nLab commutative algebraic theory

Idea

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.

Definition

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

1. applies $\alpha$ to each column of $M$ and then $\beta$ to the resulting row, or
2. applies $\beta$ to each row of $M$ and then $\alpha$ to the resulting column.

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

Note that an operation of arity $0$ or $1$ always commutes with itself; this is not necessarily the case for higher arities. Commuting nullary operations are necessarily equal.

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.

This concept may also be formalized in the language of monads, where it naturally generalizes to the concept of commutative monad. Suppose given an algebraic theory that admits the construction of a free algebra on any set, so that the theory may be equivalently described by a monad $T$ on $Set$, the category of sets. The monad $T$ is automatically strong, i.e., carries a left strength

$$\sigma_{A,B} \,\colon\, A \times (T B) \longrightarrow T(A \times B)$$

(by this example), from which a right strength

$$\tau_{A,B} \,\colon\, (T A) \times B \longrightarrow T(A \times B)$$

can be derived by exploiting the symmetry (braiding) of the cartesian product.

In this notation, we have the following definition.

Definition

(commutative monads on $Set$)
$T$ is called commutative if there is equality $\alpha = \beta$ between the two maps

$$\alpha , \beta \;\colon\; T A \times T B \rightrightarrows T(A \times B) \,,$$

where

• $\alpha$ is the composite

  (1)$$\alpha_{A,B} \;\colon\; T A \times T B \overset{ \sigma_{A, T B} }{\longrightarrow} T(A \times T B) \overset{ T(\tau_{A, B}) }{\longrightarrow} T T(A \times B) \overset{ m(A \otimes B) }{\longrightarrow} T(A \times B) \,.$$

• $\beta$ is the composite

  (2)$$\beta_{A,B} \;\colon\; T A \times T B \overset{ \tau_{T A, B} }{\longrightarrow} T(T A \times B) \stackrel{ T(\sigma_{A, B}) }{\longrightarrow} T T(A \times B) \stackrel{ m(A \times B) }{\longrightarrow} T(A \times B) \,.$$

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

$$\langle (\omega; a_1, \ldots, a_m), (\chi; b_1, \ldots, b_n)\rangle$$

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

$$(\omega(\chi, \ldots, \chi); (a_1, b_1), \ldots (a_1, b_n)), \ldots, (a_m, b_1), \ldots, (a_m, b_n))$$

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

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

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

$$(\chi(\omega, \ldots, \omega); (\vec{a}, b_1), \ldots, (\vec{a}, b_n))$$

where $\chi(\omega, \ldots, \omega)$ is the evident operation of arity $n m = m + \ldots + m$.

Relation to commutativity of operations

A classical example of a commutative algebraic theory is the theory of Abelian groups. This is a slightly misleading example: it is not true in general that the operations of a commutative algebraic theory are always commutative in the sense that they are invariant under permutation of their operands. That this is true for Abelian groups follows from the Eckmann–Hilton argument, since a binary operation $\otimes$ commutes with itself when

$$(a \otimes b) \otimes (c \otimes d) = (a \otimes c) \otimes (b \otimes d)$$

and, since the theory of groups is associative and unital, we can take $a = d = 1$ and deduce that

$$b \otimes c = c \otimes b.$$

However, if we suppose only a binary operation that commutes with itself, then the operation is not invariant under a transposition of its arguments, since there are models of that theory like $(a, b) \mapsto a \otimes b \coloneqq a$ where such invariance is violated.

There is a fairly general situation in which commutativity of an operator with itself implies that the operation is invariant under permutation of arguments.

Proposition

Let $f$ and $g$ be $n$-ary operations and let $1$ be a constant that is a unit for both $f$ and $g$, in the sense that $x \vdash f(1, \ldots, 1, x, 1, \ldots, 1) = x$ and similarly for $g$. If $f$ and $g$ commute, then $f = g$ and $f$ is invariant under permutation of its operands.

Proof

Let $\sigma$ be a permutation of $\{ 1, \ldots, n \}$. Let $y_1, \ldots, y_n$ be variables. Define a matrix $\{ x_{ij} \}_{ij}$ where $x_{i \sigma_i} = y_i$ and $x_{ij} = 1$ otherwise. Then commutativity of $f$ and $g$ implies that

$$f(g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{n1}, \ldots, x_{nn})) = g(f(x_{11}, \ldots, y_{n1}), \ldots, f(y_{1n}, \ldots, y_{nn}))$$

and hence that

$$f(g(1, \ldots, 1, y_1, 1, \ldots, 1), \ldots, g(1, \ldots, 1, y_n, 1, \ldots, 1)) = g(f(1, \ldots, 1, y_{\sigma_1}, 1, \ldots, 1), \ldots, f(1, \ldots, 1, y_{\sigma_n}, 1, \ldots, 1))$$

hence

$$f(y_1, \ldots, y_n) = g(y_{\sigma_1}, \ldots, y_{\sigma_n})$$

Taking $\sigma$ to be the identity permutation, we have that $f = g$. It then follows that $f$ is invariant under permutation of its operands.

This implies, for instance, that the $\otimes$ of an Abelian group is commutative.

Corollary

Suppose a given algebraic theory is commutative and every operation has a unit. Then every operation is invariant under permutation of its operands, and there is at most one $n$-ary operation for each $n \in \mathbb{N}$.

Proof

In a commutative algebraic theory, there is at most constant, so the units for each operator are equal. The result then follows from the previous proposition, since every operation commutes with every other operation.

Abstract formulation

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

$$\otimes \colon \mathbf{Th} \times \mathbf{Th} \to \mathbf{Th},$$

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

If $S$ and $T$ are commutative theories, then their coproduct in the category of commutative theories is $S \otimes T$.

Commutative theory as monoidal monad

Let $T$ be the $Set$-monad of a commutative theory. Then the map (1)

$$\alpha_{A, B}: T A \times T B \to T(A \times B)$$

can be shown to be the structure map for a monoidal structure on $T$, i.e., making $T$ a (symmetric) lax monoidal functor, and in fact the monad multiplication and unit become monoidal natural 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 $Set$ are precisely the same things as (finitary) symmetric monoidal monad structures on $(Set, \times)$, 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.

Properties

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 $f_1,\ldots , f_n$ are homomorphisms $A \to B$ of models (algebras) of a commutative algebraic theory, and $\omega$ is an $n$-ary operation of it, then the function $A \to B$ given by sending $a \in A$ to $\omega(f_1(a),\ldots ,f_n(a)) \in B$ is again a homomorphism, which is naturally called $\omega(f_1,\ldots ,f_n)$. In this way $Hom(A,B)$ is enriched as a model of the algebraic theory, and we have a closed category of models and homorphisms. Furthermore, this internal $Hom$ 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 “$T$-homomorphicity in separate variables”, where $T$ is the monad of the algebraic theory.

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

$$\omega(a_1, \ldots, a_m) \otimes \chi(b_1, \ldots, b_n) \sim (\omega(\chi, \ldots, \chi); a_1 \otimes b_1, \ldots, a_m \otimes b_n)$$

where the left side is represented by a composite

$$T A \times T B \stackrel{\xi_A \times \xi_B}{\to} A \times B \stackrel{u(A \times B)}{\to} T(A \times B)$$

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

$$\alpha_{A, B} \colon T A \times T B \to T(A \times B).$$

In more detail still, $A \otimes B$ is the following coequalizer in $Alg_T$:

$$\array{ T(T A \times T B) & \stackrel{T\alpha}{\to} & T T(A \times B) & \\ & ^\mathllap{T(\xi_A \times \xi_B)} \searrow & \downarrow^\mathrlap{m} & \\ & & T(A \times B) & \to & A \otimes B }$$

(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.

Examples

See (more generally) examples of commutative monads.

Related pages

• tensor product theory

References

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

• Nikolai Durov, New approach to Arakelov geometry, arXiv:0704.2030