nLab geometry of physics -- smooth homotopy types

This entry contains one chapter of the material at geometry of physics.

previous chapters: categories and toposes, smooth sets,

next chapter: groups, supergeometry, prequantum geometry

under construction

Fundamental physics is all based on the gauge principle. This says in particular that it is wrong to think of two different gauge field configurations (discussed in detail below in Fields) as being equal or not. Instead it makes sense to ask if there is (or not) a gauge transformation from one to the other that exibits an gauge equivalence between the two fields. The simplest example of this is described in detail below in Gauge transformations in electromagnetism.

But this means that the collection of gauge fields on a spacetime $X$, which we will write as a mapping space $[X, \mathbf{B}G_{conn}]$, cannot be a smooth space as considered above, for if it were such a smooth space, then we could ask if two gauge fields $\nabla_1, \nabla_2 \colon * \to [X,\mathbf{B}G_{conn}]$ were equal or not.

Notice that this already applies to a single gauge field: given any $\nabla \colon * \to [X,\mathbf{B}G_{conn}]$ it is certainly equal to itself, but is nevertheless also gauge equivalent to itself, but the latter it may be in several non-equivalent ways: there may be non-trivial auto-gauge transformations $\nabla \to \nabla$. Since these can be composed, are, by definition, invertible, and contain the trivial gauge transformaiton, these form a group, the group of auto-gauge equivalences of $\nabla$. (Groups are discussed in detail below in Groups) If that gauge field $\nabla$ is itself the trivial gauge field, $\nabla = \nabla_0$, then this group of auto-gauge equivalences is the gauge group of the given gauge theory:

$$G \simeq Aut(\nabla_0) \simeq \{\nabla_0 \stackrel{\simeq}{\to} \nabla_0\} \,.$$

For this reason, the collection of all gauge fields and all gauge transformations between them form something that is a group over ever fixed element, but which generalizes the notion of a group in that there are not only auto-equivalences, but also equivalences going from one element to another. Such a structure is called a groupoid. Gauge fields form not a set with smooth structure, a smooth space, but a groupoid with smooth structure: a smooth groupoid.

At least ordinary gauge fields do. More generally, the gauge princple goes further: it is in general also a mistake to assume that given two gauge transformations $\lambda_1, \lambda_2 \colon \nabla_1 \to \nabla_2$ between two gauge fields, it makes good sense to ask whether they are equal or not. Again, the gauge prinicle says that we should instead ask if there is a gauge-of-gauge transformation between them, that exhibits a gauge equivalence $\rho \colon \lambda_1 \simeq \lambda_2$. If these gauge-of-gauge equivalences are nontrivial it would seem that gauge fields form a generalization of the notion of a groupoid called a 2-groupoid. But in general the gauge principle goes on: we can in general never decide if two $n$-fold gauge-of-gauge transformations are actually equal, all we have is, possibly, an $(n+1)$-fold gauge transformation going between them which exhibits their gauge equivalence, this being so for all $0 \lt n \lt \infty$. One therefore says that gauge fields in general form an ∞-groupoid whose n-morphisms are $n$-fold gauge-of-gauge transformations. These $\infty$-groupoids are also called homotopy types. And since at the same time there is still a notion of gauge fields varying smoothly, these are smooth ∞-groupoids or smooth homotopy types. This is what we discuss here.

Remarkably, the same concept appears in constructive mathematics: there it is in general wrong to consider an equality between two terms $\nabla_1, \nabla_2 \colon [X, \mathbf{B}G_{conn}]$, instead one is to consider an explicit proof that they are equal, provide an explicit equivalence between them. Such a proof $\lambda$ is itself a term, of identity type, written just as before, $\lambda \colon (\nabla_1 \simeq \nabla_2)$. And, in turn, the same applies to these proofs of equivalence themselves: in constructive mathematics one demands a proof $\rho$ that two equivalences $\lambda_1, \lambda_2 \colon (\nabla_1 \simeq \nabla_2)$ are equivalent, and hence a term $\rho \colon (\lambda_1 \simeq \lambda_2)$ of a higher identity type. If this goes ever on, and one speak of intensional type theory or homotopy type theory.

This remarkable matching of higher gauge theory and homotopy type theory is what drives the discussion here.

Smooth 1-groupoids

Motivation: Fields and gauge transformations in electromagnetism

The mathematical concept of smooth groupoid is well familiar, in slight disguise, in basic gauge theory. Here we make this explicit for basic electromagnetism. For more exposition and details along these lines see (Eggertson 14).

We have seen in The electromagnetic field strength that a configuration of the electromagnetic field on $\mathbb{R}^4$ is given by a differential 1-form $A \in \Omega^1(\mathbb{R}^4)$, the “electromagnetic potential”. It describes a field configuration with field strength Lorentz tensor (with respect to the canonical coordinates $(t, x^1, x^2, x^3) \coloneqq (x^i)_{i = 1}^4 =$ on $\mathbb{R}^4$)

$$\begin{aligned} F & = \mathbf{d}A \\ & = E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + \\ & + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2 \end{aligned}$$

with electric field strength $(E_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3$ and magnetic field strength $(B_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3$ given that is given by the de Rham differential of $A$:

$$\begin{aligned} F = \mathbf{d}A \end{aligned} \,.$$

After Maxwell it was thought that $F \in \Omega^2(\mathbb{R}^4)$ alone genuinely reflects the configuration of the electromagnetic field. But with the discovery of quantum mechanics it became clear that it is indeed the potential $A$ itself that reflects the configuration of the electromagnetic field: in the presence of magnetic flux or other topoligical constraints, there can be different $A, A'$ with the same $F = \mathbf{d}A = \mathbf{d}A'$ which nevertheless describe experimentally distinguishable electromagnetic field configurations. (Distinguishable by the Aharonov-Bohm effect and also to some extent by Dirac charge quantization; this is discussed at Circle-principal connections below.)

However, not all different gauge potentials describe different physics. The actual configuration space of electromagnetism on a spacetime $X$ is finer than $\mathbf{\Omega}^2_{cl}(X)$ but coarser than $\mathbf{\Omega}^1(X)$. And it is not quite a smooth space itself, but a smooth groupoid:

one finds that two electromagnetic potentials $A, A' \in \Omega^1(\mathbb{R}^4)$ for which there is a function $\lambda \in C^\infty(\mathbb{R}^4)$ such that

$$A' = A + \mathbf{d}\lambda$$

represent different but equivalent field configurations. One says that $\lambda$ induces a gauge transformation from $A$ to $A'$. We write $\lambda \colon A \stackrel{\simeq}{\to} A'$ to reflect this.

So the configuration space of electromagnetism does not just have points and coordinate systems. But it is also equipped with the information of a space of gauge transformations between any two coordinate systems laid out in it (which may be empty).

To see what the structure of such a smooth gauge groupoid should be, notice that the above defines an action of smooth functions $\lambda$ on smooth $1$-forms $A$,

Definition

For any $U \in$ CartSp, Write $\Omega^1_{vert}(X \times U)$ for the set of differential 1-forms on $X \times U$ with no components along $U$, and write $C^\infty(X \times U , U(1))$ for the group of circle group valued smooth functions. There is an action of this group on the 1-forms

$$\rho_U \colon \Omega^1_{vert}(X \times U) \times C^\infty(X \times U, U(1)) \to \Omega^1_{vert}(X \times U)$$

given by

$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}_X \lambda \,.$$

Given such an action of a discrete group on a set, we might be demoted to form the quotient set $\Omega^1_{vert}(X\times U)/C^\infty(X \times U, U(1))$. This set contains the gauge equivalence classes of $U$-parameterized collections of electromagnetic gauge fields on $X$. But it turns out that this is too little information to correctly capture physics. For that instead we need to remember not just that two gauge fields are equivalent, but how they are equivalent. That is, we for $\lambda$ a gauge transformation from $A_1$ to $A_2$, we should have an equivalence $\lambda \colon A_1 \stackrel{\simeq}{\to} A_2$.

Definition

The action groupoid

$$\Omega^1_{vert}(X\times U)// C^\infty(X \times U, U(1)) \coloneqq ( \Omega^1_{vert}(X\times U) \times C^\infty(X \times U, U(1)) \stackrel{\overset{t = \rho}{\longrightarrow}}{\underset{s = p_1}{\longrightarrow}} \Omega^1_{vert}(X\times U) )$$

is the groupoid whose

• objects are $U$-parameterized collections of gauge potentials $A \in \Omega^1_{vert}(X \times U)$;

• morphisms are pairs $(A,\lambda)$ with $A$ an object and $\lambda \in C^\infty(X \times U, U(1))$, with domain $A$ and codomain $A + \mathbf{d}_X \lambda$;

• composition is given by multiplication of the $\lambda$-labels in the group $C^\infty(X \times U, U(1))$.

This is the discrete gauge groupoid for $U$-parameterized collections of fields. It refines the gauge group, which is recoverd as its fundamental group:

Proposition

Let $A_0 \coloneqq 0 \in \Omega^1_{vert}(X \times U)$ be the trivial gauge field. Then its automorphism group in the gauge groupoid of def. is the group of circle-group valued functions on $U$:

$$\pi_1(\Omega^1_{vert}(X\times U)// C^\infty(X \times U, U(1)), A_0) = Aut(A_0) \simeq C^\infty(U,U(1)) \,.$$

Proof

By definition, an automorphism of $A_0$ is given by a function $\lambda \in C^\infty(X \times U, U(1))$ such that $A_0 = A_0 + \mathbf{d}_X \lambda$. This is the case precisely if $\mathbf{d}_X \lambda = 0$, hence if $\lambda$ is contant along $X$. But a function on $X \times U$ which is constant along $X$ is canonically identified with a function on just $U$.

All this data in in fact natural in $U$. Recall that $\Omega^1_{vert}(X \times U) = \mathbf{\Omega}^1(X)(U)$ is the set of $U$-charts of the smooth moduli space $\mathbf{\Omega}^1(X)$ of smooth 1-forms on $X$. Similarly $C^\infty(X \times U) = \mathbf{C}^\infty(X)(U)$.

Proposition

There is a homomorphism of smooth spaces (def. )

$$\rho \colon \mathbf{\Omega}^1(X)\times \mathbf{C}^\infty(X) \to \mathbf{\Omega}^1(X)$$

from the product smooth space, def. , of the smooth moduli spaces of 1-forms and 0-forms on $X$, def. , to that of smooth functions, def. , whose component over $U \in$ CartSp is the action

$$\rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}\lambda$$

of def. .

We then also want to consider a smooth action groupoid.

Definition

Write

$$\mathbf{\Omega}^1(X) // \mathbf{C}^\infty(X, U(1)) : CartSp^{op} \to Grpd$$

for the contravariant functor from coordinate systems to the category of groupoids, which assigns to $U \in CartSp$ the discrete action groupoid of $U$-collections of gauge fields of def. .

$$U \mapsto \Omega^1_{vert}(X \times U) // C^\infty(X\times U, U(1)) \,.$$

Such a structure presheaf of groupoids is a common joint generalization of the notion of discrete groupoids and smooth spaces. We call them smooth groupoids. This is what we turn to in Smooth groupoids

Definition

Remark

on how not to define smooth groupoids

One could be tempted to define smooth groupoids to be internal groupoids in smooth sets. Restricting this definition to groupoids internal to diffeological spaces yields the concept of diffeological groupoid, restricting it further to groupoids internal to smooth manifolds yields the concept of Lie groupoid. While these are respectable definitions in themselves, care needs to be exercised in interpreting them correctly: they do not in themselves exhibit the correct homotopy theory of smooth groupoids. One may of course fix this by changing the concept of morphisms to generalized morphisms called Morita morphisms or bibundles. These are certainly useful tools for working with smooth groupoids, and they serve to present their correct homotopy theory, but they do not serve well as the definition of this homotopy theory. For instance without further insight it is rather impossible to guess form the concept of bibundles between groupoids its correct generalization to smooth 2-groupoids etc.

But just as the definition of smooth sets, contrasted with that of smooth manifolds, is not only more powerful but also simpler, so there is a definition of smooth groupoids which not only does give the correct homotopy theory, but it does so while being much more transparent than these more traditional presentations.

Now, this definition, given below, amounts to saying that smooth groupoids are stacks on the site of smooth manifolds, and that in turn may tend to not sound like a simple definition at all. But there is a further simplification at work. Traditional texts tend to define stacks in terms of the comparatively intricate structures of pseudofunctors or (dually under the Grothendieck construction) fibered categories satisfying descent, and the rich structure of these combinatorial objects tends to become unwieldy already for fairly simple examples. But the theory of localization of categories turns out to handle the same theory of stacks by much more tractable means (e.g. Hollander 01). Here a stack is presented by a plain functor with no descent condition imposed, something that is hence as simple as a pair of two presheaves. The nature of stacks is then instead just encoded in remembering that some of the morphisms of presheaves of groupoids are to be labeled as weak equivalences, namely those that locally restrict to equivalences of groupoids.

This style of definition combines the simplicity of the naive definition of Lie groupoids with the full power of homotopy theory, and it immediately generalizes to a definition of ∞-stacks which is just as simple, hence, in the present context, to a definition of smooth ∞-groupoid.

Pre-smooth groupoids

Definition

Write CartSp for the category of Cartesian spaces $\mathbb{R}^n$, $n \in \mathbb{N}$, with smooth functions between them. (The full subcategory of SmoothMfd on the Cartesian spaces.) Regard this as a site by equipping it with the coverage of (differentiably) good open covers.

For more on this see at geometry of physics – coordinate systems.

In view of the motivation for sheaves, cohomology and higher stacks and in direct analogy with the discussion at geometry of physics – smooth sets, just replacing sets by groupoids throughout, we set:

Definition

A pre-smooth groupoid $X$ is a presheaf of groupoids on CartSp, hence a functor

$$X \colon CartSp^{op} \longrightarrow Grpd$$

from CartSp to the 1-category Grpd.

See at geometry of physics – homotopy types – groupoids for more on bare groupoids. Here we will freely assume familiarity with these.

Remark

The intuition for def. is the following: a smooth structure on a groupoid is to be determined by which maps from abstract coordinate systems $\mathbb{R}^n$ into it are to be regarded as smooth maps. Since, by its groupoidal nature, two such maps may be related by a smooth homotopy/gauge transformation (in physics: “twisted sectors”), the collection of all smooth functions from $\mathbb{R}^n$ into the smooth groupoid $X$ is itself a bare groupoid. We here define a pre-smooth groupoid as something that assigns to each abstract coordinate systems a groupoid of would-be smooth maps into it

“$X(\mathbb{R}^n) = \left\{ \array{ & \nearrow \searrow^{\mathrlap{smooth}} \\ \mathbb{R}^n &\Downarrow& X \\ & \searrow \nearrow_{\mathrlap{smooth}} } \right\}$”

We sometimes therefore speak of $X(\mathbb{R}^n)$ as the groupoid of plots or of probes of $X$ (which here is only defined by these!) by $n$-dimensional coordinate systems.

The Yoneda lemma will turn this intuition into a theorem. For that we need to speak of homomorphisms of pre-smooth groupoids, hence of “smooth functors” betwen pre-smooth groupoids.

Definition

A homomorphism or smooth map between pre-smooth groupoids is a natural transformation between the presheaves that they are. We write

$$PreSmooth1Type \coloneqq Funct(CartSp^{op}, Grpd)$$

for the functor category, the category of pre-smooth groupoids, def. , regarded naturally as a Grpd-enriched category.

This means that for $X,Y \in PreSmooth1Type$ two pre-smooth groupoids, then the bare hom-groupoid

$$PreSmooth1Type(X,Y)_\bullet \in Grpd$$

between them has

• as objects the natural transformations $f \colon X \to Y$ between $X$ and $Y$ regarded as functors, hence collections $\{f(\mathbb{R}^n)\}_{n \in \mathbb{N}}$ of functors between groupoids of probes

  $$f(\mathbb{R}^n) \colon X(\mathbb{R}^n) \longrightarrow Y(\mathbb{R}^n)$$

  such that for every smooth function $\phi \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between abstract coordinate charts, these form a (strictly) commuting diagram with the probe-pullback functors of $X$ and $Y$:

  $$\array{ X(\mathbb{R}^{n_1}) &\overset{f(\mathbb{R}^n)}{\longrightarrow}& Y(\mathbb{R}^{n_1}) \\ {}^{\mathllap{X(\phi)}}\downarrow && \downarrow^{\mathrlap{Y(\phi)}} \\ X(\mathbb{R}^{n_2}) &\overset{f(\mathbb{R}^n)}{\longrightarrow}& Y(\mathbb{R}^{n_2}) } \,;$$

• as morphisms $H \colon f \rightarrow g$ the “modifications” of these, hence collections $H(\mathbb{R}^n) \colon f(\mathbb{R}^n)\to g(\mathbb{R}^n)$ of natural transformations, such that for every smooth function $\phi \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ the whiskering of these satisfies

  $$\array{ X(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{X(\phi)}} \\ & \nearrow \searrow^{\mathrlap{f(\mathbb{R}^{n_2})}} \\ X(\mathbb{R}^{n_2}) &\Downarrow^{H(\mathbb{R}^{n_2})}& Y(\mathbb{R}^{n_2}) \\ & \searrow \nearrow_{\mathrlap{g(\mathbb{R}^{n_1})}} } \;\;\; = \;\;\; \array{ & \nearrow \searrow^{\mathrlap{f(\mathbb{R}^{n_1})}} \\ X(\mathbb{R}^{n_1}) &\Downarrow^{H(\mathbb{R}^{n_1})}& Y(\mathbb{R}^{n_1}) \\ & \searrow \nearrow_{\mathrlap{g(\mathbb{R}^{n_1})}} \\ && \downarrow^{\mathrlap{Y(\phi)}} \\ && Y(\mathbb{R}^{n_2}) }$$

Remark

Despite the size of the diagrams in def. , what they encode is immediate: this just says that smooth maps between pre-smooth groupoids take all probes of $X$ by abstract coordinate charts to probes of $Y$ by these charts, and take gauge transformations between these to gauge transformations between those, in a way that it compatible with changing probes along smooth maps.

The following is the Yoneda lemma in this context, and it says that this intuition in remark is fully correct:

Proposition

Every Cartesian space $\mathbb{R}^n$ defines a pre-smooth groupoid $\underline{\mathbb{R}}^n$, def. , by the assignment

$$\underline{\mathbb{R}}^n \colon \mathbb{R}^k \mapsto C^\infty(\mathbb{R}^k, \mathbb{R}^n) \in Set \hookrightarrow Grpd$$

where the set of smooth functions on the right is regarded as a groupoid with only identity morphisms. This construction constitutes a fully faithful functor

$$\underline{(-)} \colon CartSp \hookrightarrow PreSmooth1Type$$

making CartSp a full subcategory of that of pre-smooth groupoids (the Yoneda embedding). Under this embedding for any pre-smooth groupoid $X\in PreSmooth1Type$ and any Cartesian space $\mathbb{R}^n$, there is a natural equivalence of groupoids

$$PreSmooth1Type(\underline{\mathbb{R}}^n, X) \simeq X(\mathbb{R}^n)$$

(the Yoneda lemma proper).

Remark

The last statement of the Yoneda lemma in prop. expresses just the intuition of remark and justifies removing the quotation marks displayed there. It also justifies dropping the extra underline denoting the Yoneda embedding. We will freely identify from now on $\mathbb{R}^n$ with the pre-smooth groupoid that it represents.

Example

For $X$ a smooth set (e.g. a smooth manifold), hence in particular a functor

$$X \colon CartSp^{op} \to Set$$

then its embedding into groupoids

$$X \colon CartSp^{op} \to Set \hookrightarrow Grpd$$

is a pre-smooth groupoid.

More generally:

Example

Let

$$\mathcal{G}_\bullet = \left\{ \mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\stackrel{\overset{i}{\longleftarrow}}{\underset{t}{\longrightarrow}}} \mathcal{G}_0 \right\}$$

be an internal groupoid in smooth sets, hence a pair $\mathcal{G}_0, \mathcal{G}_1 \in Smooth0Type$ of smooth sets, equipped with source, target, identity homomorphisms of smooth sets between them, and equipped with a compatibly unital, associative and invertible composition map $\mathcal{G}_1 \underset{\times}{\mathcal{G}_0} \mathcal{G}_1 \to \mathcal{G}_1$. By definition of smooth sets, this means that for every abstract coordinate system $\mathbb{R}^n$ then

$$\mathcal{G}(\mathbb{R}^n)_\bullet = \left\{ \mathcal{G}(\mathbb{R}^n)_1 \stackrel{\overset{s(\mathbb{R}^n)}{\longrightarrow}}{\stackrel{\overset{i(\mathbb{R}^n)}{\longleftarrow}}{\underset{t(\mathbb{R}^n)}{\longrightarrow}}} \mathcal{G}(\mathbb{R}^n)_0 \right\}$$

is a bare groupoid. Moreover, this assignment is functorial and hence defines a pre-smooth groupoid

$$\mathbb{R}^n \mapsto \mathcal{G}(\mathbb{R}^n)_\bullet \,.$$

This exhibits sequence of full subcategory inclusions

$$Grp \hookrightarrow LieGrpd \hookrightarrow DiffGrpd \hookrightarrow Grpd(Smooth0Type) \hookrightarrow PreSmooth1Type$$

of internal groupoids in smooth sets into pre-smooth groupoids, and hence (by further restriction) also of diffeological groupoids, of Lie groupoids and of course of just bare groupoids, too.

Regarding a bare groupoid $\mathcal{G}_\bullet$ as a pre-smooth groupoid this way means to regard it as equipped with discrete smooth structure. It is given by the constant presheaf

$$\mathbb{R}^n \mapsto \mathcal{G}_\bullet$$

exhibiting the fact that smooth functions from an $\mathbb{R}^n$ into a geometrically discrete space are constant on one of the points of this space, a situation which here is only refined by the fact that every morphism in the groupoid thus gives a homotopy/gauge transformation between the two smooth functions that are constant on the two endpoints of this morphism.

A particular case of this of special importance is this:

Example

For $G$ a Lie group, we write $(\mathbf{B}G)_\bullet \in Grpd(Smooth0Type)$ for the Lie groupoid which

$$(\mathbf{B}G)_\bullet = \left( G \stackrel{\longrightarrow}{\stackrel{\overset{i}{\longleftarrow}}{\longrightarrow}} \ast \right)$$

whose composition is the product operation in the group (the groupoidal delooping of $G$). The pre-smooth groupoid that this corresponds to under the embedding of example has groupoids of probes of the form

$$\mathbb{R}^n \mapsto B \left(C^\infty(\mathbb{R}^n,G)_{disc}\right)$$

where on the right we have the homotopy 1-type whose fundamental group is that of smooth $G$-valued functions on $\mathbb{R}^n$, under pointwise mulitplication.

More specifically, the following class of examples plays a special role in the theory, as the encode what it takes for a pre-smooth groupoid to be a genuinely smooth groupoid.

Example

For $X$ a smooth manifold, let $\{U_i \to X\}_{i \in I}$ be an open cover of $X$. Its Cech groupoid is the Lie groupoid (diffeological groupoid) $C(\{U_i\})_\bullet$ whose

• manifold of objects is $C(\{U_i\})_0 \coloneqq \underset{i \in I}{\coprod} U_i$ is the disjoint union of all the charts of the cover;

• manifold of morphisms is $C(\{U_i\})_1 \coloneqq \underset{i,j \in I}{\coprod}U_i \underset{X}{\times} U_j$ is the disjoint union of all intersections of charts.

and whose source, target and identity maps are the evident inclusions. There is then a unique composition operation.

So a global point $\ast \to C(\{U_i\})_\bullet$ may be thought of as a pair $(x,i)$ of a point in the manifold $X$ and a label $i$ of a chart $U_i$ that contains it, and there is is precisely one morphism between two such global point $(x,i)\to (y,j)$ whenever $x = y$ in $X$ and both $U_i$ as well as $U_j$ contain $x$, hence one morphism for each point in an intersection of two patches. Composition of morphism is just re-remembering which intersections they sit in, the schematic picture of the Cech groupoid is this this:

$$C(\{U_i\})_\bullet = \left\{ \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) && \longrightarrow && (x,k) } \right\} \,.$$

Under the embedding of example there is an evident morphism of pre-smooth groupoids

$$C(\{U_i\}) \longrightarrow X$$

from this Cech groupoid of the cover to the manifold that is being covered. This morphism simply forgets the information of which chart or intersection of charts a point is regarded to be in, and just remembers it as a point of $X$.

$$((x,i) \to (x,j)) \mapsto (x \stackrel{id_x}{\to} x) \,.$$

Proposition

Let $X$ a smooth manifold, $\{U_i \to X\}$ an open cover and $C(\{U_i\})$ the corresponding Cech groupoid, def. . Let $G$ be a Lie group and $\mathbf{B}G$ its groupoidal delooping according to example .

Then the hom-groupoid $PreSmooth1Type(C(\{U_i\}), \mathbf{B}G)$ of maps from $C(\{U_i\})$ to $\mathbf{B}G$, def. , has

• as objects the Cech cocycles of degree 1 on $X$ relative to the cover and with values in $G$; i.e. collections of smooth functions

  $$g_{i j} \;\colon\; U_i \underset{X}{\times} U_j \longrightarrow G$$

  satisfying on each triple intersection the cocycle condition

  $$g_{i j} g_{j k} = g_{i k}$$

• as morphisms the coboundaries between such cocycles $\{g_{i j}\}$ and $\{\tilde g_{j k}\}$, hence collections of smooth functions

  $$h_{i} \colon U_i \longrightarrow G$$

  such that on each intersection of charts

  $$g_{i j} h_j = h_i \tilde g_{i j} \,.$$

In particular the connected components of the Hom-groupoid is hence the Cech cohomology itself:

$$\pi_0 PreSmooth1Type(C(\{U_i\}), \mathbf{B}G) \simeq H^1_{Cech}(X,\{U_i\}; G) \,.$$

Proof

The components of a morphism $C(\{U_i\})\longrightarrow \mathbf{B}G$ look as follows

$$\left( \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) &&\to&& (x,k) } \right) \mapsto \left( \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow && \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\stackrel{g_{i k}(x)}{\to}&& \bullet } \right) \,.$$

This means that evaluating the morphism on the point $\ast = \mathbb{R}^0$, hence after forgetting the smooth structure, then it is a functor of groupoids as shown. Since $\mathbf{B}G(\mathbb{R}^0)$ has a single object, there is no choice for this functor on the level of objects. On morphisms it has to send each point $(x,i,j)$ in a double intersection of the cover to a group element $g_{i j}(x)\in G$. Hence the functor is given by functions

$$( g_{i j} )_{i,j \in I} \colon \underset{i,j}{\coprod} U_i \underset{X}{\times} U_j \longrightarrow G \,.$$

The Yoneda lemma gives that these must be smooth functions (as discussed at geometry of physics – smooth sets). Finally functoriality says that composition on the left needs to be taken to composition on the right, which means here that these functions satisfy

$$g_{i j}(x) g_{j k}(x) = g_{i k}(x)$$

for all $(x,i,j)$. This is precisely the data of a $G$-valued Cech cohomology cocycle.

Similarly, a homotopy between two such morphisms $h \colon g \Rightarrow \tilde g$ is in components a natural transformation of the form

$$\array{ (x,i) \\ \downarrow \\ (x,j) } \;\;\; \mapsto \;\;\; \array{ \bullet &\stackrel{h_i(x)}{\longrightarrow}& \bullet \\ {}^{\mathllap{g_{i j}(x)}}\downarrow && \downarrow^{\mathrlap{\tilde g_{i j}(x)}} \\ \bullet &\underset{h_j(x)}{\longrightarrow}& \bullet }$$

hence is given by smooth functions

$$(h_i)_{i \in I} \colon \underset{i}{\coprod} U_i \longrightarrow G$$

such that

$$h_i(x) \tilde g_{i j}(x) = g_{i j}(x) h_j(x) \,.$$

This is precisely the formula characterizing $(h_i)_{i \in I}$ as a Cech coboundary.

More along these lines is at geometry of physics – principal bundles.

Gluing

The “bootstrap”-definition of pre-smooth groupoids above works as intended, by prop. , it just needs to be restricted now to something a little less general. The issue is that while this definition consistently identifies a smooth-structure-to-be by what its possible smooth probes are, it does not enforce yet the consistent gluing of probes:

for $X$ a pre-smooth groupoid, then given a probe of it of the form $\sigma \colon \mathbb{R}^n \to X$ and given a covering $\{U_i \to \mathbb{R}^n\}$ of the probe space by other probe spaces $U_i \simeq \mathbb{R}^n$, then it should be possible to reconstruct $\sigma$ from knowing its restrictions $\sigma_{|U_i}\colon U_i \to X$ to these charts, and the information of how these are identified by gauge transformations on double overlaps.

Such a system of local data and of gauge identification on double overlaps is just what maps out of the Cech groupoid $C(\{U_i\})$, def. , encode. This is shown by prop. for the case that $X$ is of the form $(\mathbf{B}G)_\bullet$, example , but this is already the archetypical case.

In other words, the condition that smooth probes of $X$ by coordinate charts $\mathbb{R}^n$ glue along covers $\{U_i \to \mathbb{R}^n\}$ of these charts is the condition that the groupoid of smooth maps out of $\mathbb{R}^n$ itself

$$\mathbb{R}^n \longrightarrow X$$

is equivalent to the groupoid of maps out of the Cech groupoid of any cover

$$C(\{U_i\}) \longrightarrow X$$

and is so via the “restriction” map that takes the former and precomposes it with the canonical map $C(\{U_i\}) \to \mathbb{R}^n$.

$$\array{ C(\{U_i\}) &\longrightarrow& X \\ \downarrow & \nearrow \\ \mathbb{R}^n } \,.$$

Definition

A pre-smooth groupoid $X \in PreSmooth1Type$, def. , satisfies descent if for all $n \in \mathbb{N}$ and for all differentiably good open covers $\{U_i \to \mathbb{R}^n\}$ of the $n$-dimensional abstract coordinate chart, the functor

$$X(\mathbb{R}^n) \simeq PreSmooth1Type(\mathbb{R}^n, X) \longrightarrow PreSmooth1Type(C(\{U_i\}),X)$$

given by pre-composition with $C(\{U_i\}) \to \mathbb{R}^n$, is an equivalence of groupoids.

Remark

The condition in def. is called the stack condition, or the condition of descent, alluding to the fact that it says that $X$ “descends” down along the cover projection. So a smooth groupoid is a stack on the site CartSp. This is a higher analog of the sheaf condition (see the next example) and hence a more systematic terminology would be to say that such $X$ is a 2-sheaf or rather a (2,1)-sheaf (since it takes values in groupoids as opposed to in more general categories).

Proposition

Let $X \in PreSmooth0Type \hookrightarrow PreSmooth1Type$ be a pre-smooth groupoid which is really just a pre-smooth set, hence a presheaf on $CartSp$ that takes values in groupoids with only identity morphisms

$$X \colon CartSp^{op} \longrightarrow Set \hookrightarrow Grp \,.$$

Then $X$ is a smooth groupoid in the sense of def. precisely if it is a smooth set, hence precisely if, as a presheaf, it satisfies the sheaf condition.

Example

In particular, for $X \in SmoothMfd \hookrightarrow PreSmooth0Type \hookrightarrow PreSmooth1Type$ a smooth manifold, it satisfies descent as a pre-smooth groupoid.

Remark

There is an alternative formulation of the whole theory where instead of the site CartSp one uses the site SmoothMfd of all smooth manifolds. Everything discussed so far goes through verbatim for that site, too, but the descent condition in def. is a much stronger condition.

For instance the presheaves of the form $(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow} \ast)$ from example satisfy descent on $CartSp$, but not all $SmoothMfd$. Still, once we have defined the higher category of smooth groupoids, the definition wil be equivalent for both choices of sites.

The choice of the smaller site is the one that is easier to work with, and therefore we stick with that. In fact, most every example of a pre-smooth groupoid that one runs into satisfies descent on $CartSp$.

For example:

Proposition

For $G$ a Lie group, then the pre-smooth delooping groupoid $(\mathbf{B}G)_\bullet$ of example satisfies descent on $CartSp$, def. .

Proof

For $\{U_i \to \mathbb{R}^n\}$ a differentiably good open cover, then by prop. the groupoid $PreSmooth1Type(C(\{U_i\}),(\mathbf{B}G)_\bullet)$ is the groupoid of Cech 1-cocycles and coboundaries with coeffcients in $G$ on $\mathbb{R}^n$ relative to the cover. But this is equivalently the groupoid of $G$-principal bundles on $\mathbb{R}^n$. Now because the underlying topological space of $\mathbb{R}^n$ is contractible, all $G$-principal bundles on it are equivalent to the trivial one. But this is evidently represented by the image of point under the map

$$B (C^\infty(G)_{disc}) \simeq PreSmooth1Type(\mathbb{R}^n, (\mathbf{B}G)_\bullet) \longrightarrow PreSmooth1Type(C(\{U_i\}), (\mathbf{B}G)_\bullet) \,.$$

Therefore this is an essentially surjective functor of groupoids. Moreover, the automorphisms of the trivial $G$-principal bundles are precisely the smooth $G$-functions, hence this is also a fully faithful functor of groupoids. Accordingly it is an equivalence of groupoids.

The same argument however shows that on the larger site SmoothMfd this object does not satisfy descent. Put positively, this is the content of prop. . below.

Weak equivalences

While the morphisms of pre-smooth groupoids defined above correctly encode morphisms of smooth structures (by taking smooth probes compatibly to smooth probes), they are not sensitive enough yet to the required concept of equivalence. This is because smooth structure, being about existence of differentiation, is to be detected entirely locally, namely stalk-wise. If for instance $X$ is a smooth manifold, then its smooth structure is determined, around any of its points, by the smooth structure of an arbitrarily small open ball around that point.

Definition

For $n \in \mathbb{N}$, and $0 \lt r \lt 1 \in \mathbb{R}$ let

$$\mathbb{R}^n \simeq D^n_r \hookrightarrow \mathbb{R}^n$$

be the smooth function that regards the Cartesian space $\mathbb{R}^n$ as the standard $n$-disk of radius $r$ around the origin in $\mathbb{R}^n$.

For $X \in PreSmooth1Type$ we write

$$(D^n)^* X \coloneqq {\underset{\longrightarrow_r}{\lim}} X(D^n_r) \in Grpd$$

for the colimit (in the 1-category of groupoids) of the restrictions of its groupoids of plots along the inclusion of these open balls – the $n$-stalk of $X$. This extends to a functor

$$(D^n)^* \colon PreSmooth1Type \longrightarrow Grpd$$

This means that objects in $(D^n)^\ast X$ are equivalence classes of pairs $(r,x_r)$ where $0 \lt r \lt 1$ and where $x_r \in X(D^n_r)$, with two such pairs being equivalent $(r_1, x_{r_1})\sim (r_2, x_{r_2})$ precisely if there is $r_0 \lt r_1,r_2$ such that $x_{r_1}$ becomes equal to $x_{r_2}$ after restriction to $D^n_{r_0}$.

Definition

A morphism $f \colon X \longrightarrow Y$ of pre-smooth groupoids, def. , is called a local weak equivalence if for every $n \in \mathbb{N}$ the $n$-stalk, def. , is an equivalence of groupoids

$$((D^n)^* f) \colon (D^n)^* X \stackrel{}{\longrightarrow} (D^n)^* Y \,.$$

We write $X\stackrel{\simeq}{\longrightarrow} Y$ for local weak equivalences of pre-smooth groupoids. We will mostly just say weak equivalence for short.

Proposition

For $X$ a smooth manifold and $\{U_i \to X\}$ an open cover for it, then the canonical morphism from the corresponding Cech groupoid to $X$, def. , is a local weak equivalence in the sense of def. .

Proof

The $n$-stalk of the smooth manifold $X$ regarded as a presheaf is the set of equivalence classes of maps from open pointed $n$-disks into it, where two such are identified if they coincide on some small joint sub-disk of their domain. We may call this the set of germs of $X$ (but beware that this terminology is typically used for something a little bit more restrictive, namely for the case that $n$ is the dimension of $X$ and that all maps from the disks into $X$ are required to be embeddings).

On the other hand the $n$-stalk of $C(\{U_i\})$ is the groupoid whose set of objects is the set of germs, in this sense, of the disjoint union $\underset{i}{\coprod} U_i$, and whose set of morphisms is the set of germs of the disjoint union $\underset{i,j}{\coprod} U_i \underset{X}{\times} U_j$.

But now since the cover is by open subsets, it follows that for every $(x,i,j) \in \underset{i,j}{\coprod} U_i \underset{X}{\times} U_j$, then every germ of objects $[g]$ around $(x,i)$ has a representative $g$ that factors through this double intersection charts: $g \colon D^n_r \to U_i \underset{X}{\times} U_j \to \underset{i}{\coprod} U_u$. And similarly for $(x,j)$.

This means that the groupoid of $n$-stalks is a disjoint union of groupoids, one for each germ of $X$, all whose components are groupoids in which there is a unique morphism between any two objects, which are copies of this germ regarded as sitting in one of the charts of the cover. This means that each of these connected components is equivalent to the point.

Now the canonical cover projection sends each of these connected components to the germ that it corresponds to. Hence this is a an equivalence of groupoids.

Hypercovers

Definition

A morphism $p \colon Y \longrightarrow X$ of pre-smooth groupoids is called a split hypercover if

1. $Y$ is

   1. degreewise a coproduct of Cartesian spaces;

   2. such that the degenerate elements split off as a dijoint summand.

2. $p$ is a weak equivalence, def. .

Proposition

For $X$ a smooth manifold and $\{U_i \to X\}$ an open cover, then the canonical projection $C(\{U_i\}) \to X$ from the corresponding Cech groupoid, def. , is a split hypercover precisely if the cover is differentiably good.

Proof

For every cover the map is a weak equivalence, by prop. .

For the Cech groupoid the condition of cofibrancy in def. means that every non-empty finite intersection of patches is diffeomorphic to a Cartesian space, hence to an open ball. This is precisely the definition of differentialbly good open cover.

The $(2,1)$-category of smooth groupoids

The Grpd-enriched category of genuine smooth groupoids is that obtained from that of pre-smooth groupoids, def. by “universally turning local weak equivalences, def. , into actual homotopy equivalences”. This is stated formally by def. below, but for many applications in practice certain concrete presentations of what this means concretely are well sufficient, one of these we state below in prop. .

Definition

Write

$$Smooth1Type \coloneqq L_{lwe} PreSmooth1Type$$

for the (2,1)-category which is the simplicial localization of groupoid-valued presheaves at the local weak equivalences, def. .

An object $X \in Smooth1Type$ we call a smooth groupoid or smooth homotopy 1-type.

Proposition

Let $X,A \in PreSmooth1Type$ such that $A$ satisfies descent, def. . Let $Y \to X$ be a split hypercover of $X$, def. .

Then there is an equivalence of groupoids

$$Smooth1Type(X,A) \simeq PreSmooth1Type(Y,A)$$

between the hom-groupoid of smooth groupoids from $X$ to $A$, and that of pre-smooth groupoids, def. , from $Y$ to $A$.

Such statements follow with model structures on simplicial presheaves after embedding the present situation in the more general context of smooth infinity-groupoids. See there for more.

Remark

There is a canonical localization functor

$$PreSmooth1Type \longrightarrow L_{lwe} PreSmooth1Type = Smooth1Type \,.$$

which is really just the identity as a functor. Instead of doing anything to the objects, passing along this functor just means to change the definition of the hom-groupoids from the direct definition of def. to the localized definition.

When a pre-smooth groupoid is given by an internal groupoid $\mathcal{G}_\bullet$ in smooth sets via example , then we indicate its image under this functor by removing the subscript index, writing just $\mathcal{G}$. This reflects the fact that in $Smooth1Type$ it no longer makes sense to ask what the space of 0-cells and of 1-cells of an object is, as these are concepts not invariant under local weak equivalence.

In particular this means that we write $\mathbf{B}G$ for the image of $(\mathbf{B}G)_\bullet$ in $Smooth1Type$.

Proposition

Let $X$ be a smooth manifold, regarded as a smooth 0-groupoid, and let $G$ be a Lie group, with smooth delooping groupoid $\mathbf{B}G$ (example , remark ).

Then

$$Smooth1Type(X,\mathbf{B}G) \simeq G Bund(X)$$

is equivalently the groupoid of $G$-principal bundles on $X$.

Proof

By prop. the object $(\mathbf{B}G)_\bullet$ satisfies descent on CartSp. Choose $\{U_i \to X\}$ a differentiably good open cover. By prop. the correspoding Cech groupoid projection $C(\{U_i\}) \to X$ is a split hypercover, def. . Hence, by prop. , there is an equivalence of groupoids

$$Smooth1Type(X,\mathbf{B}G) \simeq PreSmooth1Type(C(\{U_i\}), (\mathbf{B}G)_\bullet) \,.$$

The groupoid on the right is, by prop. , the groupoid of Cech 1-cocycles and coboundaries with values in $G$ relative to a good open cover. This is equivalently the groupoid of $G$-principal bundles.

Remark

The content of prop. is in common jargon that: $\mathbf{B}G \in SmoothGrpd$ is the moduli stack of $G$-principal bundles“.

Smooth $\infty$-Groupoids

(…under construction…)

Simplicial presheaves

We need structures a bit richer than just bare ∞-groupoids. In generalization to Lie groupoids, we need ∞-Lie groupoids. A useful way to encode that an $\infty$-groupoid has extra structure modeled on geometric test objects that themselves form a category $C$ is to remember the rule which for each test space $U$ in $C$ produces the $\infty$-groupoid of $U$-parameterized families of $k$-morphisms in $K$. For instance for an ∞-Lie groupoid we could test with each Cartesian space $U = \mathbb{R}^n$ and find the $\infty$-groupoids $K(U)$ of smooth $n$-parameter families of $k$-morphisms in $K$.

This data of $U$-families arranges itself into a presheaf with values in Kan complexes

$$K : C^{op} \to KanCplx \hookrightarrow sSet$$

hence with values in simplicial sets. This is equivalently a simplicial presheaf of sets. The functor category $[C^{op}, sSet]$ on the opposite category of the category of test objects $C$ serves as a model for the (∞,1)-category of $\infty$-groupoids with $C$-structure.

While there are no higher morphisms in this functor 1-category that could for instance witness that two $\infty$-groupoids are not isomorphic, but still equivalent, it turns out that all one needs in order to reconstruct all these higher morphisms (up to equivalence!) is just the information of which morphisms of simplicial presheaves would become invertible if we were keeping track of higher morphism. These would-be invertible morphisms are called weak equivalences and denoted $K_1 \stackrel{\simeq}{\to} K_2$.

For common choices of $C$ there is a well-understood way to define the weak equivalences $W \subset mor [C^{op}, sSet]$, and equipped with this information the category of simplicial presheaves becomes a category with weak equivalences . There is a well-developed but somewhat intricate theory of how exactly this 1-cagtegorical data models the full higher category of structured groupoids that we are after, but for our purposes we essentially only need to work inside the category of fibrant objects of a model category structure on simplicial presheaves, which in practice amounts to the fact that we use the following three basic constructions:

1. ∞-anafunctors – A morphisms $X \to Y$ between $\infty$-groupoids with $C$-structure is not just a morphism $X\to Y$ in $[C^{op}, sSet]$, but is a span of such ordinary morphisms

   $$\array{ \hat X &\to& Y \\ \downarrow^{\mathrlap{\simeq}} \\ X }$$

   where the left leg is a weak equivalence. This is sometimes called an $\infty$-anafunctor from $X$ to $Y$.

2. homotopy pullback – For $A \to B \stackrel{p}{\leftarrow} C$ a diagram, the (∞,1)-pullback of it is the ordinary pullback in $[C^{op}, sSet]$ of a replacement diagram $A \to B \stackrel{\hat p}{\leftarrow} \hat C$, where $\hat p$ is a good replacement of $p$ in the sense of the following factorization lemma.

3. factorization lemma – For $p : C \to B$ a morphism in $[C^{op}, sSet]$, a good replacement $\hat p : \hat C \to B$ is given by the composite vertical morphism in the ordinary pullback diagram

   $$\array{ \hat C &\to& C \\ \downarrow && \downarrow^{\mathrlap{p}} \\ B^{\Delta[1]} &\to& B \\ \downarrow \\ B } \,,$$

where $B^{\Delta[1]}$ is the path object of $B$: the simplicial presheaf that is over each $U \in C$ the simplicial path space $B(U)^{\Delta[1]}$.

• simplicial presheaf

• model structure on simplicial presheaves

Good covers and split hypercovers

• Cech nerve

• good open cover

• split hypercover

Proposition

For $X$ a smooth manifold let $\{U_i \hookrightarrow X\}_i$ be an open cover. This is a differentiably good open cover, def. , precisely if the corresponding Cech nerve $C(\{U_i\}) \to X$ in $sPh(CartSp)_{proj,loc}$ is a split hypercover.

Examples

Parallel transport

We give a more intrinsic characterization of differential 1-forms.

Definition

A smooth path with sitting instants in $\mathbb{R}^k$ is a smooth function $\gamma \colon \mathbb{R} \to \mathbb{R}^k$ such

1. the value of $\gamma$ converges to two points $x = \underset{t \to -\infty}{\lim} \gamma \in \mathbb{R}^k$ and $y = \underset{t \to \infty}{\lim} \gamma \in \mathbb{R}^k$;

2. the value of all derivatives of $\gamma$ converges to 0 at $t \to \pm \infty$.

A localized diffeomorphism $\phi \colon \mathbb{R} \to \mathbb{R}$ is a diffeomorphism such that there is an an open ball in $\mathbb{R}$ outside of which $\phi$ restricts to the identity.

Write

$$[\mathbb{R}, \mathbb{R}^k] // Diff(\mathbb{R}) \in Smooth0Type$$

for the smooth quotient space of smooth paths modulo localized diffeomorphism.

Definition

Say that a smooth functor $\mathbf{P}_1(\mathbb{R}^k) \to \mathbf{B}\mathbb{R}$ from the smooth path groupoid of $\mathbb{R}^k$ is

• a homomorphism of smooth space

  $$S \colon [\mathbb{R}, \mathbb{R}^k] \to \mathbb{R}$$

• such that

  1. composition of paths $\gamma_1, \gamma_2$ is sent to addition in $\mathbb{R}$

     $$S(\gamma_2\circ \gamma_1) = S(\gamma_1) + S(\gamma_2) \,.$$

  2. the constant path is sent to 0.

Proposition

There is an equivalence

$$[\mathbf{P}_1(-), \mathbf{B}\mathbb{R}] \stackrel{\overset{\int_{(-)}(-)}{\longleftarrow}}{\underset{}{\longrightarrow}} DK(C^\infty(-) \stackrel{\mathbf{d}}{\to} \Omega^1(-) \,.$$

(SchreiberWaldorf).

Lie integration of $\infty$-Lie algebroids

In classical Lie theory, a Lie group may be (re-)constructed from infinitesimal data, namely from its Lie algebra by a process of Lie integration. We now discuss a procedure that constructs smooth homotopy types/smooth ∞-groupoids from Lie integration of infinitesimal data, namely from higher Lie algebras, called L-∞ algebras and more generally from L-∞ algebroids.

Lie algebras via their Chevalley-Eilenberg algebras

The quickest way to introduce L-∞ algebras of finite type is via theiry formally dual Chevalley-Eilenberg dg-algebras. Here we recall how this works and then below we use this to define L-∞ algebroids of finite type.

Definition

The Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of a finite dimensional Lie algebra $\mathfrak{g}$ is the semifree graded-commutative dg-algebra whose underlying graded algebra is the Grassmann algebra

$$\wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots$$

(with the $n$th skew-symmetrized power in degree $n$)

and whose differential $d$ (of degree +1) is on $\mathfrak{g}^*$ the dual of the Lie bracket

$$d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*$$

extended uniquely as a graded derivation on $\wedge^\bullet \mathfrak{g}^*$.

That this differential indeed squares to 0, $d \circ d = 0$, is precisely the fact that the Lie bracket satisfies the Jacobi identity.

Remark

If in the situation of prop. we choose a dual basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{C^a{}_{b c}\}$ be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is

$$d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,,$$

where here and in the following a sum over repeated indices is implicit.

Proposition

The construction of Chevalley-Eilenberg algebras in def. yields a fully faithful functor

$$CE(-) \colon LieAlg \longrightarrow dgAlg^{op}$$

embedding Lie algebras into formal duals of differential graded algebras. Its image consists of precisely of the semifree dg-algebras, those whose underlying graded algebra (forgetting the differential) is a Grassmann algebra generated on a vector space.

Definition

Given a Lie algebra $\mathfrak{g}$, its Weil algebra $W(\mathfrak{g})$ is the semi-free dga whose underlying graded-commutative algebra is the exterior algebra

$$\wedge^\bullet (\mathfrak{g}^* \oplus \mathfrak{g}^*[1])$$

on $\mathfrak{g}^*$ and a shifted copy of $\mathfrak{g}^*$, and whose differential is the sum

$$d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d}$$

of two graded derivations of degree +1 defined by

• $\mathbf{d}$ acts by degree shift $\mathfrak{g}^* \to \mathfrak{g}^*[1]$ on elements in $\mathfrak{g}^*$ and by 0 on elements of $\mathfrak{g}^*[1]$;

• $d_{CE(\mathfrak{g})}$ acts on unshifted elements in $\mathfrak{g}^*$ as the differential of the Chevalley-Eilenberg algebra of $\mathfrak{g}$ and is extended uniquely to shifted generators by graded-commutattivity

  $$[d_{CE(\mathfrak{g}}, \mathbf{d}] = 0$$

  with $\mathbf{d}$:

  $$d_{CE(\mathfrak{g})} \mathbf{d} \omega := - \mathbf{d} d_{CE(\mathfrak{g})} \omega$$

  for all $\omega \in \wedge^1 \mathfrak{g}^*$.

Proposition

Given a Lie algebra $\mathfrak{g}$, then a Lie algebra valued differential form on, say, a Cartesian space $\mathbb{R}^n$, is equivalently a dg-algebra homomorphims

$$\Omega^\bullet(\mathbb{R}^p) \longleftarrow W(\mathfrak{g}) \colon A \,,$$

hence there is a natural bijection

$$\Omega^1(\mathbb{R}^p, \mathfrak{g}) \simeq Hom_{dgAlg}(W(\mathfrak{g}), \Omega^\bullet(\mathbb{R}^p)) \,.$$

The form $A$ is flat in that its curvature differential 2-form $F_A$ vanishes, precisely if this morphism factors through the CE-algebra.

Remark

With a choice of basis as in remark , then the content of prop. is seen in components as follows:

a dg-algebra homomorphism is first of all a homomorphism of graded algebras, and since the domain $W(\mathfrak{g})$ is free as a graded algebra, such is entirely determined by what it does to the generators

$$\array{ t^a, &\mapsto& A^a & \in \Omega^1(\mathbb{R}^n) \\ r^a &\mapsto& F^a & \in \Omega^2(\mathbb{R}^n) } \,.$$

But being a dg-algebra homomorphism, this assignment needs to respect the differentials on both sides. For the original generators this gives

$$\array{ t^a &\mapsto&&& A^a \\ \downarrow^{\mathrlap{d_{W(\mathfrak{g})}}} &&&& \downarrow^{\mathrlap{\mathbf{d}_{dR}}} \\ - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a &\mapsto& (- \frac{1}{2} C^a{}_{b c} A^b \wedge A^c + F^a) &=& \mathbf{d}_{dR} A^a } \,.$$

With this satisfied, then, by the very nature of the Weil algebra, the differential is automatically respected also on the shifted generators. This statement is the Bianchi identity.

$L_\infty$-Algebroids via their Chevalley-Eilenberg algebras

Definition

The category of L-∞ algebras of finite type is the full subcategory of that of the opposite category of dg-algebras

$$CE(-) \;\colon\; L_\infty Alg \hookrightarrow dgAlg^{op}$$

$$(\mathfrak{g}, [-],[-,-], [-,-,-],\cdots) \mapsto (\wedge^\bullet \mathfrak{g}^\ast , d_{CE} = [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots)$$

on those whose underlying graded algebra is the Grassmann algebra $\wedge^\bullet \mathfrak{g}$ on a $\mathbb{N}$-graded vector space $\mathfrak{g}$ of finite type.

More generally, the category L-∞ algebroids is the full subcategory

$$CE(-) \;\colon\; L_\infty Algd \hookrightarrow dgAlg^{op}$$

on those dg-algebras (over $\mathbb{R}$)

$$(\mathfrak{a}_X, [-],[-,-], [-,-,-],\cdots) \mapsto (\wedge^\bullet_{C^\infty(X)} \mathfrak{a}^\ast , d_{CE} = [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots)$$

whose underlying graded algebra is the Grassmann algebra over $C^\infty(X)$ of an $\mathbb{N}$-graded projective module $\mathfrak{a}$ of finite type over $C^\infty(X)$ (i.e. by the Serre-Swan theorem, of sections of an $\mathbb{N}$-graded vector bundle).

$\infty$-Lie algebroid valued differential forms

For $X$ a smooth manifold and $\mathfrak{g}$ an ∞-Lie algebra or more generally an ∞-Lie algebroid, a ∞-Lie algebroid valued differential forms on $X$ is a morphism of dg-algebras

$$\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$$

from the Weil algebra of $\mathfrak{g}$ to the de Rham complex of $X$. Dually this is a morphism of ∞-Lie algebroids

$$A : T X \to inn(\mathfrak{g})$$

from the tangent Lie algebroid to the inner automorphism ∞-Lie algebra.

Its curvature is the composite of morphisms of graded vector spaces

$$\Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \wedge^1 \mathfrak{g}^* : F_{A} \,.$$

Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

$$(F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right)$$

in which case we call $A$ flat.

Higher dimensional paths in an $\infty$-Lie algebroid

Example

For $X$ a smooth manifold (possibly with boundary and with corners) then its tangent Lie algebroid $T X$ is the one whose Chevalley-Eilenberg algebra is the de Rham complex

$$CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.$$

Definition

For $k \in \mathbb{N}$ write $\Delta^k$ for the standard $k$-simplex regarded as a smooth manifold (with boundary and with corners).

A $k$-path in the $\infty$-Lie algebroid $\mathfrak{a}$ is a morphism of $\infty$-Lie algebroids of the form

$$\Sigma \; \coloneqq \; T \Delta^k \longrightarrow \mathfrak{a}$$

from the tangent Lie algebroid $T \Delta^k$ of the standard smooth $k$-simplex to $\mathfrak{a}$. Dually this is equivalently a homomorphism of dg-algebras

$$\Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^*$$

from the Chevalley-Eilenberg algebra of $\mathfrak{a}$ to the de Rham complex of $\Delta^d$.

See also at differential forms on simplices.

Remark

A $k$-path in $\mathfrak{a}$, def. , is equivalently

1. a flat $\mathfrak{a}$-valued differential form on $\Delta^k$;

2. a Maurer-Cartan element in $\Omega^\bullet(\Delta^k)\otimes \mathfrak{a}$.

The Lie integration of $\mathfrak{a}$ is essentially the simplicial object whose $k$-cells are the $d$-paths in $\mathfrak{a}$. However, in order for this to be well-behaved, it is possible and useful to restrict to $d$-paths that are sufficiently well-behaved towards the boundary of the simplex:

Definition

Regard the smooth simplex $\Delta^k$ as embedded into the Cartesian space $\mathbb{R}^{k+1}$ in the standard way, and equip $\Delta^k$ with the metric space structure induced this way.

A smooth differential form $\omega$ on $\Delta^k$ is said to have sitting instants along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an open neighbourhood $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face.

More generally, for any $U \in$ CartSp a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-neighbourhoods of faces.

Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra.

We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of vertical differential forms with respect to the projection $p : U \times \Delta^k \to U$, hence the coequalizer

$$\Omega^\bullet(U) \stackrel{\stackrel{p^*}{\longrightarrow}}{\underset{0}{\longrightarrow}} \Omega^\bullet_{si}(U \times \Delta^k) \to \Omega^\bullet_{si, vert}(U \times \Delta^k) \,.$$

Remark

The dimension of the normal direction to a face depends on the dimension of the face: there is one perpendicular direction to a codimension-1 face, and $k$ perpendicular directions to a vertex.

Examples

• A smooth 0-form (a smooth function) has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it is constant.

  A smooth function $f : U \times \Delta^1 \to \mathbb{R}$ is in $\Omega^0_{\mathrm{vert}}(U \times \Delta^1)$ if there is $0 \lt \epsilon \in \mathbb{R}$ such that for each $u \in U$ the function $f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R}$ is constant on $[0,\epsilon) \coprod (1-\epsilon,1)$.

• A smooth 1-form has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it vanishes.

• Let $X$ be a smooth manifold, $\omega \in \Omega^\bullet(X)$ be a smooth differential form. Let

  $$\phi \colon \Delta^k \to X$$

  be a smooth function that has sitting instants as a function: towards any $j$-face of $\Delta^k$ it eventually becomes perpendicularly constant.

  Then the pullback form $\phi^* \omega \in \Omega^\bullet(\Delta^k)$ is a form with sitting instants.

Remark

The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of $\Delta^k$. Notably for $\omega_j \in \Omega^\bullet(\Delta^{k-1})$ a collection of forms with sitting instants on the $(k-1)$-cells of a horn $\Lambda^k_i$ that coincide on adjacent boundaries, and for

$$p \colon \Delta^k \to \Lambda^{k-1}_i$$

a standard piecewise smooth retract, the pullbacks

$$p^* \omega_i$$

glue to a single smooth differential form (with sitting instants) on $\Delta^k$.

Remark

That $\omega \in \Omega^\bullet(\Delta^k)$ having sitting instants does not imply that there is a neighbourhood of the boundary of $\Delta^k$ on which $\omega$ is entirely constant. It is important for the following constructions that in the vicinity of the boundary $\omega$ is allowed to vary parallel to the boundary, just not perpendicular to it.

Higher Lie integration

Lie integration is a process that assigns to a Lie algebra $\mathfrak{g}$ – or more generally to an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by $\mathfrak{g}$. It is essentially the reverse operation to Lie differentiation, except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.

Classically, Lie integration of Lie algebras is part of Lie's three theorems, which in particular finds an unique (up to isomorphism) simply connected Lie group integrating a given finite-dimensional Lie algebra.

One may observe that the simply connected Lie group integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of equivalence classes of Lie algebra valued 1-forms on the interval where two such are identified if they are interpolated by a flat Lie-algebra valued 1-form on the disk. (Duistermaat-Kolk 00, section 1.14, see also the example below).

This path method of Lie integration stands out as having natural generalizations to higher Lie theory (Ševera 01).

In its evident generalization from Lie algebra valued differential forms to Lie algebroid valued differential forms this provides a means for Lie integration of Lie algebroids (e.g. Crainic-Fernandes 01).

In another direction, one may observe that L-∞ algebras are formally dually incarnated by their Chevalley-Eilenberg dg-algebras, and that under this identification the evident generalization of the path method to L-∞ algebra valued differential forms is essentially the Sullivan construction, known from rational homotopy theory, applied to these dg-algebras (Hinich 97, Getzler 04). Or rather, the bare such construction gives the geometrically discrete ∞-group underlying what should be the Lie integration to a smooth ∞-group. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the ∞-Lie algebroid valued differential forms (Henriques 08, Roytenberg 09, FSS 12).

Both these directions may be combined via the evident concept of ∞-Lie algebroid valued differential forms to yield a Lie integration of ∞-Lie algebroids to smooth ∞-groupoids.

While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, Lie's three theorems. But people are working on it.

Let $\mathfrak{a}$ be an ∞-Lie algebroid (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra).

For $\mathfrak{a}$ an $\infty$-Lie algebroid, the $d$-paths in $\mathfrak{a}$ naturally form a simplicial set as $d$ varies:

$$\begin{aligned} \exp(\mathfrak{a})_{bare} & \coloneqq \left( \cdots Hom_{\infty LieAlgd}(T \Delta^2, \mathfrak{a}) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{\infty LieAlgd}(T \Delta^1, \mathfrak{a}) \stackrel{\longrightarrow}{\longrightarrow} Hom_{\infty LieAlgd}(T \Delta^0, \mathfrak{g}) \right) \\ & = ( \cdots Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2)) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1)) \stackrel{\longrightarrow}{\longrightarrow} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0)) ) \end{aligned} \,.$$

(We are indicating only the face maps, not the degeneracy maps, just for notational simplicity).

If here instead of smooth differential forms one uses polynomial differential forms then this is precisely the Sullivan construction of rational homotopy theory applied to $CE(\mathfrak{a})$. We next realize smooth structure on this and hence realize this as an object in higher Lie theory.

We now discuss Lie integration of $\infty$-Lie algebroids to smooth ∞-groupoids, presented by the model structure on simplicial presheaves $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ over the site CartSp${}_{smooth}$.

For the following definition recall the presentation of smooth ∞-groupoids by the model structure on simplicial presheaves over the site CartSp${}_{smooth}$.

Definition

For $\mathfrak{a}$ an L-∞ algebra of finite type with Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ define the simplicial presheaf

$$\exp(\mathfrak{a}) \;\colon\; CartSp_{smooth}^{op} \to sSet$$

by

$$\exp(\mathfrak{a}) \;\colon\; (U,[k]) \mapsto Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(U \times \Delta^k)_{si,vert}) \,,$$

for all $U \in$ CartSp and $[k] \in \Delta$.

Remark

Compared to the integration to discrete ∞-groupoids above this definition knows about $U$-parametrized smooth families of $k$-paths in $\mathfrak{g}$.

The underlying discrete ∞-groupoid is recovered as that of the $\mathbb{R}^0 = *$-parameterized family:

$$\exp(\mathfrak{a}) \colon \mathbb{R}^0 \mapsto \exp(\mathfrak{a})_{disc} \,.$$

Proposition

The objects $\exp(\mathfrak{g})$ are indeed Kan complexes over each $U \in$ CartSp.

Proof

Observe that the standard continuous horn retracts $f : \Delta^k \to \Lambda^k_i$ are smooth away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$.

For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function

$$f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,.$$

The pullback $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$.

The resulting assignment

$$(CE(\mathfrak{g}) \stackrel{A}{\longrightarrow} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i)) \mapsto (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n))$$

provides fillers for all horns over all $U \in$ CartSp.

Definition

Write $\mathbf{cosk}_{n+1} \exp(a)$ for the simplicial presheaf obtained by postcomposing $\exp(\mathfrak{a}) : CartSp^{op} \to sSet$ with the $(n+1)$-coskeleton functor $\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\longrightarrow} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet$.

Interating Lie algebras to Lie groups

Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) Lie algebra. Standard Lie theory (see Lie's three theorems) provides a simply connected Lie group $G$ integrating $\mathfrak{g}$.

With $G$ regarded as a smooth ∞-group write $\mathbf{B}G \in$ Smooth∞Grpd for its delooping. The standard presentation of this on $[CartSp_{smooth}^{op}, sSet]$ is by the simplicial presheaf

$$\mathbf{B}G_c : U \mapsto N(C^\infty(U,G) \stackrel{\to}{\to} *) * \,.$$

See Cohesive ∞-groups – Lie groups for details.

Proposition

The operation of parallel transport $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$)

$$P \exp(\int - ) : \mathbf{cosk}_3 \exp(\mathfrak{g}) \simeq \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c \,.$$

This follows from the Steenrod-Wockel approximation theorem and the following observation.

Lemma

For $X$ a simply connected smooth manifold and $x_0 \in X$ a basepoint, there is a canonical bijection

$$\Omega^1_{flat}(X,\mathfrak{g}) \simeq C^\infty_*(X,G)$$

between the set of Lie-algebra valued 1-forms on $X$ whose curvature 2-form vanishes, and the set of smooth functions $X\to G$ that take $x_0$ to the neutral element $e \in G$.

Proof

The bijection is given as follows. For $A \in \Omega^1_{flat}(X,\mathfrak{g})$ a flat 1-form, the corresponding function $f_A : X \to G$ sends $x \in X$ to the parallel transport along any path $x_0 \to x$ from the base point to $x$

$$f_A : x \mapsto tra_A(x_0 \to x) \,.$$

Because of the assumption that the curvature 2-form of $A$ vanishes and the assumption that $X$ is simply connected, this assignment is independent of the choice of path.

Conversely, for every such function $f : X \to G$ we recover $A$ as the pullback of the Maurer-Cartan form on $G$

$$A = f^* \theta \,.$$

From this we obtain

Proof of the proposition

The $\infty$-groupoid $\mathbf{cosk}_2 \exp(\mathfrak{g})$ is equivalent to the groupoid with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths $\Delta^1 \to G$ (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy $D^2 \to G$ (with sitting instant) between them.

Since $G$ is simply connected, these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with $G$.

Remark

We do not need to fall back to classical Lie theory to obtain $G$ in the above argument. A detailed discussion of how to find $G$ with its group structure and smooth structure from $d$-paths in $\mathfrak{g}$ is in (Crainic).

Integrating to line/circle Lie $n$-groups

Definition

For $n \in \mathbb{N}, n \geq 1$ write $b^{n-1} \mathbb{R}$ for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree $n$ and vanishing differential. We may call this the line Lie $n$-algebra.

Write $\mathbf{B}^{n} \mathbb{R}$ for the smooth line (n+1)-group.

Observation

The discrete ∞-groupoid underlying $\exp(b^{n-1} \mathbb{R})$ is given by the Kan complex that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-simplex

$$\exp(b^{n-1} \mathbb{R})_{disc} : [k] \mapsto \Omega^n_{si,cl}(\Delta^k)$$

Proposition

The $\infty$-Lie integration of $b^{n-1} \mathbb{R}$ is the line Lie n-group $\mathbf{B}^{n} \mathbb{R}$.

Moreover, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the Dold-Kan correspondence by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the fiber integration of differential $n$-forms over the $n$-simplex:

$$\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \stackrel{\simeq}{\to} \mathbf{B}^{n} \mathbb{R}_{chn} \,.$$

Proof

First we observe that the map

$$\int_{\Delta^\bullet} : (\omega \in \Omega^n_{si,vert,cl}(U \times \Delta^k)) \mapsto \int_{\Delta^k} \omega \in C^\infty(U, \mathbb{R})$$

is a morphism of simplicial presheaves $\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ on CartSp${}_{smooth}$. Since it goes between presheaves of abelian simplicial groups by the Dold-Kan correspondence it is sufficient to check that we have a morphism of chain complexes of presheaves on the corresponding normalized chain complexes.

The only nontrivial degree to check is degree $n$. Let $\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1})$. The differential of the normalized chains complex sends this to the signed sum of its restrictions to the $n$-faces of the $(n+1)$-simplex. Followed by the integral over $\Delta^n$ this is the piecewise integral of $\lambda$ over the boundary of the $n$-simplex. Since $\lambda$ has sitting instants, there is $0 \lt \epsilon \in \mathbb{R}$ such that there are no contributions to this integral in an $\epsilon$-neighbourhood of the $(n-1)$-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the $(n+1)$-simplex, as indicated in the following diagram

Since $\lambda$ is a closed form on the $n$-simplex, this surface integral vanishes, by the Stokes theorem. Hence $\int_{\Delta^\bullet}$ is indeed a chain map.

It remains to show that $\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ is an isomorphism on all the simplicial homotopys group over each $U \in CartSp$. This amounts to the statement that

1. a smooth family of closed $n$-forms with sitting instants on the boundary of $\Delta^{n+1}$ may be extended to a smooth family of closed forms with sitting instants on $\Delta^{n+1}$ precisely if their smooth family of integrals over the boundary vanishes;

2. Any smooth family of closed $n \lt k$-forms with sitting instants on the boundary of $\Delta^{k+1}$ may be extended to a smooth family of closed $n$-forms with sitting instants on $\Delta^{k+1}$.

To demonstrate this, we want to work with forms on the $(k+1)$-ball instead of the $(k+1)$-simplex. To achieve this, choose again $0 \lt \epsilon \in \mathbb{R}$ and construct the diffeomorphic image of $S^k \times [1,1-\epsilon]$ inside the $(k+1)$-simplex as indicated in the above diagram: outside an $\epsilon$-neighbourhood of the corners the image is a rectangular $\epsilon$-thickening of the faces of the simplex. Inside the $\epsilon$-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this $\epsilon$-thickening of the smoothed boundary of the simplex to $S^k \times [1-\epsilon,1]$ extends to a smooth function from the $(k+1)$-simplex to the $(k+1)$-ball.

By choosing $\epsilon$ smaller than each of the sitting instants of the given $n$-form on $\partial \Delta^{k+1}$, we have that this $n$-form vanishes on the $\epsilon$-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the $(k+1)$-ball.

It is now sufficient to show: a smooth family of smooth $n$-forms $\omega \in \Omega^n_{vert,cl}(U \times S^k)$ extends to a smooth family of closed $n$-forms $\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1})$ that is radially constant in a neighbourhood of the boundary for all $n \lt k$ and for $k = n$ precisely if its smooth family of integrals vanishes, $\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R})$.

Notice that over the point this is a direct consequence of the de Rham theorem: an $n$-form $\omega$ on $S^k$ is exact precisely if $n \lt k$ or if $n = k$ and its integral vanishes. In that case there is an $(n-1)$-form $A$ with $\omega = d A$. Choosing any smoothing function $f : [0,1] \to [0,1]$ (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an $n$-form $f \wedge A$ on $(0,1] \times S^k$, vertically constant in a neighbourhood of the ends of the interval, equal to $A$ at the top and vanishing at the bottom. Pushed forward along the canonical $(0,1] \times S^k \to D^{k+1}$ this defines a form on the $(k+1)$-ball, that we denote by the same symbol $f \wedge A$. Then the form $\hat \omega := d (f \wedge A)$ solves the problem.

To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the $(n-1)$-form $A$ in a way depending smoothly on the the $n$-form $\omega$.

One way of achieving this is using Hodge theory. Fix a Riemannian metric on $S^n$, and let $\Delta$ be the corresponding Laplace operator, and $\pi$ the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator $\pi$, seen as an operator from the de Rham complex to itself, is a cochain map homotopic to the identity, via an explicit homotopy $P := d^* G$ expressed in terms of the adjoint $d^*$ of the de Rham differential and of the Green operator $G$ of $\Delta$. Since the $k$-form $\omega$ is exact its projection on harmonic forms vanishes. Therefore

$$\begin{aligned} \omega & = (Id-\pi)\omega \\ & = d (P\omega)+P (d\omega) \\ & = d (P\omega). \end{aligned}$$

Hence $A := P\omega$ is a solution of the differential equation $d A=\omega$ depending smoothly on $\omega$.

Integrating the string Lie 2-algebra to the string Lie 2-group

Let $\mathfrak{string} = \mathfrak{g}_\mu$ be the string Lie 2-algebra.

Then $\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu)$ is equivalent to the 2-groupoid $\mathbf{B}String$

• with a single object;

• whose morphisms are based paths in $G$;

• whose 2-morphisms are equivalence class of pairs $(\Sigma,c)$, where

  • $\Sigma : D^2_* \to G$ is a smooth based map (where we use a homeomorphism $D^2 \simeq \Delta^2$ which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of $D^2_*$ is the 0-vertex of $\Delta^2$)

  • and $c \in U(1)$, and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball $\phi : D^3 \to G$ filling them the labels $c_1, c_2 \in U(1)$ differ by the integral $\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z}$,,

where $\theta$ is the Maurer-Cartan form, $\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle$ the 3-form obtained by plugging it into the cocycle.

This is the string Lie 2-group. It’s construction in terms of integration by paths is due to (Henriques)