for algebraic theories are discussed in
Voevodsky proves an equivalence between Lawvere theories and l-bijective C-systems here:
Other references are
M. Jibladze, T. Pirashvili, Cohomology of Algebraic Theories , J. Algebra 137 no.2 (1991) pp.253–296.
Enrico Vitale, Localization of Algebraic Categories , JPAA 108 (1996) pp.315-320.
Enrico Vitale, Localization of Algebraic Categories 2 , JPAA 133 (1998) pp.317-326.