A typical co-sheaf condition is that underlying generalized algebras. The corresponding -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 -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 -quantity is
that identifies under the monoidal Dold-Kan correspondence with
In particular the notion of -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 -quantity of functions on the former.
This is such that for instance in the smooth context of smooth ∞-stack – i.e. a Lie ∞-groupoid – the -quantity dual to a Lie ∞-groupoid is the cosimplicial algebra of smooth functions on neighbourhoods of identities in which turns out to be the Chevalley-Eilenberg algebra of the corresponding Lie ∞-algebroid.
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 -quantities. This way we obtain model for -quantities in terms of a category of cosimplicial copresheaves
For CartSp those copresheaves that respect products in 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 -quantities are also modeled by cochain complexes of copresheaves and in this incarnation they reproduce various entities familiar in homological algebra and Lie theory.
A special case of particular interest to keep in mind is the choice CartSp. For that choice of test spaces we have that
quantities (1-quantities) modeled on are generalized smooth algebras;
natural in .
For a cosimplicial object we shall write
the degree increasing maps as
the degree lowering maps as .
for the category of cosimplicial cosheaves on , the cosimplicial objects in the category of cosheaves.
For 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 to the category 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 is a weak equivalence if for all the morphism of cosimplicial abelian groups (using the additive structure of generalized smooth algebras) induces under the dual normalized Moore complex a morphism that induces an isomorphism on cochain complex cohomology.
where is the Yoneda embedding.
This extends to a functor
from simplicial presheaves to cosimplicial smooth algebras by degreewise application: for we have
be the -quantity of local functions on .
We call the differential graded algebra given by the -quantity .
Let be the Lie ∞-groupoid that is the path ∞-groupoid of . The Moore cochain complex associtated with the -quantity of functions on is manifestly the one that computes singular cohomology (with values in ).
Let be a Lie group and let be its delooping regarded as a Kan complex valued simplicial presheaf (on Diff or CartSp or the like). The cosimplicial smooth algebra has in degree the smooth algebra of -valued functions on . The differential of the corresponding dual Moore complex is the one that computes smooth group cohomology on with coefficients in with the trivial module structure.
For more on this see
The cosimplicial copresheaf we call the -quantity of functions on infinitesimal simplices in .
For later reference we list in detail the interpretation of the face and degenercy maps in this cosimplicial object.
First think of as the space of infinitesimal -simplices in (formalized as such in some context that need not concern us here).
The maps induce the face maps
that send a -simplex to its th -face.
The maps induce the degeneracy maps
that regard a -simplex as a -simplex with degenerate th face.
Accordingly, in the cosimplicial smooth algebra of smooth algebras of functions on infinitesimal simplices
we have maps
that build a function on -simplices from one on -simplices by evaluating the latter on the th faces
that restrict functions on all -simplices to those simplices whose th face is degenerate and regard the result as a function on -simplices.
Let be a smooth manifold.
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;
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 . Therefore these are precisely the elements of the normalized Moore complex of ;
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.
There is a canonical cochain map
the vanEst morphism, that sends a function on to its differential at the identity. If is -connected, this morphism is an isomorphism on degree -cohomology. Hence the cosimplicial algebra is weakly equivalent to that truncation to the (cosimplicial version of) the Chevalley-Eilenberg algebra.
Let now be the cosimplicial algebra of function germs at the totally degenerate simplices (the identity -cells on the identity -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
a smooth functor
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 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 is well known. One reference where this is recalled is
The analog of the map of cosimplicial rings that above is called is in Lemma 3.4.2 there. The normalized Moore cochain complex of a cosimplicial ring is in definition 3.4.3 and then the isomorphism with the Chevalley-Eilenberg algebra is prop. 3.4.4.