commutative algebraic theory




An algebraic theory is said to be commutative if its operations are algebra homomorphisms under any interpretation, generalizing the familiar case of the theory of commutative monoids.

A more general notion is that of monoidal monads.


Two operations, α\alpha and β\beta, of an algebraic theory are said to commute if for any matrix MM of elements, with the number of rows given by the arity of α\alpha and the number of columns by the arity of β\beta, one gets the same result whether one

  1. applies α\alpha to each column of MM and then β\beta to the resulting row, or
  2. applies β\beta to each row of MM and then α\alpha to the resulting column.

(We formulate this notion in an element-free way below.)

Note that an operation of arity 00 or 11 always commutes with itself; this is not necessarily the case for higher arities. Commuting nullary operations are necessarily equal.

Layer 1 [] x 1 1 x 1 2 x 1 n x 2 1 x 2 2 x 2 n x m 1 x m 2 x m n ] \begin{bmatrix} x_{1 1} & x_{1 2} & \dots & x_{1 n} \ x_{2 1} & x_{2 2} & \dots & x_{2 n} \ \vdots & \vdots & \ddots & \vdots \ x_{m 1} & x_{m 2} & \dots & x_{m n} \end{bmatrix} [] β ( x 1 1 , x 1 2 , , x 1 n ) β ( x 2 1 , x 2 2 , , x 2 n ) β ( x m 1 , x m 2 , , x m n ] \begin{bmatrix} \beta(x_{1 1}, x_{1 2}, \dots, x_{1 n}) \ \beta(x_{2 1}, x_{2 2}, \dots, x_{2 n}) \ \vdots \ \beta(x_{m 1}, x_{m 2}, \dots, x_{m n} \end{bmatrix} [] α ( x 1 1 x 2 1 x m 1 ) α ( x 1 2 x 2 2 x m 2 ) α ( x 1 n x 2 n x m n ) ] \begin{bmatrix} \alpha\begin{pmatrix} x_{1 1} \ x_{2 1} \ \vdots \ x_{m 1}\end{pmatrix} & \alpha\begin{pmatrix} x_{1 2} \ x_{2 2} \ \vdots \ x_{m 2}\end{pmatrix} & \dots & \alpha\begin{pmatrix} x_{1 n} \ x_{2 n} \ \vdots \ x_{m n}\end{pmatrix} \end{bmatrix} β ( α ( x 1 1 x 2 1 x m 1 ) α ( x 1 2 x 2 2 x m 2 ) α ( x 1 n x 2 n x m n ) ) \beta\begin{pmatrix} \alpha\begin{pmatrix} x_{1 1} \ x_{2 1} \ \vdots \ x_{m 1}\end{pmatrix} & \alpha\begin{pmatrix} x_{1 2} \ x_{2 2} \ \vdots \ x_{m 2}\end{pmatrix} & \dots & \alpha\begin{pmatrix} x_{1 n} \ x_{2 n} \ \vdots \ x_{m n}\end{pmatrix} \end{pmatrix} α ( β ( x 1 1 , x 1 2 , , x 1 n ) β ( x 2 1 , x 2 2 , , x 2 n ) β ( x m 1 , x m 2 , , x m n ) \alpha\begin{pmatrix} \beta(x_{1 1}, x_{1 2}, \dots, x_{1 n}) \ \beta(x_{2 1}, x_{2 2}, \dots, x_{2 n}) \ \vdots \ \beta(x_{m 1}, x_{m 2}, \dots, x_{m n} \end{pmatrix} β \beta β \beta α \alpha α \alpha

The operations that commute with a given set of operations in an algebraic theory form a subtheory. The centre of an algebraic theory is given by the operations that commute with all the operations of the theory. An algebraic theory is commutative if every pair of its operations commute. Another way of describing the centre is to say that it consists of those operations which are also homomorphisms; an algebraic theory is commutative if all of its operations are homomorphisms.

Here is a more formal definition, expressed in terms of structure on the monad T:SetSetT \colon Set \to Set associated with the algebraic theory.


TT is a commutative monad if there is an equality between two maps

α=β:TA×TBT(A×B)\alpha = \beta \colon T A \times T B \stackrel{\to}{\to} T(A \times B)


  • α\alpha is the composite

    TA×TBσ A,TBT(A×TB)T(τ A,B)TT(A×B)m(A×B)T(A×B).T A \times T B \stackrel{\sigma_{A, T B}}{\to} T(A \times T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \times B) \stackrel{m(A \times B)}{\to} T(A \times B).

    Here σ\sigma denotes the strength of the monad TT, and τ\tau its symmetric counterpart.

  • β\beta is the composite

    TA×TBτ TA,BT(TA×B)T(σ A,B)TT(A×B)m(A×B)T(A×B).T A \times T B \stackrel{\tau_{T A, B}}{\to} T(T A \times B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \times B) \stackrel{m(A \times B)}{\to} T(A \times B).

It is worth checking what this description gives more explicitly. Starting with a pair of elements in TA×TBT A \times T B,

(ω;a 1,,a m),(χ;b 1,,b n)\langle (\omega; a_1, \ldots, a_m), (\chi; b_1, \ldots, b_n)\rangle

(and here we should be considering equivalence classes of such formal operations), the map α\alpha sends this to

(ω(χ,,χ);(a 1,b 1),(a 1,b n)),,(a m,b 1),,(a m,b n))(\omega(\chi, \ldots, \chi); (a_1, b_1), \ldots (a_1, b_n)), \ldots, (a_m, b_1), \ldots, (a_m, b_n))

where ω(χ,,χ)\omega(\chi, \ldots, \chi) is the evident operation of arity mn=n++nm n = n + \ldots + n. In more detail, α((ω;a),(χ;b)\alpha(\langle (\omega; \vec{a}), (\chi; \vec{b} \rangle) is the end result of the sequence

(ω;a),(χ;b)σ(ω;(a 1,(χ;b)),,(a m,(χ;b)))Tτ(ω;(χ;(a 1,b)),,(χ;(a m,b)))m(ω(χ,,χ);(a 1,b),,(a m,b)).\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle \stackrel{\sigma}{\mapsto} (\omega; (a_1, (\chi; \vec{b})), \ldots, (a_m, (\chi; \vec{b}))) \stackrel{T\tau}{\mapsto} (\omega; (\chi; (a_1, \vec{b})), \ldots, (\chi; (a_m, \vec{b}))) \stackrel{m}{\mapsto} (\omega(\chi, \ldots, \chi); (a_1, \vec{b}), \ldots, (a_m, \vec{b})).

Similarly, the map β\beta sends the pair (ω;a),(χ;b)\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle to

(χ(ω,,ω);(a,b 1),,(a,b n))(\chi(\omega, \ldots, \omega); (\vec{a}, b_1), \ldots, (\vec{a}, b_n))

where χ(ω,,ω)\chi(\omega, \ldots, \omega) is the evident operation of arity nm=m++mn m = m + \ldots + m.

Abstract formulation

The category Th\mathbf{Th} of Lawvere theories is endowed with a symmetric monoidal product (called Kronecker product; see Freyd’s article in the references),

:Th×ThTh,\otimes \colon \mathbf{Th} \times \mathbf{Th} \to \mathbf{Th},

whereby (ST)(S \otimes T)-algebras are SS-algebras internal to TT-algebras, or equally well TT-algebras internal to SS-algebras. A commutative theory is tantamount to a commutative monoid in the symmetric monoidal category Th\mathbf{Th}.

If SS and TT are commutative theories, then their coproduct in the category of commutative theories is STS \otimes T.

Commutative theory as monoidal monad

Let TT be the SetSet-monad of a commutative theory. Then the map

α A,B:TA×TBT(A×B)\alpha_{A, B}: T A \times T B \to T(A \times B)

as defined above can be shown to be the structure map for a monoidal structure on TT, i.e., making TT a lax (symmetric) monoidal functor, and in fact the monad multiplication and unit become monoidal transformations. In other words, we get a monad in the 2-category of symmetric monoidal categories, lax symmetric monoidal functors, and monoidal transformations: a monoidal monad.

In fact, it may be shown that commutative Lawvere theories on SetSet are precisely the same things as (finitary) symmetric monoidal monad structures on (Set,×)(Set, \times), as shown by Anders Kock. For more on this, see monoidal monad.

As commutative monoid in a duoidal category

The commutativity of a theory can also be expressed as an abstract property of a monoid in a duoidal category, specialized to the duoidal category of finitary endofunctors.


Closed monoidal structure on algebras

We discuss that the category of algebras for an algebraic theory over a commutative algebraic theory is canonically a closed symmetric monoidal category (Keigher 78, Seal 12).

If f 1,,f nf_1,\ldots , f_n are homomorphisms ABA \to B of models (algebras) of a commutative algebraic theory, and ω\omega is an nn-ary operation of it, then the function ABA \to B given by sending aAa \in A to ω(f 1(a),,f n(a))B\omega(f_1(a),\ldots ,f_n(a)) \in B is again a homomorphism, which is naturally called ω(f 1,,f n)\omega(f_1,\ldots ,f_n). In this way Hom(A,B)Hom(A,B) is enriched as a model of the algebraic theory, and we have a closed category of models and homorphisms. Furthermore, this internal HomHom has a left adjoint \otimes for which the free model on one generator is a unit, so we have a closed monoidal category, in fact a closed symmetric monoidal category.

The monoidal structure \otimes can be extracted by a straightforward generalization of the usual tensor product of abelian groups (or of commutative monoids), where “bilinearity conditions” = “linearity in separate variables” is replaced by “TT-homomorphicity in separate variables”, where TT is the monad of the algebraic theory.

In slightly more detail, if AA and BB are TT-algebras, the tensor product ABA \otimes B ought to be T(A×B)T(A \times B) modulo equivalences which we may write suggestively as

ω(a 1,,a m)χ(b 1,,b n)(ω(χ,,χ);a 1b 1,,a mb n)\omega(a_1, \ldots, a_m) \otimes \chi(b_1, \ldots, b_n) \sim (\omega(\chi, \ldots, \chi); a_1 \otimes b_1, \ldots, a_m \otimes b_n)

where the left side is represented by a composite

TA×TBξ A×ξ BA×Bu(A×B)T(A×B)T A \times T B \stackrel{\xi_A \times \xi_B}{\to} A \times B \stackrel{u(A \times B)}{\to} T(A \times B)

(the ξ\xi‘s are TT-algebra structures), and the right side by the monoidal structure map on TT,

α A,B:TA×TBT(A×B).\alpha_{A, B} \colon T A \times T B \to T(A \times B).

In more detail still, ABA \otimes B is the following coequalizer in Alg TAlg_T:

T(TA×TB) Tα TT(A×B) T(ξ A×ξ B) m T(A×B) AB\array{ T(T A \times T B) & \stackrel{T\alpha}{\to} & T T(A \times B) & \\ & ^\mathllap{T(\xi_A \times \xi_B)} \searrow & \downarrow^\mathrlap{m} & \\ & & T(A \times B) & \to & A \otimes B }

(Seal 12, section 2.2 and theorem 2.5.5)

This construction carries over to the wider context of monoidal monads, see tensor product of algebras over a commutative monad.


See (more generally) examples of commutative monads.


The notion of commutative algebraic theory was introduced by Fred Linton:

  • Fred Linton, Autonomous equational categories, Journal of Mathematics and Mechanics 15.4 (1966), 637-642.

It was formulated in terms of monads by Anders Kock.

  • Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970), 1–10.

The closed category-structure on the EM-category of monoidal monads was studied in

  • Anders Kock, Strong functors and monoidal monads, Arhus Universitet, Various Publications Series No. 11 (1970). PDF.

  • Anders Kock, Closed categories generated by commutative monads (pdf)

and the monoidal category-structure in

  • William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293 (NUMDAM, pdf)

  • Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)

There is related MO discussion.

The Kronecker product of theories was introduced in an article of Freyd:

  • Peter Freyd, Algebra valued functors in general and tensor products in particular, Colloq. Math. 14 (1966), 89-106.

Recently Nikolai Durov rediscovered that notion for the purposes of geometry (under the name commutative algebraic monad), constructed their spectra (generalizing the spectrum of Grothendieck) and theory of generalized schemes on this basis. There is a generalized version of the Eckmann–Hilton argument concerning commutative finitary monads. Much detail including many examples and further constructions are in his thesis

Last revised on November 28, 2021 at 12:30:32. See the history of this page for a list of all contributions to it.