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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI was asked by email about the claim at _[[geometry of physics]]_ that _[[integration]]_ can be axiomatized in [[cohesive homotopy type theory]] simply a truncation operation (followed by concretification, for the right cohesive structure). That may sound surprising. So I have started to work on the section _[geometry of physics -- integration](http://ncatlab.org/nlab/show/geometry+of+physics#Integration)_. So far I have there the following introductory text, which is supposed to already indicate at least while the claim is plausible. Eventually maybe I can move parts of this to the entry _[[integration]]_ proper. Here is what it currently has in the intro-paragraph of [geometry of physics - integration](http://ncatlab.org/nlab/show/geometry+of+physics#Integration):

   I was asked by email about the claim at geometry of physics that integration can be axiomatized in cohesive homotopy type theory simply a truncation operation (followed by concretification, for the right cohesive structure). That may sound surprising.

   So I have started to work on the section geometry of physics – integration. So far I have there the following introductory text, which is supposed to already indicate at least while the claim is plausible. Eventually maybe I can move parts of this to the entry integration proper.

   Here is what it currently has in the intro-paragraph of geometry of physics - integration:

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2012
   • (edited Oct 26th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex---[begin forwarded text]--- By the discussion in _[Differential forms](#DifferentialForms)_ and _[Principal connections](#PrincipalConnections)_, [[differential forms]] and more generally [[connection on a bundle|connections]] may be regarded as [[infinitesimal space|infinitesimal]] [[measures]] of _change_, of _displacement_. The discussion in _[Differentiation](#Differentiation)_ showed how to extract from a _finite_ but [[cohesion|cohesive]] (e.g. smoothly continuous) displacement all its infinitesimal measures of displacements by [[differentiation]]. Here we discuss the reverse operation: _[[integration]]_ is a construction from a [[differential form]] of the corresponding finite cohesive displacement. More generally this applies to any [[connection on a bundle|connection]] and is then called the _[[nLab:parallel transport]]_ of the connection, a term again referring to the idea that a finite displacement proceeds pointwise in parallel to a given infinitesimal displacement. Under good conditions this construction can proceed literally by "adding up all the infinitesimal contributions" and therefore [[integration]] is traditionally thought of as a generalization of forming [[sums]]. Therefore one has the notation "$\int_{\Sigma} \omega$" for the [[integral]] of a [[differential form]] $\omega$ over a space $\Sigma$, as a variant of the notation "$\sum_{S} f$" for the [[sum]] of values of a [[function]] on a [[set]] $S$. For the case of integrals of connections the corresponding parallel transport expression is often denoted by an exponentiated integral sign "$\mathcal{P} \exp(\int_\Sigma \omega)$", referring to the fact that the passage from infinitesimal to finite quantities involves also the passage from [[Lie algebra]] data to [[Lie group]] data ("[[Lie integration|exponentiated Lie algebra data]]"). However, both from the point of view of [[gauge theory]] [[physics]] as well as from the [[general abstract]] perspective of [[cohesive homotopy type theory]] another characterization of integration is more fundamental: the [[integral]] $\int_\Sigma \omega$ of a [[differential form]] $\omega$ (or more generally of a [[circle n-bundle with connection|connection]]) is an [[invariant]] under those [[gauge transformations]] of $\omega$ that are trivial on the [[boundary]] of $\Sigma$, and it is the _[[universal construction|universal]]_ such invariant, hence is uniquely characterized by this property. In traditional accounts this fact is referred to via the _[[Stokes theorem]]_ and its generalizations (such as the [[nonabelian Stokes theorem]]), which says that the [[integral]]/[[parallel transport]] is indeed _[[invariant]]_ under gauge transformations of differential forms/connections. That this invariance actually characterizes the integral and the parallel transport is rarely highlighted in traditional texts, but it is implicit for instance in the old "path method" of [[Lie integration]] (discussed below in _[Lie integration](#LieIntegration)_) as well as in the famous characterization of [[flat connections]], discussed above in _[Flat 1-connections](#Flat1Connections)_: for $X$ a [[connected topological space|connected]] [[manifold]] and for $G$ a [[Lie group]], the operation of sending a [[flat connection|flat]] $G$-[[principal connection]] $\nabla$ to its [[parallel transport]] $\gamma \mapsto hol_{\gamma}(\nabla)$ around [[loops]] $\gamma\colon S^1 \to X$, hence to the [[integral]] of the connection around all possible loops (its [[holonomy]]), for any fixed basepoint $$ hol \coloneqq \mathcal{P} \exp(\int_{(-)} (-)) \;\colon\; H^1_{conn, flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X) , G)/G $$ exhibits a [[bijection]] between gauge [[equivalence classes]] of connections and [[group]] homomorphisms from the [[fundamental group]] $\pi_1(X)$ of $X$ to the [[gauge group]] $G$ (modulo [[adjoint action|adjoint]] $G$-action from gauge transformations at the base point, hence at the integration boundary). This is traditionally regarded as a property of the definition of the [[parallel transport]] $\mathcal{P} \exp(\int_{(-)}(-))$ by [[integration]]. But being a [[bijection]], we may read this fact the other way round: it says that forming equivalence classes of flat $G$-connections is a way of computing their [[integral]]/[[parallel transport]]. ---[continued in next message]---

   —[begin forwarded text]—

   By the discussion in Differential forms and Principal connections, differential forms and more generally connections may be regarded as infinitesimal measures of change, of displacement. The discussion in Differentiation showed how to extract from a finite but cohesive (e.g. smoothly continuous) displacement all its infinitesimal measures of displacements by differentiation.

   Here we discuss the reverse operation: integration is a construction from a differential form of the corresponding finite cohesive displacement. More generally this applies to any connection and is then called the parallel transport (nlab) of the connection, a term again referring to the idea that a finite displacement proceeds pointwise in parallel to a given infinitesimal displacement.

   Under good conditions this construction can proceed literally by “adding up all the infinitesimal contributions” and therefore integration is traditionally thought of as a generalization of forming sums. Therefore one has the notation “$\int_{\Sigma} \omega$” for the integral of a differential form $\omega$ over a space $\Sigma$, as a variant of the notation “$\sum_{S} f$” for the sum of values of a function on a set $S$. For the case of integrals of connections the corresponding parallel transport expression is often denoted by an exponentiated integral sign “$\mathcal{P} \exp(\int_\Sigma \omega)$”, referring to the fact that the passage from infinitesimal to finite quantities involves also the passage from Lie algebra data to Lie group data (“exponentiated Lie algebra data”).

   However, both from the point of view of gauge theory physics as well as from the general abstract perspective of cohesive homotopy type theory another characterization of integration is more fundamental: the integral $\int_\Sigma \omega$ of a differential form $\omega$ (or more generally of a connection) is an invariant under those gauge transformations of $\omega$ that are trivial on the boundary of $\Sigma$, and it is the universal such invariant, hence is uniquely characterized by this property.

   In traditional accounts this fact is referred to via the Stokes theorem and its generalizations (such as the nonabelian Stokes theorem), which says that the integral/parallel transport is indeed invariant under gauge transformations of differential forms/connections. That this invariance actually characterizes the integral and the parallel transport is rarely highlighted in traditional texts, but it is implicit for instance in the old “path method” of Lie integration (discussed below in Lie integration) as well as in the famous characterization of flat connections, discussed above in Flat 1-connections:

   for $X$ a connected manifold and for $G$ a Lie group, the operation of sending a flat $G$-principal connection $\nabla$ to its parallel transport $\gamma \mapsto hol_{\gamma}(\nabla)$ around loops $\gamma\colon S^1 \to X$, hence to the integral of the connection around all possible loops (its holonomy), for any fixed basepoint

   $$hol \coloneqq \mathcal{P} \exp(\int_{(-)} (-)) \;\colon\; H^1_{conn, flat}(X,G) \stackrel{\simeq}{\to} Hom_{Grp}(\pi_1(X) , G)/G$$

   exhibits a bijection between gauge equivalence classes of connections and group homomorphisms from the fundamental group $\pi_1(X)$ of $X$ to the gauge group $G$ (modulo adjoint $G$-action from gauge transformations at the base point, hence at the integration boundary). This is traditionally regarded as a property of the definition of the parallel transport $\mathcal{P} \exp(\int_{(-)}(-))$ by integration. But being a bijection, we may read this fact the other way round: it says that forming equivalence classes of flat $G$-connections is a way of computing their integral/parallel transport.

   —[continued in next message]—

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2012
   • (edited Oct 26th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex---[continued from previous message]--- We saw a generalization of this fact to non-closed forms and non-flat connections already in the discussion at _[Differential 1-forms as smooth incremental path measures](##1FormsAsSmoothFunctors)_, where gauge equivalence classes of differential forms are shown to be equivalently assignments of [[parallel transport]] to [[path groupoid|smooth paths]]. This is also implied by the above discussion: for $\nabla \in H^1_{conn}(X,G) $ any non-flat connection and $\gamma \colon S^1 \to X$ a trajectory in $X$, we may form the [[pullback of differential forms|pullback]] of $\nabla$ to $S^1$. There it becomes a necessarily flat connection $\gamma^* \nabla \in H^1_{conn, flat}(S^1,G)$, since the [[curvature]] [[differential 2-form]] necessarily vanishes on the 1-[[dimension|dimensional]] [[manifold]] $S^1$. Accordingly, by the above bijection, forming the gauge [[equivalence class]] of $\gamma^* \nabla$ means to find a group homomorphism $$ \mathbb{Z} \simeq \pi_1(S^1) \to G $$ modulo conjugation (modulo nothing if $G$ is [[abelian group|abelian]], such as $G = U(1)$) and since $\mathbb{Z}$ is the [[free group]] on a single generator this is the same as finding an element $$ hol_\gamma(\nabla) = \mathcal{P} \exp(\int_\gamma \gamma^*\nabla) \in G \,. $$ This total operation of first pulling back the connection and then forming its [[integration]] (by taking gauge equivalence classes) is called the _[[transgression]]_ of the original 1-form connection on $X$ to a 0-form connection on the [[loop space]] $[S^1,X]$. Below in the [Model Layer](#IntegrationModelLayer) we discuss the classical examples of [[integration]]/[[parallel transport]] and their various generalizations in detail. Then in the [Semantic Layer](#IntegrationSemanticLayer) we show how indeed all these constructions are obtained forming [[equivalence classes]] in the [[(∞,1)-topos]] of [[smooth ∞-groupoids|smooth homotopy types]], hence by _[[truncated object in an (∞,1)-category|truncation]]_ (followed, to obtain the correct cohesive structure, by _concretification_, def. \ref{ConcreteObjectsAndConcretification}). ---[end forwarded text]---

   —[continued from previous message]—

   We saw a generalization of this fact to non-closed forms and non-flat connections already in the discussion at Differential 1-forms as smooth incremental path measures, where gauge equivalence classes of differential forms are shown to be equivalently assignments of parallel transport to smooth paths.

   This is also implied by the above discussion: for $\nabla \in H^1_{conn}(X,G)$ any non-flat connection and $\gamma \colon S^1 \to X$ a trajectory in $X$, we may form the pullback of $\nabla$ to $S^1$. There it becomes a necessarily flat connection $\gamma^* \nabla \in H^1_{conn, flat}(S^1,G)$, since the curvature differential 2-form necessarily vanishes on the 1-dimensional manifold $S^1$. Accordingly, by the above bijection, forming the gauge equivalence class of $\gamma^* \nabla$ means to find a group homomorphism

   $$\mathbb{Z} \simeq \pi_1(S^1) \to G$$

   modulo conjugation (modulo nothing if $G$ is abelian, such as $G = U(1)$) and since $\mathbb{Z}$ is the free group on a single generator this is the same as finding an element

   $$hol_\gamma(\nabla) = \mathcal{P} \exp(\int_\gamma \gamma^*\nabla) \in G \,.$$

   This total operation of first pulling back the connection and then forming its integration (by taking gauge equivalence classes) is called the transgression of the original 1-form connection on $X$ to a 0-form connection on the loop space $[S^1,X]$.

   Below in the Model Layer we discuss the classical examples of integration/parallel transport and their various generalizations in detail. Then in the Semantic Layer we show how indeed all these constructions are obtained forming equivalence classes in the (∞,1)-topos of smooth homotopy types, hence by truncation (followed, to obtain the correct cohesive structure, by concretification, def. \ref{ConcreteObjectsAndConcretification}).

   —[end forwarded text]—

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeOct 26th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn connection to categorically interesting new ideas about functorial ideas on integration in geometry, this may interest some of you. But being in Utrecht, Urs might have seen this. Alan Weinstein (UC Berkeley) - Talk at Poisson 2012, Utrecht, July 2012 [video](http://youtu.be/TJ004CWDDLI) __Functorial integration of Poisson manifolds and Lie algebroid comorphisms__ >Abstract: I will report on ongoing work with Alberto Cattaneo and Benoît Dherin, in which we construct true functors from categories of Poisson manifolds and Poisson maps to categories of (possibly micro-) symplectic groupoids. Some of this consists of a rediscovery of announced results (with details unpublished) of Pierre Dazord on the integration of complete comorphisms between Lie algebroids.

   In connection to categorically interesting new ideas about functorial ideas on integration in geometry, this may interest some of you. But being in Utrecht, Urs might have seen this.

   Alan Weinstein (UC Berkeley) - Talk at Poisson 2012, Utrecht, July 2012 video

   Functorial integration of Poisson manifolds and Lie algebroid comorphisms

   Abstract: I will report on ongoing work with Alberto Cattaneo and Benoît Dherin, in which we construct true functors from categories of Poisson manifolds and Poisson maps to categories of (possibly micro-) symplectic groupoids. Some of this consists of a rediscovery of announced results (with details unpublished) of Pierre Dazord on the integration of complete comorphisms between Lie algebroids.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 31st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexZoran, we should discuss this in its own thread, not here in the thread on latest changes of the entry _[[integration]]_.

   Zoran, we should discuss this in its own thread, not here in the thread on latest changes of the entry integration.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeOct 31st 2012
   • (edited Oct 31st 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexDiscussion in 3 is about some special ideas about integration which are not in the entry [[integration]] (they are in the entry on geometry of integration). Discussion in 4 is a short remark about some other special ideas about integration which are not in the entry [[integration]] (they could be put into the entry on geometry of integration). I see no difference except that your current mood maybe relates the thoughts in your post in 3 to the nonexisting part of [[integration]] but it does not relate equally in phase with the thoughts in 4. What is the point in discouraging me to write a short remark with another short remark. I spent some time to find the reference which I seen few days ago, exclusively to serve your current interest in geometry in integration by writing a remark and the appreciation is not proportional.

   Discussion in 3 is about some special ideas about integration which are not in the entry integration (they are in the entry on geometry of integration). Discussion in 4 is a short remark about some other special ideas about integration which are not in the entry integration (they could be put into the entry on geometry of integration). I see no difference except that your current mood maybe relates the thoughts in your post in 3 to the nonexisting part of integration but it does not relate equally in phase with the thoughts in 4. What is the point in discouraging me to write a short remark with another short remark. I spent some time to find the reference which I seen few days ago, exclusively to serve your current interest in geometry in integration by writing a remark and the appreciation is not proportional.