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I apologize in advance for my ignorance about universal algebra, but I have just recently found out about the beautiful notion of Lawvere theory, and the idea to do algebra without ever talking about underlying sets appeals to me a lot. Unfortunately I am not able to find any references that take this point of view and develop basic algebra in this way. So is it possible to define “(commutative) ring” and “module over a (commutative) ring” from scratch using Lawvere theories? If so, does anyone know a reference where this is done?
More generally, suppose that I have some Lawvere theory of “commutative ring”-like structures (I have in mind things like smooth algebras or even log-rings). Is there a formal way to define a notion of “module” in any such context?
My motivation would be to define simplicial commutative rings and simplicial modules over them as the models of the respective theories in simplicial sets, and then to try to define cotangent complexes in this language, in a way that hopefully generalizes to other contexts like logarithmic geometry, etc.
I’m actually not sure what you mean by “(n)ever talking about underlying sets”, since the underlying-set functor $U: Alg_T \to Set$ plays an important role in the general theory. A commutative ring can be described as a product-preserving functor $Poly^{op} \to Set$ where $Poly$ is a Kleisli category whose morphisms are functions $S \to \mathbb{Z}[T]$ into polynomial rings. But maybe this isn’t what you’re asking for? There’s also an option to replace the codomain $Set$ by any category $C$ with finite products, where the result is a general internal ring object in $C$. For example, taking $C$ to be the category of simplicial sets.
A Lawvere theory can also be viewed as a “cartesian operad”. I once wrote up some notes on this here, although I think Mike might know of more standard sources. There’s also a general notion of module over an algebra over an operad. Putting these two together would give you some sort of notion of module over an algebra of a Lawvere theory, although I haven’t thought about this much or what it gives exactly in special cases. The more classical set-up for this would be to take an ordinary operad taking values in a symmetric monoidal category like the category of abelian groups or vector spaces, for example the associative algebra operad or Lie algebra operad, and there the general notion of module over an operad specializes to the correct notion (of module over an algebra or over a Lie algebra).
One way to define modules over algebras over some Lawvere theory is to use the characterization of the ordinary category of modules over commutative rings as the “tangent category” of that of rings, e.g. here and here. This tangent category construction directly gives a concept of cotangent complexes, as here.
You may be interested in taking a look at the related entries (∞,1)-algebraic theory and (2,1)-algebraic theory of E-infinity algebras.
@Todd:
I’m actually not sure what you mean by “(n)ever talking about underlying sets” […]
I suppose what I meant was not talking about elements in the underlying sets.
A commutative ring can be described as a product-preserving functor $Poly^{op} \to Set$ where $Poly$ is a Kleisli category whose morphisms are functions $S \to \mathbb{Z}[T]$ into polynomial rings.
That is the type of thing I am looking for, except I would like to be able to define $Poly$ without having yet defined “polynomial ring”. I suppose I can take formal symbols $\mathbf{Z}[T]$ as objects, and define the morphism sets appropriately. Does a similar thing work for (left) $R$-modules though, for $R$ a unital ring?
A Lawvere theory can also be viewed as a “cartesian operad”.
Thanks, this is interesting! I wanted to avoid operads though, if possible (mainly because this approach is technically very complicated in the $(\infty,1)$-case; cf. Lurie’s Higher Algebra…).
@Urs:
Thanks! I had thought about simply defining R-mod as the category of abelian group objects in CRing/R, but I wasn’t sure it was reasonable for other Lawvere theories than ordinary commutative rings. Now that I think about it though, I think this equivalence holds in the cases I care about. And as you say, one gets a definition of cotangent complex directly from this, which is nice.
The Lawvere theory for left $R$-modules is very easy to construct: you just take the free additive category generated by $R$ (considered as a one-object $\mathbf{Ab}$-enriched category).
Incidentally, the category of internal abelian groups in the category of $\mathbb{T}$-algebras for a Lawvere theory $\mathbb{T}$ is equivalent to the category of $\mathbb{T}^{ab}$-algebras, where $\mathbb{T}^{ab}$ is the Lawvere theory obtained by “forcing” $\mathbb{T}$ to be additive. In particular, it is a category of left modules for a certain ring.
@Zhen Lin, that’s very useful, thanks. Do you know a reference for these facts?
They’re both easy. You might find some information by searching for Beck modules.
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