(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
∞-Lie theory (higher geometry)
Formal Lie groupoids
For any abelian Lawvere theory, here we discuss – in a variation of the theme of Isbell conjugation, in generalization of (Toën) and following (Stel) – a simplicial Quillen adjunction between model category structures on cosimplicial -algebras and on simplicial presheaves over duals of -algebras. We find mild general conditions under which this descends to the local model structure that models ∞-stacks over duals of -algebras. In these cases the left adjoint of the Quillen adjunction is given by sending -stacks to their cosimplicial -algebras of functions with values in the canonical -line object, and the adjunction models small objects relative to a choice of a small full subcategory of the localization
of the -topos of -sheaves over duals of -algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical -line object.
For the special case where is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that is the theory of smooth algebras (-rings) we obtain a refinement of this to the context of synthetic differential geometry. In these cases, in as far as objects in may be understood as ∞-Lie groupoids, the objects in may be understood as ∞-Lie algebroids.
As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable -stack of infinitesimal paths of a manifold. This construction is an -categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the -stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic -vector bundles with flat connection.
Models for -stacks and their function algebras
A good general notion of function algebras on generalized spaces are -algebras, for a Lawvere theory. A good general notion of function algebras on internal ∞-groupoids in such spaces are cosimplicial -algebras.
We recall some basics and then discuss a model category structure on cosimplicial -algebras for the cases that contains the theory of abelian groups.
A Lawvere theory may be thought of as a generalization of the theory of ordinary associative algebras.
A Lawvere theory is encoded in its syntactic category , which by definition is a category with finite products such that every object is (isomorphic to) a finite cartesian power of a fixed object . We are to think of the hom-set as the set of -ary operations of the algebras defined by the theory. A -algebra is accordingly a product-preserving functor . Its image is the underlying set, and its value on an element is the -ary operation as implemented by .
The category of -algebras is the full subcategory of the category of presheaves on on these product-preserving functors.
The category of free finitely generated abelian groups is the syntactic category of the Lawvere theory whose algebras are abelian groups.
For a field, the category of free finitely generated -algebras is the Lawvere theory whose algebras are -associative algebras;
The category CartSp is the syntactic category whose algebras are smooth algebras.
A morphism of Lawvere theories is again a product-preserving functor.
An abelian Lawvere theory is a morphism of Lawvere theories .
For abelian, -algebras have an underlying abelian group, given by the functor
This functor is a right adjoint.
For example associative algebras and smooth algebras are algebras over an abelian Lawvere theory, and their underlying abelian groups are the evident ones.
Similarly, the forgetful functor has a left adjoint, the free -algebra functor . By the Yoneda lemma this sends the -element set to .
More generally, for any the copresheaf
is isomorphic to .
The free -algebra on a single generator may be thought of as the -algebra of functions on the -line. For instance
We say more on the canonical -line object below in The Line object
Model structure on cosimplicial -algebras
The proof of the existence of the model structure on cochain complexes in non-negative degree – – whose fibrations are the degreewise surjections (and weak equivalences the usual quasi-isomorphism)s is spelled out here.
By the dual Dold-Kan correspondence this induces the model structure on cosimplicial abelian groups whose fibrations are the degreewise surjections (using that the normalized cochain complex functor sends surjections to surjections).
That with the standard structure of an sSet-enriched category on this constitutes a simplicial model category-structure is proven here.
Now we use the basic fact of Lawvere theories that any morphism of such induces a pair of adjoint functors
between their categories of algebras: the adjunction of relatively free algebras.
Since by assumption that our is an abelian Lawvere theory we are given a morphism from the theory of abelian groups, this means that we have an adjunction
and hence also an adjunction
We need to check that the right adjoint induces the transferred model structure on from the above model structure .
By the facts recalled at transferred model structure, we need to check that
and for the simplicial enrichment that
- preserves the powering.
The first condition is trivial, since all objects are fibrant. The last condition is evidently satisfied, since
Using this, we claim that we can take the path space object functor to be given by powering with the simplicial interval
This is because factors the co-diagonal in by a cofibration followed by a weak equivalence between cofibrant objects. Accordingly the induced
factors the digonal, and the morphism on the left is a weak equivalence, since it is the image under the left Quillen functor of a weak equivalence between cofibrant objects (and by the factorization lemma such weak equivalences are preserved by left Quillen functors).
Simplicial presheaves on duals of -algebras
A good notion of a generalized space modeled on objects in a category is a sheaf on . A good notion of an ∞-groupoid in such generalized spaces is an (∞,1)-sheaf on . Such objects are modeled by the model structure on simplicial presheaves on .
We are interested here in that case that
is a small full subcategory of the opposite category of -algebras, for an abelian Lawvere theory. In the remainder of this section we assume such a choice to be fixed. Below in the section on Examples and applications we discuss concrete choices of interest.
Notice that such a choice induces also a full subcategory of (co)simplicial objects
The prolonged Yoneda embedding
for the ordinary Yoneda embedding and
for its degreewise simplicial prolongation
For and denoting the simplicially enriched category of -algebras, we have a natural identification
Using end/coend-calculus for handling the canonical enrichment of , we have for and natural isomorphisms
where in the last step we used the isomorphism (described at coend)
The line object
The adjunction that we shall be concerned with is essentially Isbell conjugation. We recall some basics of Function T-algebras on presheaves.
Recall from above that we write for the free -algebra on a single generator.
We call the line object in .
As a presheaf, the line object sends a -algebra to its underlying set
This characterization may look simpler, but does not capture the important fact that homming into produces -algebras of functions . This is what the following definition deals with.
(-algebras of functions)
For , the cosimplicial set
we call the cosimplicial set of -valued functions on . This is naturally the cosimplical set underlying the cosimplicial -algebra
We call the -algebra of functions on . This extends to a functor
In the next section we see that forms a simplicial Quillen adjunction.
Model structure on simplicial presheaves
Write for the global projective model structure on simplicial presheaves over . With the simplicial enrichment this is naturally a simplicial model category.
Let be a class of split hypercovers.
Write for the left Bousfield localization at this class.
By general results on left Bousfield localization, this exists always for a small set, notably for the set of Cech nerve projections for covers of the Grothendieck topology on . By general results on the local model structure on simplicial presheaves, the localization also exists for the class of all (split) hypercovers.
We relate now the model structure on cosimplicial T-algebras with the model structure on simplicial presheaves over using the function algebra functor and the prolonged Yoneda embedding .
The functors and constitute a simplicial Quillen adjunction
We first establish the adjunction itself: using end-calculus for expressing hom-sets in functor categories we have for and natural isomorphisms
where the crucial step is the isomorphism for the line object discussed above. This computation is just simplicial-degreewise the adjunction discussed at Isbell duality – Function T-algebras on presheaves.
That this lifts to an -enriched adjunction follows with the prolonged Yoneda lemma and the -tensoring and cotensoring of and :
By the pushout-product axiom satisfied by the -enriched model category and using that in every object is cofibrant, we have that for a fibration (acyclic fibration) in and for any object, the morphism is a fibration (acyclic fibration) in . Therefore is a fibration (acyclic fibration) in .
This establishes that is a right Quillen functor and completes the proof.
The following theorems say that the obstructions to making this Quillen adjunction descent to local model structures on simplicial presheaves are mild.
Let be a subcanonical coverage on , and a split hypercover with respect to .
Then for we have that induces an isomorphism in -cohomology in degree : .
Regard as a simplicial object in the overcategory
for the degreewise free abelian group object of that, a simplicial object in the category of abelian group objects in the sheaf topos over . The chain homology of the corresponding normalized chain complex vanishes in positive degree (as discussed here):
Let now by the Freyd-Mitchell embedding theorem
be a full and faithful functor from the abelian category of abelian group object into the category of -module over some ring .
Write for the canonical -line object regarded first as the abelian group object and then injected with into .
Using this, the cochain cohomology that we are after is equivalently the cohomology of
To compute this, we use the universal coefficient theorem, which says that we have an exact sequence
By the above fact that the homology vanishes in positive degree, this gives finally that
vanishes in degree . That it also vanishes in degree 0 is seen to be equivalent to the sheaf condition on , which is true by the assumption that we are working with a subcanonical coverage.
(passage to local model structure)
If for all split (hyper-)covers we have that is an isomorphisms then is a simplicial Quillen adjunction to the local model structure on simplicial presheaves.
By the previous proposition we have that under the given assumptions every (hyper-)cover in is taken by to a weak equivalence.
Using this we can follow the remainder of the argument of Toën, prop. 2.2.2:
Since the model structure on simplicial presheaves is a left proper model category and since left Bousfield localization preserves left properness, we have that is left proper. Since moreover left Bousfield localization does not change the class of cofibrations, we know that still preserves cofibrations.
Then by the recognition theorem for simplicial Quillen adjunction it is sufficient to check that sends fibrant objects to local objects with respect to the morphisms .
Since by definition of hypercovers, their domain and codomain is cofibrant (codomain because it is a representable, domain by assumption that it is a degreewise coproduct of representables with disjoint degeneracies, see the discussion of cofibrancy in the projective structure at model structure on simplicial presheaves), this means that it is sufficient to check that for all and fibrant we have that is a weak equivalence. But by the adjunction this is isomorphically .
Now by the above propositions and assumptions, we have that is a weak equivalence. Since all objects in are cofibrant, it is a weak equivalence between cofibrant objects. With the factorization lemma it follows that in an enriched model category the enriched hom of a weak equivalence into a fibrant object is a weak equivalence.
The following proposition asserts that the Quillen adjunction that we have established is very special, in that it is the model-category theoretic analog of a reflective subcategory. Below in the section Localization of the (∞,1)-topos at R-cohomology we see that this indeed presents such a reflective inclusion in (∞,1)-category theory.
When restricted along the functor is homotopy full and faithful in that for all we have that the canonical morphism
into the image of the derived functors of and is an isomorphism in the homotopy category .
With the above results, this follows verbatim as the proof of the analogous (Toën, corollary 2.2.3).
Localization of the -topos at -cohomology
We consider now the cohomology localization of at the canonical line object.
In this section we discuss that in terms of the (∞,1)-category theory that is presented by the model category theoretic structures above, these serve to establish the following intrinsic statement.
We obtain a proof of this after the following discussions.
Since is assumed to be an abelian Lawvere theory, the T-line object canonically has the structure of an abelian group object in . As such it presents a 0-truncated ∞-group in the , and so we may consider its Eilenberg-MacLane objects for .
The following proposition provides a model for these Eilenberg-MacLane objects.
Write for the dual Dold-Kan correspondence map. Notice that the free -algebra is , the free abelian group on a single generator, the integer. Write for the cochain complex concentrated in degree on . For the left adjoint to the underlying abelian group functor we have then thagt is the cosimplicial -algebra which in degree is a product of copies of the free -algebra corresponding to the product of copies in .
For the object is presented in by
Every (∞,1)-topos such as comes with its intrinsic notion of abelian cohomology: for any object and for a ∞-group object with arbitrary deloopings , the th cohomology group of with coefficients in is
In terms of the model category presentation by and writing for a representative of this is the hom-set in the homotopy category
For representing an object , the intrinsic -cohomology of coincides with the cochain cohomology of its cosimplicial function algebra :
Notice that , being the image of a cofibrant object in , is cofibrant in , hence fibrant in .
Using this, we compute as follows
This is essentially the argument of (Toën, corollary 2.2.6).
We say a morphism in is an -equivalence if it induces isomorphisms in -cohomology.
By the above proposition this is equivalent to saying that the derived functor takes to a weak equivalence.
We say an object is an -local object if for all -equivalences we have that
is an equivalence, equivalently if the derived hom-space functor produces a weak equivalence (of Kan complexes).
The -local objects of that are equivalent to those in the image of span precisely the homotopy-essential image of the restriction of to
We may explicitly see this by observing that the proof of (Toën, theorem 2.2.9) goes through verbatim: it only uses the general properties of the -adjunction that we have established above, as well as the fact that is a cofibrantly generated model category for the theory of ordinary commutative algebras. But by our result on the model structure on TAlg we have that for general this is the transferred model structure of the model structure on cosimplicial abelian groups, which is cofibrantly generated. Hence by the general properties of transferred model structures, also is.
But more abstractly, we can also simply use the general theory of reflective sub-(∞,1)-categories and their characterization as the reflective localization of an (∞,1)-category at a set of weak equivalences: from the above we know that on the full sub--category of on the objects in is a reflective sub--category
and that the left adjoint to the embedding inverts precisely the -equivalences. Hence is the full sub--category of on -local objects.
In derived geometry
We now discuss function algebras on -stacks more generally in the context of derived geometry, meaning that we we pass in the above from sites inside the opposite of a 1-category of -algebras to an (∞,1)-site inside the opposite of an (∞,1)-category of ∞-algebras over an (∞,1)-algebraic theory.
Over ordinary associative algebras
Let be a field of characteristic .
Let be a small full sub--category equipped with the structure of a subcanonical (∞,1)-site.
for the -Yoneda extension of the inclusion .
By construction is a colimit-preserving -functor between locally presentable (∞,1)-categories. Accordingly, by the adjoint (∞,1)-functor theorem is has a right adjoint (∞,1)-functor.
This is given by
This follows by the general yoga of Kan extensions. Explcitly, we check the hom-equivalence
This is considered in (Ben-Zvi/Nadler, prop. 3.1).
The above Yoneda-Quillen adjunction for the theory of commutative -algebras is compatible with this in that it also does model the -Yoneda extension of the inclusion
By the general discussion of cofibrant replacement in the projective model structure on simplicial presheaves we have that every has a cofibrant resolution of the form , where the integrand the integrand we have the fat simplex tensored degreewise with a coproduct of representables such that the degenerate cells split off as direct summands (a split hypercover). This makes Reedy cofibrant an therefore the whole coend is a model for its homotopy colimit.
Since both the simplex as well as the fat simplex are Reedy cofibrant cosimplicial simplicial sets, this is moreover equivalent to and this is still cofibrant. Now the left Quillen functor takes this to . Since every object in is cofibrant, this coend is still a homotopy colimit.
This shows that the derived functor of the left Quillen functor sends the decomposition of any -stack as the -colimit over representable to the -colimit of the images of these representables.
Examples and applications
The conditons of the above theorem are satisfied for instance for
the theory of ordinary commutative algebras over a field and the fpqc topology.
In this case the adjunction is that considered in (Toën).
the theory of smooth algebras and the site of the Cahiers topos. This is what we discuss in more detail below.
Rational homotopy theory
the Lawvere theory of -algebras. Then reproduces the setup discussed at rational homotopy theory in an (∞,1)-topos.
-Lie theory in the -Cahiers topos
In this section we study the general theory for the case that
Write for the category of smooth algebras. Sheaf toposes on sub-sites are well known to provide smooth toposes that are well adapted models for synthetic differential geometry.
We consider here the choice
Write for the left Bousfield localization of the global projective model structure at the Cech nerves of good open covers in ThCartSp.
We have that
this presents the -Cahiers topos ;
the fibrant objects of are precisely those fibrant objects such that for all goop open covers with Cech nerve we have that
is a weak equivalence (of Kan complexes).
The Cech nerves projections induce isomorphisms on the cohomology of their cosimplicial function algebras: is an isomorphism, for all .
This means that the assumptions of the Theorem on passage to the local model structure are satisfied.
We have a simplicial Quillen adjunction
The objects of the -Cahiers topos we call synthetic differential ∞-Lie groupoids.
The objects of the reflective sub--category of -local objects in the -Cahiers topos
we call ∞-Lie algebroids.
A connected -Lie algebroid we call an ∞-Lie algebra.
Passing along the embedding we may compute ∞-Lie algebra cohomology in .
The infinitesimal path -groupoid of a manifold
be the simplicial object of infinitesimal simplices in .
the infinitesimal path -Lie groupoid of .
Or the path -Lie algebroid .
The tangent category of smooth algebras
The tangent category of the category of smooth algebras is the category of modules over -rings.
Proposition This abstract definition of module over -rings reproduces the definition given by Kock.
The tangent category of the category of simplicial -rings is …
This serves the purpose of presenting the -stack of -vector bundles on .
Enrichment of categories of simplicial objects
We make use of the canonical structure of an sSet-enriched category on any category of cosimplicial objects in a category with all limits and colimits (see there).
For and we have that the tensoring is given by
with the product taken in .
Model structure on cosimplicial abelian groups
We use the model category structure on whose fibratin are the degreewise surjections, and whose weak equivalences are the usual quasi-isomorphisms under the dual Dold-Kan correspondence .
The model structure is described in detail at model structure on chain complexes - the projective structure.
The structure of a simplicial model category is described in detail at model structure on cosimplicial abelian groups.
The Quillen adjunction over abelian -algebras that we consider generalizes that discussed in
over ordinary commutative -algebras. See also rational homotopy theory in an (infinity,1)-topos.
The generalization to arbitrary abelian -algebras and the application to synthetic differential geometry is the content of
Herman Stel, -Stacks and their function algebras – with applications to -Lie theory , master thesis (2010) (web)
Herman Stel, Cosimplicial -rings and the de Rham complex of Euclidean space (arXiv:1310.7407)
on which this entry here is based.
The considerations in
- David Spivak, Derived smooth manifolds Duke Math. J. Volume 153, Number 1 (2010), 55-128. (pdf)
on derived smooth manifolds may be considered as complementary to the approach taken here: there simplicial -rings are considered, instead of cosimplicial ones. A fully comprehensive treatment of derived synthetic differential geometry would consider the combination of both aspects: simplicial presheaves on duals of simplicial -rings with a functor taking them to cosimplicial-simplicial -rings.
For ordinary commutative algebras the generalizaton of Toen’s setup to geometry over duals of simplicial algebras is used for instance in