September 9, 2008

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.)

Posted at September 9, 2008 9:33 AM UTC

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

Posted by: David Corfield on September 9, 2008 12:07 PM | Permalink | Reply to this

Lawvere’s treatment of intensive and extensive quantities

So, from p. 14, we have

“extensive quantity” = homology cycles

“intensive quantitiy” = cohomology cocycle

?

On page 15 Grassman’s “extensive quantities” are mentioned. Now, that’s interesting, since these are of course the elemens of a Grassman algbra $\wedge^\bullet V$ which play a central role in the “$C^\infty qDGCAs$” mentioned above.

Posted by: Urs Schreiber on September 9, 2008 12:29 PM | Permalink | Reply to this

Perhaps a candidate for a tac reprint. It would be nice to know what happens on pp. 18-19 and pp. 25-26.

Posted by: David Corfield on September 9, 2008 12:33 PM | Permalink | Reply to this

I did some searching. If I understand correctly then an “intensive quantity” is taken to be essentially one that depends contravariantly on Spaces, while an “extensive quantity” is essentially one that depends covariantly.

So function algebras, $C^\infty : Spaces \to Algebras$ is an intensive quantity, while linear duals of such (distributions) are “extensive”.

I must admit that I haven’t figured out yet in which sense the words “intensive” and “extensive” are supposed to be suggestive here.

But in any case I suppose I can now answer your question: the quantities I was talking about in the above entry are “intensive”, in this sense (essentially being functions and sections on spaces).

Posted by: Urs Schreiber on September 9, 2008 12:50 PM | Permalink | Reply to this

A classic example of the difference between intensive and extensive properties is mass (extensive) and density (intensive).

Intensive quantities work well with products via projection.

Extensive quantities work well with coproducts via sum.

Posted by: David Corfield on September 9, 2008 1:20 PM | Permalink | Reply to this

Is it an issue that normally one thinks of multiplying an intensive quantity by an extensive quantity to yield an extensive quantity (density $\times$ volume = mass or integrating a density against a measure), where you are multiplying intensive quantities?