nLab
commutative algebraic theory

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.

(It is left as an exercise to the reader to formulate this notion in an element-free way.) 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.

If $f_1,\dots ,f_n$ 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 $\omega (f_1(a),\dots ,f_n(a))\in B$ is again a homomorphism, which is naturally called $\omega (f_1,\dots ,f_n)$. In this way $\mathrm{Hom}(A,B)$ is enriched as a model of the algebraic theory, and we have a closed category of models and homorphisms. Furthermore, this internal $\mathrm{Hom}$ has a left adjoint $\otimes$ for which the free model on one generator is a unit, so we have a closed monoidal category.

The notion of commutative algebraic theory was formulated in terms of monads by Anders Kock.

References

Can we have some please! This is something I want to use soon so it’d be nice to know where the details were originally worked out. —Andrew

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

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

See also here.

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