This is a sub-section of the entry cohesive (∞,1)-topos . See there for background and context
Structure in a cohesive -topos
A cohesive -topos is a general context for higher geometry with good cohomology and homotopy properties. We list fundamental structures and constructions that exist in every cohesive -topos.
The cohesive structure on an object in a cohesive -topos need not be supported by points. We discuss a general abstract characterization of objects that do have an interpretation as bare -groupoids equipped with cohesive structure.
Compare with the section Quasitoposes of concrete objects at cohesive topos.
On a cohesive -topos both and are full and faithful (∞,1)-functors and exhibits ∞Grpd as a sub--topos of by an
We say an object is -concrete if the canonical morphism is (n-1)-truncated.
If a 0-truncated object is -concrete, we call it just concrete.
See also concrete (∞,1)-sheaf.
For and , the -concretification of is the morphism
that is the left factor in the decomposition with respect to the n-connected/n-truncated factorization system of the -unit
Every (∞,1)-topos comes with a notion of ∞-group objects that generalizes the traditional notion of grouplike spaces in Top ∞Grpd. For more details on the following see also looping and delooping.
For an object and a point, the loop space object of is the (∞,1)-pullback :
This object is canonically equipped with the structure of an ∞-group obect.
This follows from the above proposition which says that necessarily has homotopy dimension .
for the inverse to . For we call the delooping of .
Notice that since the cohesive -topos has homotopy dimension by the above proposition every 0-connected object has an essentially unique point, but nevertheless the homotopy type of may differ from that of .
The delooping object is concrete precisely if is.
We may therefore unambiguously speak of concrete cohesive -groups.
For any morhism in and a point, the ∞-fiber or of over this point is the (∞,1)-pullback
This means that we have a diagram
where the outer rectangle is an (∞,1)-pullback if the left square is an (∞,1)-pullback. This follows from the pasting law for -pullbacks in any (∞,1)-category.
If the cohesive -topos has an ∞-cohesive site of definition , then
every ∞-group object has a presentation by a presheaf of simplicial groups
which is fibrant in ;
the corresponding delooping object is presented by the presheaf
which is given over each by (see simplicial group for the notation).
Let be a locally fibrant representative of . Since the terminal object is indeed presented by the presheaf constant on the point we have functorial choices of basepoints in all the for all and by assumption that is connected all the are connected. Hence without loss of generality we may assume that is presented by a presheaf of reduced simplicial sets .
Then notice the Quillen equivalence between the model structure on reduced simplicial sets and the model structure on simplicial groups
In particular its unit is a weak equivalence
for every and is always a Kan complex. Therefore
is an equivalent representative for , fibrant at least in the global model structure. Since the finite (∞,1)-limit involved in forming loop space objects is equivalently computed in the global model structure, it is sufficient to observe that
a pullback diagram in (because it is so over each by the general discussion at simplicial group);
a homotopy pullback of the point along itself (since is objectwise a fibration resolution of the point inclusion).
Cohomology and principal -bundles
There is an intrinsic notion of cohomology and of principal ∞-bundles in any (∞,1)-topos .
For two objects, we say that
is the cohomology set of with coefficients in . If is an ∞-group we write
for cohomology with coefficients in its delooping. Generally, if has a -fold delooping, we write
In the context of cohomology on with coefficients in we we say that
the hom-space is the cocycle -groupoid;
a morphism is a cocycle;
a 2-morphism : is a coboundary between cocycles.
a morphism represents the characteristic class
For every morphism define the long fiber sequence to the left
to be the given by the consecutive pasting diagrams of (∞,1)-pullbacks
The long fiber sequence to the left of becomes constant on the point after iterations if is -truncated.
For every object we have a long exact sequence of pointed sets
The first statement follows from the observation that a loop space object is a fiber of the free loop space object and that this may equivalently be computed by the (∞,1)-powering , where is the circle. (See Hochschild cohomology for details.)
The second statement follows by observing that the -hom-functor preserves all (∞,1)-limits, so that we have (∞,1)-pullbacks
etc. in ∞Grpd at each stage of the fiber sequence. The statement then follows with the familiar long exact sequence for homotopy groups in Top ∞Grpd.
To every cocycle is canonically associated its homotopy fiber , the (∞,1)-pullback
We discuss now that canonically has the structure of a -principal ∞-bundle and that is the fine moduli space for -principal -bundles.
Let be a group object in the (∞,1)-topos . A principal action of on an object is a groupoid object in the (∞,1)-topos that sits over in that we have a morphism of simplicial diagrams
We say that the (∞,1)-colimit
is the base space defined by this action.
We may think of as the action groupoid of the -action on . For us it defines this -action.
The -principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects
For any morphism, its homotopy fiber is canonically equipped with a principal -action with base space .
By the above we need to show that we have a morphism of simplicial diagrams
where the left horizontal morphisms are equivalences, as indicated. We proceed by induction through on the height of this diagram.
The defining (∞,1)-pullback square for is
To compute this, we may attach the defining -pullback square of to obtain the pasting diagram
and use the pasting law for pullbacks, to conclude that is the pullback
By defnition of as the homotopy fiber of , the lower horizontal morphism is equivalent to , so that is equivalent to the pullback
This finally may be computed as the pasting of two pullbacks
of which the one on the right is the defining one for and the remaining one on the left is just an (∞,1)-product.
Proceeding by induction from this case we find analogousy that : suppose this has been shown for , then the defining pullback square for is
We may again paste this to obtain
and use from the previous induction step that
to conclude the induction step with the same arguments as before.
We say a -principal action of on over is a -principal ∞-bundle if the colimit over produces a pullback square: the bottom square in
Of special interest are principal -bundles of the form :
We say a sequence of cohesive ∞-groups
exhibits as an extension of by if the corresponding delooping sequence
if a fiber sequence. If this fiber sequence extends one step further to the right to a morphism , we have by def. 8 that is the -principal ∞-bundle classified by the cocycle ; and is its fiber over the unique point of .
Given an extension and a a -principal ∞-bundle in we say a lift of to a -principal -bundle is a factorization of its classifying cocycle through the extension
Let be an extension of -groups, def. 9 in and let be a -principal ∞-bundle.
Then a -extension of is in particular also an -principal -bundle over with the property that its restriction to any fiber of is equivalent to .
We may summarize this as saying:
An extension of -bundles is an -bundle of extensions.
This follows from repeated application of the pasting law for (∞,1)-pullbacks: consider the following diagram in
The bottom composite is a cocycle for the given -principal -bundle and it factors through by assumption of the existence of the extension .
Since also the bottom right square is an -pullback by the given -group extension, the pasting law asserts that the square over is also a pullback, and then that so is the square over . This exhibits as an -principal -bundle over .
Now choose any point of the base space as on the left of the diagram. Pulling this back upwards through the diagram and using the pasting law and the definition of loop space objects the diagram completes by -pullback squares on the left as indicated, which proves the claim.
For the moment see the discussion at ∞-gerbe .
Twisted cohomology and section
A slight variant of cohomology is often relevant: twisted cohomology.
For an (∞,1)-topos let a morphism representing a characteristic class . Let be pointed and write for its homotopy fiber.
We say that the twisted cohomology with coefficients in relative to is the intrinsic cohomology of the over-(∞,1)-topos with coefficients in .
If is understood and is any morphism, we write
and speak of the cocycle ∞-groupoid of twisted cohomology on with coefficients in and twisting cocycle relative to .
For short we often say twist for twisting cocycle .
We have the following immediate properties of twisted cohomology:
The -twisted cohomology relative to depends, up to equivalence, only on the characteristic class represented by and also only on the equivalence class of the twist.
If the characteristic class is terminal, we have and the corresponding twisted cohomology is ordinary cohomology with coefficients in .
Let the characteristic class and a twist be given. Then the cocycle -groupoid of twisted -cohomology on is given by the (∞,1)-pullback
This is an application of the general pullback-formula for hom-spaces in an over-(∞,1)-category. See there for details.
If the twist is trivial, (meaning that it factors as through the point of the pointed object ), the corresponding twisted -cohomology is equivalent to ordinary -cohomology
In this case we have that the characterizing -pullback diagram from prop. 12 is the image under the hom-functor of the pullback diagram . By definition of as the homotopy fiber of , its pullback is . Since the hom-functor preserves (∞,1)-pullbacks the claim follows:
Often twisted cohomology is formulated in terms of homotopy classes of sections of a bundle. The following asserts that this is equivalent to the above definition.
By the discussion at Cohomology and principal ∞-bundles we may understand the twist as the cocycle for an -principal ∞-bundle over , being the (∞,1)-pullback of the point inclusion along , where the point is the homotopy-incarnation of the universal -principal -bundle. The characteristic class in turn we may think of as an -bundle associated to this universal bundle. Accordingly the pullback of is the associated -bundle over classified by .
Let be (∞,1)-pullback of the characteristic class along the twisting cocycle
Then the -twisted -cohomology of is equivalently the space of sections of over :
where on the right we have the (∞,1)-pullback
Consider the pasting diagram
By the fact that the hom-functor preserves (∞,1)-limits the bottom square is an (∞,1)-pullback. By the pasting law for (∞,1)-pullbacks so is then the total outer diagram. Noticing that the right vertical composite is the claim follows with prop. 12.
In applications one is typically interested in situations where the characteristic class and the domain is fixed and the twist varies. Since by prop. 11 only the equivalence class matters, it is sufficient to pick one representative in each equivalence class. Such as choice is equivalently a choice of section
of the 0-truncation projection from the cocycle -groupoid to the set of cohomology classes. This justifies the following terminology.
With a characteristic class with homotopy fiber understood, we write
for the union of all twisted cohomology cocycle -groupoids.
We have that is the (∞,1)-pullback
where the right vertical morphism in any section of the projection from -cocycles to -cohomology.
When the (∞,1)-topos is presented by a model structure on simplicial presheaves and model for and is chosen, then the cocycle ∞-groupoid is presented by an explicit simplicial presheaf . Once these choices are made, there is therefore the inclusion of simplicial presheaves
where on the left we have the simplicially constant object on the vertices of . This morphism, in turn, presents a morphism in that in general contains a multitude of copies of the components of any : a multitude of representatives of twists for each cohomology class of twists. Since by the above the twisted cohomology does not depend, up to equivalence, on the choice of representative, the coresponding -pullback yields in general a larger coproduct of -groupoids as the corresponding twisted cohomology. This however just contains copies of the homotopy types already present in as defined above.
-Group representations and associated -bundles
The material to go here is at Schreiber, section 2.3.7.
Since is an (∞,1)-topos it carries canonically the structure of a cartesian closed (∞,1)-category. For
, write for the corresponding internal hom.
Since ∞Grpd preserves products, we have for all canonically induced a morphism
This should yield an (∞,1)-category with the same objects as and hom--groupoids defined by
We have that
is the -groupoid whose objects are -principal ∞-bundles on and whose morphisms have the interpretaton of -principal bundles on the cylinder . These are concordances of -bundles.
Geometric homotopy / étale homotopy
We discuss canonical internal realizations of the notions of étale homotopy, geometric homotopy groups in an (infinity,1)-topos and local systems .
We say a geometric homotopy between two morphism in is a diagram
such that is geometrically connected, .
If are geometrically homotopic in , then their images are equivalent in .
By the condition that preserves products in a cohesive -topos we have that the image of the geometric homotopy in is a diagram of the form
Now since is connected by assumption, there is a diagram
Taking the product of this diagram with and pasting the result to the above image of the geometric homotopy constructs the equivalence in .
For a locally ∞-connected (∞,1)-topos, also all its objects are locally -connected, in the sense their petit over-(∞,1)-toposes are locally -connected.
The two notions of fundamental -groupoids of induced this way do agree, in that there is a natural equivalence
By the general facts recalled at étale geometric morphism we have a composite essential geometric morphism
and is given by sending to .
Cohesive -homotopy / The Continuum
An object is called a line object exhibiting the cohesion of if the shape modality (hence the reflector ) exhibits the localization of an (∞,1)-category of at the class of morphisms .
This is (dcct, 3.9.1).
See also at
We discuss a canonical internal notion of Galois theory in .
We call this the -groupoid of locally constant ∞-stacks on .
Since is left adjoint and right adjoint it commutes with coproducts and with delooping and therefore
Therefore a cocycle may be identified on each geometric connected component of as a -principal ∞-bundle over for the ∞-group object . We may think of this as an object in the little topos over . This way the objects of are indeed identified -stacks over .
The following proposition says that the central statements of Galois theory hold for these canonical notions of geometric homotopy groups and locally constant -stacks.
For locally ∞-connected and ∞-connected, we have
a natural equivalence
of locally constant -stacks on with -permutation representations of the fundamental ∞-groupoid of (local systems on );
for every point a natural equivalence of the endomorphisms of the fiber functor and the loop space of at
The first statement is just the adjunction .
Using this and that preserves the terminal object, so that the adjunct of is
the second statement follows with an iterated application of the (∞,1)-Yoneda lemma (this is pure Tannaka duality as discussed there):
The fiber functor evaluates an -presheaf on at . By the (∞,1)-Yoneda lemma this is the same as homming out of , where is the (∞,1)-Yoneda embedding:
This means that itself is a representable object in . If we denote by the corresponding Yoneda embedding, then
With this, we compute the endomorphisms of by applying the (∞,1)-Yoneda lemma two more times:
van Kampen theorem
A higher van Kampen theorem asserts that passing to fundamental ∞-groupoids preserves certain colimits.
On a cohesive -topos the fundamental -groupoid functor is a left adjoint (∞,1)-functor and hence preserves all (∞,1)-colimits.
More interesting is the question which -colimits of concrete spaces in
are preserved by . These colimits are computed by first computing them in and then applying the concretization functor. So we have
Let be a diagram such that the (∞,1)-colimit is concrete, .
Then the fundamental ∞-groupoid of is computed as the -colimit
In the Examples we discuss the cohesive -topos of topological ∞-groupoids For that case we recover the ordinary higher van Kampen theorem:
We inject the topological space via the external Yoneda embedding
as a 0-truncated topological ∞-groupoid in the cohesive -topos . Being an (∞,1)-category of (∞,1)-sheaves this is presented by the left Bousfield localization of the injective model structure on simplicial sheaves on (as described at models for ∞-stack (∞,1)-toposes).
Notice that the injection of topological spaces as concrete sheaves on the site of open balls preserves the pushout . (This is effectively the statement that as a representable on Diff is a sheaf.) Accordingly so does the further inclusion into as simplicially constant simplicial sheaves.
Since cofibrations in that model structure are objectwise and degreewise injective maps, it follows that the ordinary pushout diagram
in has all objects cofibrant and is the pushout along a cofibration, hence is a homotopy pushout (as described there). By the general theorem at (∞,1)-colimit homotopy pushouts model -pushouts, so that indeed is the -pushout
The proposition now follows with the above observation that preserves all -colimits and with the statement (from topological ∞-groupoid) that for a topological space (locally contractible or paracompact) we have .
Paths and geometric Postnikov towers
The above construction of the fundamental ∞-groupoid of objects in as an object in ∞Grpd may be reflected back into , where it gives a notion of homotopy path n-groupoids and a geometric notion of Postnikov towers of objects in .
for the reflective sub-(∞,1)-category of n-truncated objects and
for the truncation-localization funtor.
is the homotopy path n-groupoid functor.
We say that the (truncated) components of the -unit
are the constant path inclusions. Dually we have canonical morphism
If is cohesive, then has a right adjoint
and this makes be -connected and locally -connected over itself.
In a cohesive -topos , if is small-projective then so is its path ∞-groupoid .
Because of the adjoint triple of adjoint (∞,1)-functors we have for diagram that
where in the last step we used that is small-projective by assumption.
For we say that the geometric Postnikov tower of is the Postnikov tower in an (∞,1)-category of :
Universal coverings and geometric Whitehead towers
We discuss an intrinsic notion of Whitehead towers in a locally ∞-connected ∞-connected (∞,1)-topos .
For a pointed object, the geometric Whitehead tower of is the sequence of objects
in , where for each the object is the homotopy fiber of the canonical morphism to the path n+1-groupoid of .
We call the -fold universal covering space of .
We write for the homotopy fiber of the untruncated constant path inclusion.
Here the morphisms are those induced from this pasting diagram of (∞,1)-pullbacks
where the object is defined as the homotopy fiber of the bottom right morphism.
Every object is covered by objects of the form for different choices of base points in , in the sense that every is the (∞,1)-colimit over a diagram whose vertices are of this form.
Consider the diagram
The bottom morphism is the constant path inclusion, the -unit. The right morphism is the equivalence in an (∞,1)-category that is the image under of the decomposition of every ∞-groupoid as the (∞,1)-colimit (see there) over itself of the (∞,1)-functor constant on the point.
The left morphism is the (∞,1)-pullback along of this equivalence, hence itself an equivalence. By universal colimits in the (∞,1)-topos the top left object is the (∞,1)-colimit over the single homotopy fibers of the form as indicated.
The inclusion of the fundamental ∞-groupoid of each of these objects into is homotopic to the point.
We apply to the above diagram over a single vertex and attach the -counit to get
Then the bottom morphism is an equivalence by the -zig-zag-identity.
Flat -connections and local systems
We describe for a locally ∞-connected (∞,1)-topos a canonical intrinsic notion of flat ∞-connections, flat higher parallel transport and higher local systems.
Write for the adjunction given by the path ∞-groupoid. Notice that this comes with the canonical -unit with components
and the -counit with components
For we write
and call the flat (nonabelian) differential cohomology of with coefficients in .
We say a morphism is a flat ∞-connnection on the principal ∞-bundle corresponding to , or an-local system** on .
The induced morphism
we say is the forgetful functor that forgets flat connections.
By the -adjunction we have a natural equivalence
A cocycle for a principal ∞-bundle on is in the image of
precisely if there is a lift in the diagram
We call the coefficient object for flat -connections.
The following lists some basic properties of objects of the form and their interpretation in terms of flat -connections.
For a discrete ∞-group the canonical morphism is an equivalence.
Since is a full and faithful (∞,1)-functor we have that the unit is a natural equivalence. It follows that on also the counit is a weak equivalence (since by the triangle identity we have that is the identity).
de Rham cohomology
In every locally ∞-connected (∞,1)-topos there is an intrinsic notion of nonabelian de Rham cohomology.
For an object, write for the (∞,1)-pushout
For any pointed object in , write for the (∞,1)-pullback
We also say is the dR-flat modality and is the dR-shape modality.
The construction in def. 24 yields a pair of adjoint (∞,1)-functors
We check the defining natural hom-equivalence
The hom-space in the under-(∞,1)-category is (as discussed there), computed by the (∞,1)-pullback
By the fact that the hom-functor preserves limits in both arguments we have a natural equivalence
We paste this pullback to the above pullback diagram to obtain
By the pasting law for (∞,1)-pullbacks the outer diagram is still a pullback. We may evidently rewrite the bottom composite as in
This exhibits the hom-space as the pullback
where we used the -adjunction. Now using again that preserves pullbacks, this is
If is also local, then there is a further right adjoint
where is the triple of adjunctions discussed at Paths.
This follows by the same kind of argument as above.
For we write
A cocycle we call a flat -valued differential form on .
We say that is the de Rham cohomology of with coefficients in .
This follows by the fact that the hom-functor preserves the defining (∞,1)-pullback for .
Just for emphasis, notice the dual description of this situation: by the universal property of the (∞,1)-colimit that defines we have that corresponds to a diagram
The bottom horizontal morphism is a flat connection on the -bundle given by the cocycle . The diagram says that this is equivalent to the trivial bundle given by the trivial cocycle .
The de Rham cohomology with coefficients in discrete objects is trivial: for all we have
Using that in a ∞-connected (∞,1)-topos the functor is a full and faithful (∞,1)-functor so that the unit is an equivalence and using that by the zig-zag identity we have then that the counit component is also an equivalence, we have
since the pullback of an equivalence is an equivalence.
In a cohesive pieces have points precisely if for all , the de Rham coefficient object is globally connected in that .
If has at least one point () and is geometrically connected () then is also locally connected: .
Since preserves (∞,1)-colimits in a cohesive -topos we have
where in the last step we used that is a full and faithful, so that there is an equivalence .
To analyse this (∞,1)pushout we present it by a homotopy pushout in the standard model structure on simplicial sets . Denoting by and any representatives in of the objects of the same name in , this may be computed by the ordinary pushout in sSet
where on the right we have inserted the cone on in order to turn the top morphism into a cofibration. From this ordinary pushout it is clear that the connected components of are obtained from those of by identifying all those in the image of a connected component of . So if the left morphism is surjective on then . This is precisely the condition that pieces have points in .
For the local analysis we consider the same setup objectwise in the injective model structure on simplicial presheaves . For any we then have the pushout in
as a model for the value of the simplicial presheaf presenting . If is geometrically connected then and hence for the left morphism to be surjective on it suffices that the top left object is not empty. Since the simplicial set contains at least the vertices of which there is by assumption at least one, this is the case.
Integration of differential forms and Stokes lemma
See at integration of differential forms – In cohesive homotopy-type theory
Exponentiated -Lie algebras
We now use the intrinsic non-abelian de Rham cohomology in the cohesive -topos discussed above to see that there is also an intrinsic notion of exponentiated higher Lie algebra objects in . (The fact that for Smooth∞Grpd these abstractly defined objects are indeed presented by L-∞ algebras is discussed at smooth ∞-groupoid -- structures.)
The idea is that for an ∞-group, a -valued differential form on some , which by the above is given by a morphism
maps “infinitesimal paths” to elements of , and hence only hits “infinitesimal elements” in . Therefore the object that such forms universally factor through we write and think of as the formal Lie integration of the -Lie algebra of .
The reader should note here that all this is formulated without an explicit (“synthetic”) notion of infinitesimals. Instead, it is infinitesimal in the same sense that is the schematic de Rham homotopy type of , as discussed above. But if we add a bit more structure to the cohesive -topos , then these infinitesimals can be realized also synthetically. That extra structure is that of infinitesimal cohesion. See there for more details.
For a connected object in that is geometrically contractible
we call its loop space object the Lie integration of an ∞-Lie algebra in .
If is cohesive, then is a left adjoint.
When is cohesive we have the de Rham triple of adjunction . Accordingly then is part of an adjunction
For all the object is geometrically contractible.
Since on the locally ∞-connected (∞,1)-topos and ∞-connected the functor preserves (∞,1)-colimits and the terminal object, we have
where we used that in the ∞-connected the functor is full and faithful.
We have for every that is geometrically contractible.
We shall write for , when the context is clear.
Every de Rham cocycle factors through the ∞-Lie algebra of
By the universality of the counit of we have that factors through the counit .
Therefore instead of speaking of a -valued de Rham cocycle, it is less redundant to speak of an -valued de Rham cocycle. In particular we have the following.
Every morphism from an exponentiated -Lie algebra to an -group factors through the exponentiated -Lie algebra of that -group
If is cohesive then we have
First observe that for all we have
This follows using
using that the (∞,1)-pullback of an equivalence is an equivalence.
From this we deduce that
by computing for all
Also observe that by a proposition above we have
for all .
Finally to obtain we do one more computation of this sort, using that
Maurer-Cartan forms and curvature characteristic forms
In the intrinsic de Rham cohomology of a locally ∞-connected ∞-connected there exist canonical cocycles that we may identify with Maurer-Cartan forms and with universal curvature characteristic forms.
For an ∞-group, write
for the -valued de Rham cocycle on which is induced by the (∞,1)-pullback pasting
and the above proposition.
We call the Maurer-Cartan form on .
For an ∞-group, there are canonical -∞-actions on and on . By the discussion at ∞-action these are exhibited by the defining homotopy fiber sequences
respectively, and they identify the homotopy quotients of the action as
By the discussion at ∞-action the equivariant structure in question is a morphism of the form
such that it induces on homotopy fibers.
By remark 12 the above diagram is equivalently
There is an essentially unique horizontal morphism making this commute (up to homotopy). To see that this does induce the Maurer-Cartan form on homotopy fibers, notice that the morphism on homotopy fibers is the universal one from the total homotopy pullback diagram to the bottom homotopy pullback diagram labeled in
The pasting law implies that also the top rectangle here, is a homotopy pullback, hence this identifies in this diagram indeed as the MC form.
For an Eilenberg-MacLane object, we also write
for the intrinsic Maurer-Cartan form and call this the intrinsic universal curvature characteristic form on .
Flat Ehresmann connections
We discuss now a general abstract notion of flat Ehresmann connections in a cohesive -topos .
Let be an ∞-group. For a cocycle that modulates a -principal ∞-bundle , we saw above that lifts
modulate flat -connections in .
We can think of as the cocycle datum for the connection on base space, in generalization of the discussion at connection on a bundle. On the other hand, there is the classical notion of an Ehresmann connection, which instead encodes such connection data in terms of differential form data on the total space .
We may now observe that such differential form data on is identified with the twisted ∞-bundle induced by the lift, with respect to the local coefficient ∞-bundle given by the fiber sequence
that defines the de Rham coefficient object, discussed above.
Notice also that the -twisted cohomology defined by this local coefficient bundle says that: flat -connections are locally flat -valued forms that are globally twisted by by a -principal -bundle.
By the general discussion at twisted ∞-bundle we find that the flat connection induces on the structure
In the model = Smooth∞Grpd one finds that the last condition reduces indeed to that of an Ehresmann connection for on (this is discussed here). One of the two Ehresmann conditions is manifest already abstractly: for every point of base space, the restriction of to the fiber of over is the Maurer-Cartan form
on the -group , discussed above.
Ordinary differential cohomology
In every locally ∞-connected ∞-connected (∞,1)-topos there is an intrinsic notion of ordinary differential cohomology.
Fix a 0-truncated abelian group object . For all we have then the Eilenberg-MacLane object .
For any object and write
for the cocycle -groupoid of twisted cohomology, def. 10, of with coefficients in and with twist given by the canonical curvature characteristic morphism . This is the (∞,1)-pullback
where the right vertical morphism is any choice of cocycle representative for each cohomology class: a choice of point in every connected component.
the degree- differential cohomology of with coefficient in .
For a cocycle, we call
the class of the underlying -principal ∞-bundle;
the curvature class of .
We also say is an -connection on (see below).
The differential cohomology does not depend on the choice of morphism (as long as it is an isomorphism on , as required). In fact, for different choices the corresponding cocycle ∞-groupoids are equivalent.
is, as a 0-truncated ∞-groupoid, an (∞,1)-coproduct of the terminal object in ∞Grpd. By universal colimits in this (∞,1)-topos we have that (∞,1)-colimits are preserved by (∞,1)-pullbacks, so that is the coproduct
of the homotopy fibers of over each of the chosen points . These homotopy fibers only depend, up to equivalence, on the connected component over which they are taken.
When restricted to vanishing curvature, differential cohomology coincides with flat differential cohomology:
Moreover this is true at the level of cocycle ∞-groupoids
By the pasting law for (∞,1)-pullbacks the claim is equivalently that we have an -pullback diagram
By definition of flat cohomology and of intrinsic de Rham cohomology in , the outer rectangle is
Since the hom-functor preserves (∞,1)-limits this is a pullback if
is. Indeed, this is one step in the fiber sequence
that defines (using that preserves limits and hence looping and delooping).
The following establishes the characteristic short exact sequences that characterizes intrinsic differential cohomology as an extension of curvature forms by flat -bundles and of bare -bundles by connection forms.
Let be the image of the curvatures. Then the differential cohomology group fits into a short exact sequence
Apply the long exact sequence of homotopy groups to the fiber sequence
of prop. 32 and use that is, as a set, a homotopy 0-type to get the short exact sequence
The differential cohomology group fits into a short exact sequence of abelian groups
This is a general statement about the definition of twisted cohomology. We claim that for all we have a fiber sequence
in ∞Grpd. This implies the short exact sequence using that by construction the last morphism is surjective on connected components (because in the defining -pullback for the right vertical morphism is by assumption surjective on connected components).
To see that we do have the fiber sequence as claimed consider the pasting composite of (∞,1)-pullbacks
The square on the right is a pullback by the above definition. Since also the square on the left is assumed to be an -pullback it follows by the pasting law for (∞,1)-pullbacks that the top left object is the -pullback of the total rectangle diagram. That total diagram is
because, as before, this -pullback is the coproduct of the homotopy fibers, and they are empty over the connected components not in the image of the bottom morphism and are the loop space object over the single connected component that is in the image.
Finally using that (as discussed at cohomology and at fiber sequence)
since both as well as preserve (∞,1)-limits and hence formation of loop space objects, the claim follows.
In view of the second of these points one can make a choice of cover in order to present the twisting cocycles functorially. To that end, let
denote a choice of effective epimorphism out of a 0-truncated object which we suggestively denote by .
With a choice fixed, we say an object is dR-projective if the induced morphism
is itself an effective epimorphism (of ∞-groupoid)s.
Write for the -pullback
We say that this is the differential coefficient object of .
Consider the diagram
The bottom square is an ∞-pullback? by definition. A morphism as in the top right exists by assumption that is dR-prohective. Let also the top square be an -pullback. Then by the pasting law so is the total rectangle, which identifies the top left object as indicated, since preserves -pullbacks.
Since the top right morphism is in injection of sets, it is a monomorphism of -groupoids. These are stable under -pullback, which proves the claim.
Generalized differential cohomology
For cohesive stable homotopy types the above discussion may be refined and stream-lined considerably. For more on this see at differential cohomology diagram.
Chern-Weil homomorphism and -connections
Induced by the intrinsic differential cohomology in any ∞-connected and locally ∞-connected (∞,1)-topos is an intrinsic notion of Chern-Weil homomorphism.
Let be the chosen abelian ∞-group as above. Recall the universal curvature characteristic class
for all .
For an ∞-group and
a representative of a characteristic class we say that the composite
represents the corresponding differential characteristic class or curvature characteristic class .
The induced map on cohomology
we call the (unrefined) ∞-Chern-Weil homomorphism induced by .
The following construction universally lifts the -Chern-Weil homomorphism from taking values in intrinsic de Rham cohomology to values in intrinsic differential cohomology.
For any object, define the ∞-groupoid as the (∞,1)-pullback
a cocycle in is an ∞-connection
on the principal ∞-bundle ;
a morphism in is a gauge transformation of connections;
for each the morphism
is the (full/refined) ∞-Chern-Weil homomorphism induced by the characteristic class .
Under the curvature projection the refined Chern-Weil homomorphism for projects to the unrefined Chern-Weil homomorphism.
This is due to the existence of the pasting composite
of the defining -pullback for with the products of the defining -pullbacks for the .
As before for abelian coefficients, we introduce differential coefficient objects that represent these differential cohomology classes over dR-projective objects
The notion of intrinsic ∞-connections in a cohesive -topos induces a notion of higher holonomy
We say an object has cohomological dimension if for all -connected and -truncated objects the corresponding cohomology on is trivial
Let be the maximum for which this is true.
Since is a right adjoint it preserves delooping and hence . It follows that
Let now again be fixed as above.
Let , with .
We say that the composite
of the adjunction equivalence followed by truncation is the flat holonomy operation on flat -connections.
More generally, let
be a differential coycle on some
(using the above proposition) for the morphism on -pullbacks induced by the morphism of diagrams
The holonomy of over is the flat holonomy of
Transgression in differential cohomology
We discuss an intrinsic notion of transgression/fiber integration in ordinary differential cohomology internal to any cohesive -topos. This generalizes the notion of higher holonomy discussed above.
Fix an abelian group object as above and a corresponding differential coefficient object. Then for of cohomological dimension consider the map
In typical models we have an equivalence
In this case we say that for
a differential characteristic map, that the composite
is the transgression of to the mapping space .
For the reproduces, on the underlying -groupoids, the higher holonomy discussed above.
The notion of intrinsic ∞-connections and their higher holonomy in a cohesive -topos induces an intrinsic notion of and higher Chern-Simons functionals.
Let be of cohomological dimension and let a representative of a characteristic class for some object . We say that the composite
where denotes the refined Chern-Weil homomorphism induced by , is the extended Chern-Simons functional induced by on .
The cohesive refinement of this (…more discussion required…)
denotes the cartesian internal hom;
is the concretification projection in degree
is the truncation projection in the same degree
we call the smooth extended Chern-Simons functional.
In the language of sigma-model quantum field theory the ingredients of this definition have the following interpretation
is the worldvolume of a fundamental -brane ;
is the target space;
is the background gauge field on ;
is the space of worldvolume field configurations or trajectories of the brane in ;
is the value of the action functional on the field configuration .
In suitable situations this construction refines to an internal construction.
Assume that has a canonical line object and a natural numbers object . Then the action functional may lift to the internal hom with respect to the canonical cartesian closed monoidal structure on any (∞,1)-topos to a morphism of the form
We call the configuration space of the ∞-Chern-Simons theory defined by and the action functional in codimension defined on it.
See ∞-Chern-Simons theory for more discussion.
graded differential cohesion
For general references on cohesive (∞,1)-toposes see there.
The above list of structures in any cohesive -topos is the topic of section 2.3 of
Formulation in homotopy type theory
For formalizations of some structures in cohesive -toposes in terms of homotopy type theory see cohesive homotopy type theory.