nLab
Lawvere theory

Idea

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

A Lawvere theory or finite-product theory is a category $T$ with finite cartesian products in which every object is isomorphic to a finite cartesian power $x^n$ of a distinguished object $x$ (called the generic object for the theory $T$). It is common to adopt the (slightly evil) convention that every object of $T$ is equal to a chosen power of $x$. Thus, if $\mathrm{Fin}$ is the category of finite cardinals and functions between them, then the unique (up to isomorphism) product-preserving functor

$${\mathrm{Fin}}^{\mathrm{op}}\to T$$Fin^{op} \to T

that takes the 1-element cardinal to $x$ is commonly supposed to be surjective on objects (rather than, less evilly, essentially surjective), or even an isomorphism on objects so that each morphism $x^n\to x$ has a well-defined arity $n$.

A model of $T$ is a product-preserving functor $T\to \mathrm{Set}$, and homomorphism of models is a natural transformation between such functors. A morphism of theories $T\to T\prime$ is again a product-preserving functor. Thus, ${\mathrm{Fin}}^{\mathrm{op}}$ (the “theory of equality”) is initial in the category of Lawvere theories.

Examples

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

$$\mathrm{FreeGroup}\hookrightarrow \mathrm{Group}$$FreeGroup \hookrightarrow Group

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

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

The other direction is more interesting. Let

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

be a model of $T_{\mathrm{Grp}}$, 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 $T_{\mathrm{Grp}}$ is a “generic group”, so we first need to understand how multiplication works there. The idea is that the product in the generic group

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

corresponds to a homomorphism

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

which by freeness corresponds to an element $1\to F(2)$, and the element we are after should the product $ab$ 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

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

$$G\times G=M(x)\times M(x)\cong M(x^2)\stackrel{M(m)}{\to }M(x)=G$$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 $T_{\mathrm{Grp}}$-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 $T_{\mathrm{Grp}}$.

In particular, any hom-functor

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

$${\mathrm{hom}}_{T_{\mathrm{Grp}}}(x,-)\to {\mathrm{hom}}_{T_{\mathrm{Grp}}}(x^n,-)$$\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 $T\to \mathrm{Set}$, and

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

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.

Remarks

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

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 $\mathrm{Ob}(C)$ 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.

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 $i:S\to \mathrm{Ob}(C)$ 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.

• A Fermat theory is a Lawvere theory equipped with a notion of differentiation.