nLab Lawvere theory

Idea

The notion of Lawvere theory is a joint generalization of the notions of group, ring, associative algebra, etc.

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.

Definition

There are two related definitions of Lawvere theory in the literature. Both definitions are widely used and both have a substantial history.

Definition A

A Lawvere theory or finite-product theory is (equivalently encoded by its syntactic category which is) a category $T$ with finite products in which every object is isomorphic to a finite cartesian power $x^n = x \times x \times \cdots \times x$ of a distinguished object $x$ (called the generic object for the theory $T$).

A homomorphism of such theories $T \to T'$ is a product-preserving functor which also preserves the generic object.

Definition B

Let $\mathbb{F}$ be the category of natural numbers and functions between them (a skeleton of the category of finite sets). This category has canonical chosen coproducts.

A Lawvere theory is a category $T$ equipped with an identity-on-objects strictly power-preserving functor $J:\mathbb{F}^{\mathrm{op}} \to T$. (Power preservation is here equivalent to product preservation.)

In this definition, a homomorphism is a product-preserving functor making the obvious triangle commute strictly.

Remark

Clearly, if $T$ is part of a structure in the form of Definition B, then $T$ is of the form in Definition A, taking $x=J(1)$. See further remarks below.

For $T$ a Lawvere theory, we are to think of the hom-set $T(n,1) := T(x^n, x)$ as the set of $n$-ary operations definable in the theory. For instance for $T$ the theory of abelian groups, $T(2,1)$ includes operations like $+ \colon (x, y) \mapsto x + y$, $- \colon (x, y) \mapsto x-y$, and $(x, y) \mapsto 2 x - 3 y$. For $0$-ary or nullary operations, we have $T(0,1) = \{0\}$.

Definition

A model of $T$ – an algebra over the Lawvere theory or simply $T$-algebra – is a product-preserving functor

$$A : T \to Set \,,$$

and homomorphism of $T$-algebras is a natural transformation between such functors.

Remark

Such a functor picks out a single set $U(A) := A(1)$ and picks for every $n$-ary abstract operation $\phi \in T(n,1)$ an actual $n$-ary operation on the elements of this set

$$A(\phi) : U(A)^n \to U(A)$$

such that these operations are all compatible with each other in the way governed by the composition rules of morphisms in $T$.

When working with Definition B of Lawvere theory, it would be normal to require that $A$ strictly preserves products.

Remarks

1. From some perspectives, Definition B violates the principle of equivalence, although from another perspective it should be regarded as a basic structural definition that is equivalent in content to an abstract clone, or equivalently a relative monad over $\mathbb{F}$. On the other hand, given a structure of the form Definition A, we may choose a power of $x$ for each natural number, and thus derive an essentially surjective product-preserving functor $\mathbb{F}^{op} \to T$, and then we can always factor this as an identity-on-objects functor to reach Definition B.

2. 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 $x$. Finite-product theories $C$ can be regarded as a special case of multisorted Lawvere theories (see below) where the set of sorts is $Ob(C)$ itself. Some, but not all, the above discussion generalizes to this case.

3. 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.

4. If $C$ is a category with finite products, then a group (object) in $C$ may be defined as a product-preserving functor $T_{Grp} \to C$. For example, a topological group may be identified with a functor $T_{Grp} \to Top$, and a Lie group with a product-preserving functor $T_{Grp} \to Man$ into the category of smooth manifolds. An analogous statement holds for any finitary algebraic theory, when formulated in terms of its Lawvere theory $T$.

Variations

• A multisorted or multityped Lawvere theory for a given set of sorts $S$ is a category with finite products $C$ together with a function $i: S \to Ob(C)$ such that every object of $C$ is isomorphic to a finite product of objects of the form $i(S)$. 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 $n$, an operation to take the supremum of $n$ elements. While Lawvere theories correspond to finitary monads on $Set$, 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.

Examples

The theory of sets

The tautological example of a Lawvere theory is the algebraic theory of no operations. This is also called the theory of equality.

Its syntactic category is the category $\mathcal{S}$ on objects $n \in \mathbb{N}$ with morphisms generated by the projections $\pi_i : n \to 1$. This is the opposite category of the category FinSet

$$\mathcal{S} \simeq FinSet^{op} \,.$$

This is the initial object in the category of Lawvere theories.

An algebra over this theory is just a bare set:

$$\mathcal{S} Alg \simeq Set \,.$$

For $T$ any Lawvere theory, there is a canonical morphism $\mathcal{S} \to T$. On categories of algebras this induces the functor

$$U_T : T Alg \to \mathcal{S}Alg \simeq Set \,.$$

This sends each algebra to its underlying set . For more on this see the section Free T-algebras below.

The theory of groups

We consider here the theory of groups (defined however you like). To get the corresponding Lawvere theory $T$, let $F(n)$ (for any natural number $n \geq 0$) be a free group on $n$ generators, and define the Lawvere theory $T_{Grp}$ to be the category opposite to the category of free groups $F(n)$ and group homomorphisms. The generic object $x$ of $T_{Grp}$ is taken to be $F(1)$.

The category of free groups has finite coproducts since $F(m) + F(n) \cong F(m+n)$ (in other words, the inclusion

$$FreeGroup \hookrightarrow Group$$

creates coproducts in $FreeGroup$), so $T_{Grp}$ has finite products, and we have $F(n) = x^n$ in $T_{Grp}$. Any group $G$ defines a product-preserving functor

$$T_{Grp} = FreeGroup^{op} \hookrightarrow Group^{op} \stackrel{\hom(-, G)}{\to} Set$$

since contravariant hom-functors take coproducts to products. Thus any group gives a model of $T_{Grp}$.

The other direction is more interesting. Let

$$M: T_{Grp} \to Set$$

be a model of $T_{Grp}$, i.e., a product-preserving functor. We will define a group structure on $G = M(x)$, the underlying set of the group.

To understand this, let’s consider how group multiplication would be defined. The idea is that $x$ in $T_{Grp}$ is a “generic group”, so we first need to understand how multiplication works there. The idea is that the product in the generic group

$$m: x^2 \to x^1$$

corresponds to a homomorphism

$$F(1) \to F(2)$$

which by freeness corresponds to an element $1 \to F(2)$, and the element we are after should be the product $a b$ of the generators $a, b$ of the free group $F(2) = F(a, b)$. The generators $a, b$ themselves correspond to the two coproduct inclusions

$$i_1: F(1) \to F(1) + F(1) = F(2) \qquad i_2: F(1) \to F(1) + F(1) = F(2)$$

Then, since $M$ is assumed to preserve products, we obtain a map

$$G \times G = M(x) \times M(x) \cong M(x^2) \stackrel{M(m)}{\to} M(x) = G$$

and this defines the group multiplication on $G$. The group identity and group inversion on $G$ are defined by following similar recipes.

It may be checked that the notion of homomorphism of $T_{Grp}$-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 $T_{Grp}$.

In particular, any hom-functor

$$\hom_{T_{Grp}}(x^n, -): T_{Grp} \to Set$$

preserves products, and so defines a group. This group is precisely the free group on $n$ generators, and a little thought shows that the $n$ generators correspond to the natural transformations

$$\hom_{T_{Grp}}(x, -) \to \hom_{T_{Grp}}(x^n, -)$$

induced by the $n$ projection maps $x^n \to x$.

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 $T$ 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 $T \to Set$, and

• The free algebras are retrieved as the representable functors $T \to Set$.

As discussed in the article on operads, the notion of Lawvere theory may also be formulated in terms of operads relative to the theory of cartesian monoidal categories.

Other examples

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.

• For $k$ a field, the category of free $k$-associative algebras is the (syntactic category of the) theory of ordinary associative algebras over $k$.

• for $G$ a fixed group, then $G$-actions (permutation representations) are an example of algebras over a Lawvere theory

• for $R$ a fixed ring, then $R$-modules are an example

• for $k$ a field, then Lie algebras over $k$ are an example

• The category of say distributive lattices is the category of algebras of a Lawvere theory. So is the category of Heyting algebras.

• The category CartSp is the (syntactic category of the) theory of smooth algebras (as used in synthetic differential geometry). This is also a Fermat theory.

• The terminal Lawvere theory has exactly one morphism $f \colon m \to n$ for each $m,n$. Since there is a unique morphism between each pair of objects, each object is in fact isomorphic. Thus, this theory is a terminal category. Algebras of the terminal Lawvere theory are terminal sets, singletons.

Characterization of examples

If $T$ 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 $T$ 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 $C$, with underlying set-functor $U: C \to Set$, to be the category of algebras of a Lawvere theory.

Theorem

A concrete category $U \colon C \to Set$ is a category of algebras over a Lawvere theory precisely if $U$

1. is monadic

2. is finitary in that it preserves filtered colimits.

Another characterization is:

Theorem

(Birkhoff's HSP theorem)

Suppose given a language $L$ generated by a set of (single-sorted) finitary operations, and a class $C$ of structures for $L$. Then $C$ is the class of models for a set of universally quantified equations between terms of $L$ if and only if

1. (H) The class is closed under homomorphic images,

2. (S) The class is closed under subalgebras,

3. (P) The class is closed under taking products.

Some non-examples

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 $0$). But one should be careful that there isn’t some sneaky alternative axiomatization for these structures which counters these objections!

The first clause in Theorem already rules out fields, since $U$ in that case will create limits in $C$, but the category Field of fields does not even have products. Similarly for integral domains.

The second clause in Theorem suggests another type of non-example:

• The category $SLat$ of sup-lattices is not the category of algebras of a (finitary) Lawvere theory.

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 $U: SLat \to Set$ created filtered colimits, then the countable copower $\mathbb{N} \cdot \mathbf{2}$ of 2-element sup-lattices (which turns out to be the power set $P(\mathbb{N})$ with its usual order) would be the filtered colimit (in fact a union) over finite subsets $S$ of finite copowers $S \cdot \mathbf{2}$, hence a countable union of finite sup-lattices, which is clearly impossible.

Properties

Coequalizers of $T$-algebras

Definition (congruence)

Let $T$ be a Lawvere theory and $A$ a $T$-algebra. A congruence on $A$ is an equivalence relation on the set $A(1)$ such that whenever for all $n \in \mathbb{N}$ whenever any $(a_i \in A(1))_{i=1}^n$ and $(b_i \in A(1))_{i=1}^n$ are pairwise equivalent, $a_i \sim b_i$, then also for every operation $f \in T(n,1)$ the results are equivalent: $f(a_1, \cdot, a_n) \sim f(b_1, \cdots , b_n)$.

Observation/Definition

For $A$ a $T$-algebra and for every relation $R \subset A(1) \times A(1)$ there is a smallest congruence on $A$ containing $R$. Write $\langle R \rangle \subset A(1) \times A(1)$ for this smallest congruence.

Proposition

For $A$ a $T$-algebra and $C$ a congruence on $A$, the relation $A/C(f) \subset (A/C)^n \times A/C$ induced by $A(f)$ for each $f : \in T(n,1)$ are functions and define a $T$-algebra structure on $A(1)/C$.

Proposition

For $f,g : A \to B$ two morphisms of $T$-algebras, the canonical morphism $B \to B/\langle \f(a)\sim g(a) | a \in A(1)\rangle$ is the coequalizer of $f$ and $g$.

Free $T$-algebras and underlying sets

Write $\mathcal{S}$ for the (syntactic category of the) algebraic theory of sets (described above). Then for $T$ any other (syntactic category of a) Lawvere theory, there is a canonical morphism

$$i_T : \mathcal{S} \to T \,.$$

By precomposition with $i_T$ we obtain a corresponding functor on $T$-algebras, which we write

$$U_T : T Alg \to Set$$

and call the underlying set functor.

Lemma

The functor $U_T$ has a left adjoint $F_T : Set \to T Alg$.

This is a standard example of a free functor, called the free $T$-algebra functor.

Proof

For $S \in$ Set, let $F_T(S)$ be the $T$-algebra whose underlying set is the set of formal expressions $\{f(s_1, \cdots, s_n) | f \in T(n,1), s_i \in S\}$ with the evident composition operation.

Observation

For $S = (n) \in$ FinSet a finite set with $n$ elements , the free $T$-algebra on $S$ is just the representable

$$F_T(S) = T(n,-) : T \to Set \,.$$

The adjunction isomorphism

$$T Alg(F_T(S), A) \simeq Set(S, U_T(A)) \simeq (A(1))^n \simeq A(n)$$

is in this case just the Yoneda lemma.

$$T Alg(T(n,-),A) \simeq A(n) \,.$$

Notice that this extends to a functor

$$F_T(-) : FinSet \to T Alg$$

which is the composite

$$F_T(-) : FinSet \stackrel{i_T^{op}}{\to} T^{op} \stackrel{j}{\to} T Alg$$

of the Yoneda embedding with the opposite of the canonical functor $i_T : FinSet^{op} \simeq \mathcal{S} \to T$ from the theory of sets, described above.

More generally, for $S \in Set$ not necessarily finite, let $Sub(S)$ be the poset of finite subsets of $S$ and their inclusions.

Then $F_T(S)$ is the filtered colimit of the representables corresponding to the finite subsets

$$F_T(S) \simeq {\lim_{\to}}_{{n \in Sub(S)}} T(n,-) \,.$$

As discussed below, these filtered colimits of $T$-algebras are computed objectwise.

Adjunctions of relatively free $T$-algebras

The following establishes that more generally any morphism of Lawvere theories leads to an adjunction between their categories of algebras.

Let $T_1$ and $T_2$ be Lawvere theories and $f : T_1 \to T_2$ a morphism. Write $f^* : T_2 Alg \to T_1 Alg$ for the functor on categories of algebras induced by precomposition with $f$.

Lemma

The functor $f^*$ has a left adjoint $f_* : T_1 Alg \to T_2 Alg$.

Proof

Here is an elementary proof:

Let $F_{T_i} \dashv U_{T_i}$ be the two free algebra/underlying-set adjunctions. For $A$ a $T_1$-algebra there is a $T_1$-congruence $\Gamma$ such that

$$A \simeq F_{T_1} U_{T_1}(A) / \Gamma \,.$$

Since for any set $S$ we have $U_{T_1} F_{T_1}(S) \subset U_{T_2} F_{T_2}(S)$ it follows that $\Gamma \subset T_{T_2} F_{T_2} U_{T_1}(A)$. For $\langle \Gamma \rangle$ the smallest $T_2$-congruence containing $\Gamma$ we have that $F_{T_2} U_{T_1}(A) / \langle \Gamma \rangle$ is a $T_2$-algebra.

This one checks is $f_* A$.

Here is a more high-powered way to obtain this using the monads $K_i$ whose algebras are $T_i$-algebras:

for $A$ a $T_1$-algebra let $f_* A := K_2 \circ_{K_1} A$ the the evident reflexive coequalizer

$$K_2 K_1 A \stackrel{\to}{\to} K_2 A \to K_2 \circ_{K_1} A$$

in $T_2 Alg$.

Limits and colimits of $T$-algebras

Proposition

For $T$ a Lawvere theory, the category $T Alg$ has all small limits and colimits.

The limits and the filtered colimits in $T Alg$ are computed pointwise.

The category of algebras $T Alg$ has a zero object if and only if $T$ has a single nullary operation (Proposition 3.3 of Wraith).

An open problem

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. While in fact the conjecture is not true for all single-sorted algebraic theories (see Boone conjecture for further details), as was known by Soviet mathematicians, the general question “In which algebraic categories does (an analogue of) Higman’s embedding theorem hold?” is still interesting. In fact, it has been urged by F. W. Lawvere on several occasions that this Boone conjecture should be settled by the category theory community.

Related concepts

• algebraic theory / generalized algebraic theory / Lawvere theory / second order algebraic theory / 2-Lawvere theory / (∞,1)-algebraic theory

• internalization

• enriched Lawvere theory

• globular theory

• lambda theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

• PRO

• PROP

• C-systems

• Boone conjecture

• Freyd category

References

The origin of the categorical formulation of algebraic theories as Lawvere theories is in

• William Lawvere, Functorial Semantics of Algebraic Theories, Ph.D. thesis Columbia University (1963). Published with an author’s comment and a supplement in: Reprints in Theory and Applications of Categories 5 (2004) pp 1–121. (tac:tr5)

The concept was then streamlined in

• Samuel Eilenberg, J. B. Wright, Automata in General Algebras , Information and Control 11 (1967) pp.452-470.

Also still worthwhile reading are the following early papers:

• Gavin Wraith, Algebraic Theories , Aarhus Lecture Notes Series no.22 (1969). (pdf, 3,7MB)

• Gavin Wraith, Algebras over Theories , Colloquium Mathematicum XXIII no.2 (1971) pp.181-190. (link)

An early textbook treatment was in

• Bodo Pareigis, Categories and Functors , New York: Academic Press 1970. (chap. 3; link)

Modern 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

• J. Adámek, J. Rosicky, E. M. Vitale, Algebraic Theories - a Categorical Introduction to General Algebra , Cambrige UP 2010. (draft)

The concept of an internal algebraic theory in topos theory is dicussed in

• Peter Johnstone, Gavin Wraith, Algebraic theories in toposes , pp.141-242 in LNM 661 Springer Heidelberg 1978.

• Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014; sec. 6.4, pp.190-198)

Comparison with the notion of monads:

• Martin Hyland, John Power, The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theor. Comp. Sci. 172 (2007) 437-458 [doi:10.1016/j.entcs.2007.02.019, preprint]

An elementary proof of the equivalence between infinitary Lawvere theories and monads on the category of sets is given in Appendix A of

• Martin Brandenburg, Large limit sketches and topological space objects, arXiv:2106.11115.

Distributive laws for algebraic theories are discussed in

• Eugenia Cheng, Distributive laws for Lawvere theories , arXiv:1112.3076 (2011). (abstract)

Voevodsky proves an equivalence between Lawvere theories and l-bijective C-systems here:

• Vladimir Voevodsky, Lawvere theories and C-systems, arXiv:1512.08104 (2015). (abstract)

Other references are

• M. Jibladze, T. Pirashvili, Cohomology of Algebraic Theories , J. Algebra 137 no.2 (1991) pp.253–296.

• Steve Lack, Jiri Rosicky, Notions of Lawvere theory , arXiv:0810.2578. (abstract)

• 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.

Lawvere theories, and finitary monads, are identified with certain enriched categories and the passage from one to the other with a cocompletion, in

• Richard Garner, Lawvere theories, finitary monads and Cauchy-completion, 2013, arxiv

• Richard Garner and John Power, An enriched view on the extended finitary monad–Lawvere theory correspondence, 2017, arxiv

Some properties of Lawvere theories:

• V. Trnková, Representability and local representability of algebraic theories, algebra universalis 39.3 (1998): 121-144.

A short introductory/overview note:

• Andreas Nuyts, Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams, note