The n-Category Café

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.)