### Smooth Differential Graded Algebra?

#### Posted by Urs Schreiber

A while ago I had a discussion with Todd Trimble about how to define “generalized smooth” differential graded-commutative algebras (DGCAs), generalizing the “generalized smooth”-algebras, called $C^\infty$-algebras, discussed in the book by Moerdijk & Reyes. I think back then we fell short of arriving at a satisfactory conclusion.

As I mentioned, I would like to pick up that thread again and chat about some ideas.

This here is the general motivation:

given a category $S$ of “test objects”, generalized *spaces* are presheaves on $S$, namely things that can be *probed* by throwing objects of $S$ into them, and generalized *quantities* (to be thought of as numbers, functions, sections, etc. as discussed in more detail below) are co-presheaves on $S$.

If $S := CartesianSpaces$ is the full subcategory of Manifolds on the manifolds $\mathbb{R}^n$, for all $n \in \mathbb{N}$, then the spaces in question are something like “smooth spaces” (in particular, if their underlying presheaves happen to be *concrete sheaves*, these these are diffeological or Chen-smooth spaces) and then the quantities are something like “smooth quantities” (in particular, if the underlying co-presheaves happen to be monoidal, these are those $C^\infty$-algebras).

Given such a notion of spaces, there is an obvious notion of *higher* spaces: pick your favorite definition of $\infty$-groupoid. Then a *higher degree space*, an $\infty$-space, should be an $\infty$-groupoid internal to the above spaces.

What is the analog of this on the side of “quantities”? What is an $\infty$-quantity? There are several possible answers one could come up with, I suppose, such as the answer by David Spivak, who replaces co-presheaves by simplicial co-presheaves and hence essentially follows the above $\infty$-ization of spaces.

But here I am interested in a different kind of answer which supposes that $\infty$-quantities corresponding to $\infty$-spaces in the above sense are something like “**quasi-free differential graded commutative $C^\infty$-algebras**” , $C^\infty$qDGCAs – to be be determined.

There is a reason for this assumption, namely $\infty$-Lie theory, but that is not of concern right now. Here I just want to talk about possible definitions of $C^\infty$-qDGCAs, an interesting question in its own right.

The idea I want to propose is simple. The goal is to have it “as simple as possible but no simpler”. Maybe you can help me check if that’s achieved, especially concerning the “no simpler”-part (i.e. the mistakes).

Here goes:

Write $Quantities := Set^{CartesianSpaces}$ for the category of co-presheaves on CartesianSpaces. As a co-presheaf category, this is a monoidal category with tensor product of $A,B \in Quantities$ given by $A \times B : \mathbb{R}^k \mapsto A(\mathbb{R}^k) \times B(\mathbb{R}^k) \,.$

In any monoidal category we can consider monoids:

Write $Algebras := Monoids(Quantities)$

for the category of monoids internal to Quantities. Every $C^\infty$-algebra of Moerdijk-Reyes canonically becomes an object of Algebras by using postcomposition with the maps $\cdot^k : \mathbb{R}^k \times \mathbb{R}^k \to \mathbb{R}^k$ of componentwise multiplication in $\mathbb{R}$.

Write
$Spaces := Sheaves(CartesianSpaces)$
for the category of sheaves on CartesianSpaces. For every $X \in Spaces$ we get an object $C^\infty(X) \in Algebras$, the *algebra of functions* whose underlying co-presheaf is
$C^\infty(X) : \mathbb{R}^k \mapsto Hom_{Spaces}(X, \mathbb{R}^k)$
and whose monoidal structure comes from postcomposition with the $\cdot^k$ from above:
$Hom(X, \mathbb{R}^k)
\times
Hom(X, \mathbb{R}^k)
\stackrel{\simeq}{\to}
Hom(X, \mathbb{R}^k \times \mathbb{R}^k)
\stackrel{Hom(-,\cdot^k)}{\to}
Hom(X, \mathbb{R}^k)
\,.$
For $X$ an ordinary manifold, $C^\infty(X)$ is its ordinary algebra of smooth functions.

Now comes the point: Since the $Algebras$ above are monoid objects, we can consider *modules* internal to $Quantities$ of objects in $Algebras$. (In fact, there should be a monoidal bicategory $Bimod(Quantities)$.)

Let $E \to X$ be a vector bundle internal to $Spaces$ and consider the set $Sections(E) \in Sets$ of its sections. The assignment $\Gamma(E) : \mathbb{R}^k \mapsto Sections(E \otimes \mathbb{R}^k)$ extends naturally to a co-presheaf on $CartesianSpaces$, hence to an object in $Quantities$. This naturally comes with an action of $C^\infty(X) \in Algebras$, where in components the action is given by postcomposition with $\cdot^k$ acting on the trivial bundle part: $\array{ Hom(X,\mathbb{R}^k) \times Sections(E \otimes \mathb{R}^k) \\ \downarrow^{\subset} \\ Hom(X,\mathbb{R}^k) \times Hom(X,E \otimes \mathbb{R}^k) \\ \downarrow^\simeq \\ Hom(X,( E \otimes \mathbb{R}^k) \times \mathbb{R}^k ) \\ \downarrow \\ Hom(X,E \otimes \mathbb{R}^k) }$

For $C \in Quantities$ an $(A \in Algebras)$-module, There is the obvious notion of dual-over-$A$ module $C^* := Hom_{A-Modules}(C,A)$. Using all this, the standard definition of qDGCAs should now straightforwardly generalize to the generalized smooth context:

Definition: A *quasi-free differential graded-commutative algebra* (qDGCA) over $A \in Algebras$ is a non-positively graded cochain complex $V$ of $A$-modules internal to $Quantities$ together with a degree +1 differential
$d : \wedge^\bullet_A V^* \to \wedge^\bullet_A V^*$.

I am thinking that all the ingredients I glossed over here have the obvious straightforward definition. But maybe I should check this in more detail.

(One might want to add to the above definition the condition that $V$ is degree-wise projective.)

## Re: Smooth Differential Graded Algebra?

How does Lawvere’s treatment of intensive and extensive quantities fit with what you’re saying?