nLab
Lawvere theory

Contents

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

Definition

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

A homomorphism of such theories TTT \to T' is a product-preserving functor.

Remark

For TT a Lawvere theory, we are to think of the hom-set T(n,1):=T(x n,x)T(n,1) := T(x^n, x) as the set of nn-ary operations definable in the theory. For instance for TT the theory of abelian groups, T(2,1)T(2,1) includes operations like +:(x,y)x+y+ \colon (x, y) \mapsto x + y, :(x,y)xy- \colon (x, y) \mapsto x-y, and (x,y)2x3y(x, y) \mapsto 2 x - 3 y. For 00-ary or nullary operations, we have T(0,1)={0}T(0,1) = \{0\}.

Definition

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

A:TSet, A : T \to Set \,,

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

Remark

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

A(ϕ):U(A) nU(A) 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 TT.

Remarks

  1. It is common to adopt the (slightly evil) convention that every object of TT is equal to a chosen power of xx. Thus, if FinFin is the category of finite cardinals and functions between them, then the unique (up to isomorphism) product-preserving functor Fin opTFin^{op} \to T that takes the 1-element cardinal to xx 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 nxx^n \to x has a well-defined arity nn.

  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 xx. Finite-product theories CC can be regarded as a special case of multisorted Lawvere theories (see below) where the set of sorts is Ob(C)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 CC is a category with finite products, then a group (object) in CC may be defined as a product-preserving functor T GrpCT_{Grp} \to C. For example, a topological group may be identified with a functor T GrpTopT_{Grp} \to Top, and a Lie group with a product-preserving functor T GrpManT_{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 TT.

Variations

  • A multisorted or multityped Lawvere theory for a given set of sorts SS is a category with finite products CC together with a function i:SOb(C)i: S \to Ob(C) such that every object of CC is isomorphic to a finite product of objects of the form i(S)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 nn, an operation to take the supremum of nn elements. While Lawvere theories correspond to finitary monads on SetSet, 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 𝒮\mathcal{S} on objects nn \in \mathbb{N} with morphisms generated by the projections π i:n1\pi_i : n \to 1. This is the opposite category of the category FinSet

𝒮FinSet op. \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:

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

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

U T:TAlg𝒮AlgSet. 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 TT, let F(n)F(n) (for any natural number n0n \geq 0) be a free group on nn generators, and define the Lawvere theory T GrpT_{Grp} to be the category opposite to the category of free groups F(n)F(n) and group homomorphisms. The generic object xx of T GrpT_{Grp} is taken to be F(1)F(1).

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

FreeGroupGroupFreeGroup \hookrightarrow Group

creates coproducts in FreeGroupFreeGroup), so T GrpT_{Grp} has finite products, and we have F(n)=x nF(n) = x^n in T GrpT_{Grp}. Any group GG defines a product-preserving functor

T Grp=FreeGroup opGroup ophom(,G)SetT_{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 GrpT_{Grp}.

The other direction is more interesting. Let

M:T GrpSetM: T_{Grp} \to Set

be a model of T GrpT_{Grp}, i.e., a product-preserving functor. We will define a group structure on G=M(x)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 xx in T GrpT_{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 2x 1m: x^2 \to x^1

corresponds to a homomorphism

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

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

i 1:F(1)F(1)+F(1)=F(2)i 2:F(1)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 MM is assumed to preserve products, we obtain a map

G×G=M(x)×M(x)M(x 2)M(m)M(x)=GG \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 GG. The group identity and group inversion on GG are defined by following similar recipes.

It may be checked that the notion of homomorphism of T GrpT_{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 GrpT_{Grp}.

In particular, any hom-functor

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

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

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

induced by the nn projection maps x nxx^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 TT 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 TSetT \to Set, and

  • The free algebras are retrieved as the representable functors TSetT \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

Properties

Coequalizers of TT-algebras

Definition (congruence)

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

Observation/Definition

For AA a TT-algebra and for every relation RA(1)×A(1)R \subset A(1) \times A(1) there is a smallest congruence on AA containing RR. Write RA(1)×A(1)\langle R \rangle \subset A(1) \times A(1) for this smallest congruence.

Proposition

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

Proposition

For f,g:ABf,g : A \to B two morphisms of TT-algebras, the canonical morphism BB/f(a)g(a)|aA(1)B \to B/\langle \f(a)\sim g(a) | a \in A(1)\rangle is the coequalizer of ff and gg.

Free TT-algebras and underlying sets

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

i T:𝒮T. i_T : \mathcal{S} \to T \,.

By precomposition with i Ti_T we obtain a corresponding functor on TT-algebras, which we write

U T:TAlgSet U_T : T Alg \to Set

and call the underlying set functor.

Lemma

The functor U TU_T has a left adjoint F T:SetTAlgF_T : Set \to T Alg.

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

Proof

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

Observation

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

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

The adjunction isomorphism

TAlg(F T(S),A)Set(S,U T(A))(A(1)) nA(n) 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.

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

Notice that this extends to a functor

F T():FinSetTAlg F_T(-) : FinSet \to T Alg

which is the composite

F T():FinSeti T opT opjTAlg 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 op𝒮Ti_T : FinSet^{op} \simeq \mathcal{S} \to T from the theory of sets, described above.

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

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

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

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

Adjunctions of relatively free TT-algebras

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

Let T 1T_1 and T 2T_2 be Lawvere theories and f:T 1T 2f : T_1 \to T_2 a morphism. Write f *:T 2AlgT 1Algf^* : T_2 Alg \to T_1 Alg for the functor on categories of algebras induced by precomposition with ff.

Lemma

The functor f *f^* has a left adjoint f *:T 1AlgT 2Algf_* : T_1 Alg \to T_2 Alg.

Proof

Here is an elementary proof:

Let F T iU T iF_{T_i} \dashv U_{T_i} be the two free algebra/underlying-set adjunctions. For AA a T 1T_1-algebra there is a T 1T_1-congruence Γ\Gamma such that

AF T 1U T 1(A)/Γ. A \simeq F_{T_1} U_{T_1}(A) / \Gamma \,.

Since for any set SS we have U T 1F T 1(S)U T 2F T 2(S)U_{T_1} F_{T_1}(S) \subset U_{T_2} F_{T_2}(S) it follows that ΓT T 2F T 2U T 1(A)\Gamma \subset T_{T_2} F_{T_2} U_{T_1}(A). For Γ\langle \Gamma \rangle the smallest T 2T_2-congruence containing Γ\Gamma we have that F T 2U T 1(A)/ΓF_{T_2} U_{T_1}(A) / \langle \Gamma \rangle is a T 2T_2-algebra.

This one checks is f *Af_* A.

Here is a more high-powered way to obtain this using the monads K iK_i whose algebras are T iT_i-algebras:

for AA a T 1T_1-algebra let f *A:=K 2 K 1Af_* A := K_2 \circ_{K_1} A the the evident reflexive coequalizer

K 2K 1AK 2AK 2 K 1A K_2 K_1 A \stackrel{\to}{\to} K_2 A \to K_2 \circ_{K_1} A

in T 2AlgT_2 Alg.

Limits and colimits of TT-algebras

Proposition

For TT a Lawvere theory, the category TAlgT Alg has all small limits and colimits.

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

References

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

Textbook treatments are

  • Francis Borceux, Handbook of categorical algebra 2 – Categories and structures , Encyclopedia of Mathematics and its Applications, Cambridge University Press (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

Other references are

Revised on November 7, 2014 20:56:27 by Thomas Holder (89.204.139.237)