symmetric monoidal (∞,1)-category of spectra
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, $\alpha$ 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
(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.
Here is a more formal definition, expressed in terms of structure on the monad $T \colon Set \to Set$ associated with the algebraic theory.
$T$ is a commutative monad if there is an equality between two maps
where
$\alpha$ is the composite
Here $\sigma$ denotes the strength of the monad $T$, and $\tau$ its symmetric counterpart.
$\beta$ is the composite
It is worth checking what this description gives more explicitly. Starting with a pair of elements in $T A \times T B$,
(and here we should be considering equivalence classes of such formal operations), the map $\alpha$ sends this to
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
Similarly, the map $\beta$ sends the pair $\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle$ to
where $\chi(\omega, \ldots, \omega)$ is the evident operation of arity $n m = m + \ldots + m$.
The category $\mathbf{Th}$ of Lawvere theories is endowed with a symmetric monoidal product (called Kronecker product; see Freyd’s article in the references),
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 $S \otimes T$.
Let $T$ be the $Set$-monad of a commutative theory. Then the map
as defined above can be shown to be the structure map for a monoidal structure on $T$, i.e., making $T$ 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 $Set$ 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.
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.
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
where the left side is represented by a composite
(the $\xi$’s are $T$-algebra structures), and the right side by the monoidal structure map on $T$,
In more detail still, $A \otimes B$ is the following coequalizer in $Alg_T$:
(Seal 12, section 2.2 and theorem 2.5.5)
This construction carries over to the wider context of monoidal monads.
The notion of commutative algebraic theory was formulated in terms of monads by Anders Kock.
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:
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