The duality of space and quantity is usefully refined to a higher category theoretical context:
while the idea of ∞-space is well established in terms of the notion of ∞-stack, under an $\infty$-quantity we here understand the corresponding dual notion.
As
so
A typical co-sheaf condition is that underlying generalized algebras. The corresponding $\infty$-quantities in this case are therefore generalized cosimplicial algebras. Under the monoidal Dold-Kan correspondence these are identified with differential graded algebras.
Therefore, to a large extent the theory of $\infty$-quantities turns out to be just the theory of differential graded algebras reinterpreted from a more abstract nonsense perspective that we regard as helpful for making the relation to the theory of ∞-stacks usefully transparent.
A typical $\infty$-quantity is
that identifies under the monoidal Dold-Kan correspondence with
In particular the notion of $\infty$-quantity serves for us the purpose of providing the bridge between the definition of ∞-Lie algebroids as infinitesimal ∞-Lie groupoids and their widely used description in terms of differential graded algebra: the latter is the image under the monoidal Dold-Kan correspondence of the $\infty$-quantity of functions on the former.
The categorification of the notion of space is that of ∞-space – a higher categorical presheaf usually called an ∞-stack. These may be modeled by simplicial presheaves.
Here we discuss the notion dual to the notion $\infty$-space/$\infty$-stack/simplicial presheaf in the sense of space and quantity: that of $\infty$-quantity .
This is such that for instance in the smooth context of smooth ∞-stack – i.e. a Lie ∞-groupoid – the $\infty$-quantity $C^\infty(A)$ dual to a Lie ∞-groupoid $A$ is the cosimplicial algebra of smooth functions on neighbourhoods of identities in $A$ which turns out to be the Chevalley-Eilenberg algebra of the corresponding Lie ∞-algebroid.
For instance
the $\infty$-quantity of functions on the path ∞-groupoid $\Pi(X)$ is the deRham complex of differential forms on $X$;
the $\infty$-quantity of functions on the delooping $\mathbf{B}G$ of a Lie group $G$ is the Chevalley-Eilenberg algebra of its Lie algebra.
Recall from the discussion at models for ∞-stack (∞,1)-toposes that simplicial presheaves model generalized spaces, in the form of ∞-groupoids with extra structure (smooth structure, for instance, in the case of smooth ∞-stacks).
In the sense of space and quantity the concrete dual notion obtained by homming into objects of the underlying site should be tought of as $(\infty,1)$-quantities. This way we obtain model for $(\infty,1)$-quantities in terms of a category of cosimplicial copresheaves $[C,[\Delta, Set]]\,.$
For $C =$ CartSp those copresheaves that respect products in $C$ are generalized smooth algebras. By the dual monoidal Dold-Kan correspondence cosimplicial smooth algebras are equivalent to differential graded smooth algebras (in non-negative degree), namely differential graded algebras in the context of generalized smooth algebras. Therefore our $(\infty,1)$-quantities are also modeled by cochain complexes of copresheaves and in this incarnation they reproduce various entities familiar in homological algebra and Lie theory.
Let $C$ be a site with products.
We model $\infty$-quantities on $C$ the way simplicial presheaves on $C$ are models for ∞-space (∞,1)-toposes.
A special case of particular interest to keep in mind is the choice $C =$ CartSp. For that choice of test spaces we have that
∞-spaces modeled on $C$ are Lie ∞-groupoids
quantities (1-quantities) modeled on $C$ are generalized smooth algebras;
$\infty$-quantities modeled on $C$ include examples such as smooth singular cohomology and Lie ∞-algebroids.
We shall write $CoSh(C) \subset CoPSh(C) := [C,Set]$ for the full subcategory of that on those co-presheaves $A$ on $C$ that satisfy the “co-sheaf” condition that for all there is an isomorphism
natural in $U,V$.
For $C =$ CartSp the cosheaves on $C$ are the generalized smooth algebras
For $K$ a cosimplicial object we shall write
the degree increasing maps as $d_i : K^n \to K^{n+1}$
the degree lowering maps as $s_i : K^{n+1} \to K^n$.
for the category of cosimplicial cosheaves on $C$, the cosimplicial objects in the category of cosheaves.
We regard $CoSCoSh(C)$ as a category with weak equivalences by declaring a morphism to be a weak equivalence if it induces an isomorphism on cohomotopy groups. (…explain…)
Under the forgetful functor from generalized smooth algebras to the underlying algebras, these cosimplicial cosheaves map to cosimplicial algebras.
For $C =$ CartSp the condition on weak equivalences above becomes under the Dold-Kan correspondence the condition that a morphism is a weak equivalence precisely if under the Moore cochain complex functor it induces an isomorphism on cochain cohomology.
Moreover, in this case the forgetful functor from generalized smooth algebras to ordinary algebras sends $CoSCoSh(C)$ to the category $[\Delta,Algebras]$ of cosimplicial algebras. There is a standard model category structure on these, with the weak equivalences as above, and the fibrations the objectwise surjections. See definition 9.1 of
In more detail this means that a morphism $f : A \to B$ is a weak equivalence if for all $\mathbb{R}^n \in C$ the morphism $f(U) : A(U) \to B(U)$ of cosimplicial abelian groups (using the additive structure of generalized smooth algebras) induces under the dual normalized Moore complex a morphism $N^\bullet f(U) : N^\bullet A(U) \to N^\bullet(B(U))$ that induces an isomorphism on cochain complex cohomology.
Recall that for $X$ a presheaf on $C$ its generalized smooth algebra is its Isbell dual copresheaf
where $Y$ is the Yoneda embedding.
This extends to a functor
from simplicial presheaves to cosimplicial smooth algebras by degreewise application: for $X_\bullet \in SPSh(X)$ we have
For $X_\bullet \in SConSh(C)$ a simplicial concrete sheaf (a simplcial diffeological space) there is an obvious notion of open simplicial neighbourhood $V_\bullet \subset X_\bullet$ of the entirely degenerate simplices (those in the image of $X([k]\to [0])$). Let
be the $(\infty,1)$-quantity of local functions on $X$.
Let $C =$ CartSp, so that an $\infty$-quantity modeled on $C$ is a generalized smooth algebra.
As described at monoidal Dold-Kan correspondence, the Moore cochain complex $C(K)$ of a cosimplicial algebra $K$ is a differential graded algebra, whose product is the cup product
We call $C(K)$ the differential graded algebra given by the $\infty$-quantity $K$.
Let $C =$ CartSp.
Let $\Pi(X)$ be the Lie ∞-groupoid that is the path ∞-groupoid of $X$. The Moore cochain complex associtated with the $\infty$-quantity $C^\infty(\Pi(X)) := C^\infty(X^{\Delta^\bullet_C})$ of functions on $\Pi(X)$ is manifestly the one that computes singular cohomology (with values in $\mathbb{R}$).
The monoidal structure induced in $C^\infty(\Pi(X))$ under the monoidal Dold-Kan correspondence is manifestly the ordinary cup product on singular cohomology.
Let $G$ be a Lie group and let $\mathbf{B}G$ be its delooping regarded as a Kan complex valued simplicial presheaf (on Diff or CartSp or the like). The cosimplicial smooth algebra $C^\infty(\mathbf{B}G)$ has in degree $k$ the smooth algebra of $\mathbb{R}$-valued functions on $G^{\times_k}$. The differential of the corresponding dual Moore complex $N^\bullet(C^\infty(\mathbf{B}G))$ is the one that computes smooth group cohomology on $G$ with coefficients in $\mathbb{R}$ with the trivial module structure.
For more on this see
See also
Consider again the example of singular cohomology of a smooth space $X$ above. In the sense of synthetic differential geometry we have a natural restriction map
from functions on simplices in $X$ to functions on infinitesimal simplices., i.e. from functions on the full path ∞-groupoid to functions on just the infinitesimal singular simplicial complex of $X$.
The cosimplicial copresheaf $C^\infty(X^{\Delta^\bullet_{inf}})$ we call the $\infty$-quantity of functions on infinitesimal simplices in $X$.
For later reference we list in detail the interpretation of the face and degenercy maps in this cosimplicial object.
First think of $X^{\Delta^k_{inf}} := [\Delta^k_{inf}, X]$ as the space of infinitesimal $k$-simplices in $X$ (formalized as such in some context that need not concern us here).
The maps $\delta_i : [k] \to [k+1]$ induce the face maps
that send a $(k+1)$-simplex to its $i$th $k$-face.
The maps $\sigma_i : [k+1] \to [k]$ induce the degeneracy maps
that regard a $k$-simplex as a $(k+1)$-simplex with degenerate $i$th face.
Accordingly, in the cosimplicial smooth algebra $C^\infty([\Delta^\bullet_{inf},X])$ of smooth algebras of functions on infinitesimal simplices
we have maps
that build a function on $(k+1)$-simplices from one on $k$-simplices by evaluating the latter on the $i$th faces
and maps
that restrict functions on all $(k+1)$-simplices to those simplices whose $i$th face is degenerate and regard the result as a function on $k$-simplices.
Let $X$ be a smooth manifold.
The normalized Moore DGA of the $\infty$-quantity $C^\infty([\Delta_{inf}^\bullet,X])$ of functions on infinitesimal simplices in $X$ is isomorphic, as a differential graded algebra to the differential algebra of differential forms on $X$.
Unwrapping what this means in detail, it turns out that this is item-per-item the characterization of differential forms as functions on infinitesimal simplices as given by Anders Kock in his work on synthetic differential geometry. See differential forms in synthetic differential geometry.
Anders Kock’s crucial insight in this context has been that the description of differential forms simplifies notably when considering them in terms of functions on infinitesimal simplices. He noticed that
plain functions on infinitesimal simplices are automatically alternating if they have the property that they vanish on degenaret simplices and hence are isomorphic to differential forms;
the coboundary operator on differential forms is given by the expression that defines the diferential of the Moore cochain complex on functions on simplices;
the ordinary cup product on such functions on infinitesimal simplicies is already the wedge product on the differential forms represented by them.
But notice that
those functions on infinitesimal simplices that vanish on degenerate simplices are precisely those that are in the joint kernel of the degeneracy maps of the cosimplicial ring $C^\infty(X^{\Delta^\bullet_{inf}})$. Therefore these are precisely the elements of the normalized Moore complex $N^\bullet(C^\infty(X^{\Delta^\bullet_{inf}}))$ of $C^\infty(X^{\Delta^\bullet_{inf}})$;
the induced monoidal structure on the Moore complex is, by the above, precisely the cup product.
The relevant theorems by Anders Kock are found here:
the identification of the deRham complex as functions on infinitesimal simplices that vanish on degenerate simplices is theorem 18.3 in
That the coboundary operator on such simplicial differential forms is precisely the differential in the Moore cochain complex is around equation (3.2.1) in
That the wedge product on differential forms is then just the cup product of these functions on infinitesimal simplices is in section 3.5 of that book.
Let $g$ be the Lie algebra of $G$ and let $CE(g)$ denote the Chevalley-Eilenberg cochain complex that computes Lie algebra cohomology with coefficients in the corresponding trivial Lie algebra module.
There is a canonical cochain map
the vanEst morphism, that sends a function on $G^{\times k}$ to its differential at the identity. If $G$ is $k$-connected, this morphism is an isomorphism on degree $(n \leq k+1)$-cohomology. Hence the cosimplicial algebra $C^\infty(\mathbf{B}G)$ is weakly equivalent to that truncation to the (cosimplicial version of) the Chevalley-Eilenberg algebra.
Let now $C^\infty_{loc}(\mathbf{B}G) \subset C^\infty(\mathbf{B}G)$ be the cosimplicial algebra of function germs at the totally degenerate simplices (the identity $n$-cells on the identity $(n-1)$-cells on the…).
The cohomology of the corresponding cochain complex is the local Lie group cohomology. It coincides with the Lie algebra cohomology. Therefore it should be true that under the cosimplicial Dold-Kan correspondence we have a weak equivalence
Let $X$ be a manifold and $G$ a Lie group with Lie algebra $g$. A flat $g$-valued differential 1-form $A \in \Omega^1(X,g)$ with $d A + [A \wedge A ] = 0$ is the same as
a smooth functor
from the path ∞-groupoid to $\mathbf{B}G$ (as described at connection on a bundle)
a morphism of differential graded algebras
(The emphasis of this simple but far-reaching observation goes back to Cartan. For a detailed account of this in its wider context see for instance LInfCon)
Using the above we find that the systematic relation between these two points of view is that the latter is the image under $C^\infty(-) [C^{op},[\Delta^{op}, Set]] \to [C,[\Delta, Set]]$ in that
At least for algebraic groups the statement that the Chevalley-Eilenberg complex of a Lie algebra is the normalized Moore cochain complex of the cosimplicial algebra of functions on neighbourhoods of the identity in $\mathbf{B}G$ is well known. One reference where this is recalled is
The analog of the map of cosimplicial rings that above is called $C^\infty(\mathbf{B}G) \to C^\infty_{loc}(\mathbf{B}G)$ is in Lemma 3.4.2 there. The normalized Moore cochain complex $N^\bullet(C^\infty(\mathbf{B}G)_{loc})$ of a cosimplicial ring is in definition 3.4.3 and then the isomorphism with the Chevalley-Eilenberg algebra $N^\bullet(C^\infty(\mathbf{B}G)) \simew CE(Lie(G))$ is prop. 3.4.4.