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geometry of physics -- smooth homotopy types

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

previous chapters: smooth sets, homotopy types

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Context

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theory (physics), model (physics)

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Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

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Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Smooth homotopy types

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 XX, which we will write as a mapping space [X,BG conn][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 1, 2:*[X,BG conn]\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 :*[X,BG conn]\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, = 0\nabla = \nabla_0, then this group of auto-gauge equivalences is the gauge group of the given gauge theory:

GAut( 0){ 0 0}. 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 λ 1,λ 2: 1 2\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 ρ:λ 1λ 2\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 nn-fold gauge-of-gauge transformations are actually equal, all we have is, possibly, an (n+1)(n+1)-fold gauge transformation going between them which exhibits their gauge equivalence, this being so for all 0<n<0 \lt n \lt \infty. One therefore says that gauge fields in general form an ∞-groupoid whose n-morphisms are nn-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 1, 2:[X,BG conn]\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 λ\lambda is itself a term, of identity type, written just as before, λ:( 1 2)\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 ρ\rho that two equivalences λ 1,λ 2:( 1 2)\lambda_1, \lambda_2 \colon (\nabla_1 \simeq \nabla_2) are equivalent, and hence a term ρ:(λ 1λ 2)\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.

Contents:
  1. Model layer

    This introduces groupoids, then smooth groupoids and eventually ∞-groupoids modeled as Kan complexes and then smooth ∞-groupoids.

  2. Semantics layer

    This introduces the general abstract framework for smooth ∞-groupoids which is cohesion of (∞,1)-toposes.

  3. Syntactic layer

    This introduces the internal language of cohesive (∞,1)-toposes, called cohesive homotopy type theory.

Remark The reader who is after elementary exposition may want to skip over this discussion of homotopy theory on first reading to the next chapterGroups_ and only come back here for reference as need arises.

Model Layer

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 4\mathbb{R}^4 is given by a differential 1-form AΩ 1( 4)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)(x i) i=1 4=(t, x^1, x^2, x^3) \coloneqq (x^i)_{i = 1}^4 = on 4\mathbb{R}^4)

F =dA =E 1dx 1dt+E 1dx 1dt+E 1dx 1dt+ +B 1dx 2dx 3+B 2dx 3dx 1+B 3dx 1dx 2 \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 iC ( 4)) i=1 3(E_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3 and magnetic field strength (B iC ( 4)) i=1 3(B_i \in C^\infty(\mathbb{R}^4))_{i = 1}^3 given that is given by the de Rham differential of AA:

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

After Maxwell it was thought that FΩ 2( 4)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 AA itself that reflects the configuration of the electromagnetic field: in the presence of magnetic flux or other topoligical constraints, there can be different A,AA, A' with the same F=dA=dAF = \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 XX is finer than Ω cl 2(X)\mathbf{\Omega}^2_{cl}(X) but coarser than Ω 1(X)\mathbf{\Omega}^1(X). And it is not quite a smooth space itself, but a smooth groupoid:

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

A=A+dλ A' = A + \mathbf{d}\lambda

represent different but equivalent field configurations. One says that λ\lambda induces a gauge transformation from AA to AA'. We write λ:AA\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 λ\lambda on smooth 11-forms AA,

Definition

For any UU \in CartSp, Write Ω vert 1(X×U)\Omega^1_{vert}(X \times U) for the set of differential 1-forms on X×UX \times U with no components along UU, and write C (X×U,U(1))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

ρ U:Ω vert 1(X×U)×C (X×U,U(1))Ω vert 1(X×U) \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

ρ U:(A,λ)A+d Xλ. \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 Ω vert 1(X×U)/C (X×U,U(1))\Omega^1_{vert}(X\times U)/C^\infty(X \times U, U(1)). This set contains the gauge equivalence classes of UU-parameterized collections of electromagnetic gauge fields on XX. 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 λ\lambda a gauge transformation from A 1A_1 to A 2A_2, we should have an equivalence λ:A 1A 2\lambda \colon A_1 \stackrel{\simeq}{\to} A_2.

Definition

The action groupoid

Ω vert 1(X×U)//C (X×U,U(1))(Ω vert 1(X×U)×C (X×U,U(1))s=p 1t=ρΩ vert 1(X×U)) \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 UU-parameterized collections of gauge potentials AΩ vert 1(X×U)A \in \Omega^1_{vert}(X \times U);

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

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

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

Proposition

Let A 00Ω vert 1(X×U)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. 2 is the group of circle-group valued functions on UU:

π 1(Ω vert 1(X×U)//C (X×U,U(1)),A 0)=Aut(A 0)C (U,U(1)). \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 0A_0 is given by a function λC (X×U,U(1))\lambda \in C^\infty(X \times U, U(1)) such that A 0=A 0+d XλA_0 = A_0 + \mathbf{d}_X \lambda. This is the case precisely if d Xλ=0\mathbf{d}_X \lambda = 0, hence if λ\lambda is contant along XX. But a function on X×UX \times U which is constant along XX is canonically identified with a function on just UU.

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

Proposition

There is a homomorphism of smooth spaces (def. \ref{HomomorphismOfSmoothSpaces})

ρ:Ω 1(X)×C (X)Ω 1(X) \rho \colon \mathbf{\Omega}^1(X)\times \mathbf{C}^\infty(X) \to \mathbf{\Omega}^1(X)

from the product smooth space, def. \ref{ProductOfSmoothSpaces}, of the smooth moduli spaces of 1-forms and 0-forms on XX, def. \ref{SmoothSpaceOfFormsOnSmoothSpace}, to that of smooth functions, def. \ref{SmoothSpaceOfSmoothFunctions}, whose component over UU \in CartSp is the action

ρ U:(A,λ)A+dλ \rho_U \colon (A,\lambda) \mapsto A + \mathbf{d}\lambda

of def. 1.

We then also want to consider a smooth action groupoid.

Definition

Write

Ω 1(X)//C (X,U(1)):CartSp opGrpd \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 UCartSpU \in CartSp the discrete action groupoid of UU-collections of gauge fields of def. 2.

UΩ vert 1(X×U)//C (X×U,U(1)). 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 n\mathbb{R}^n, nn \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 XX is a presheaf of groupoids on CartSp, hence a functor

X:CartSp opGrpd 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. 5 is the following: a smooth structure on a groupoid is to be determined by which maps from abstract coordinate systems n\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 n\mathbb{R}^n into the smooth groupoid XX 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( n)={ smooth n X smooth}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( n)X(\mathbb{R}^n) as the groupoid of plots or of probes of XX (which here is only defined by these!) by nn-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

PreSmooth1TypeFunct(CartSp op,Grpd) PreSmooth1Type \coloneqq Funct(CartSp^{op}, Grpd)

for the functor category, the category of pre-smooth groupoids, def. 5, regarded naturally as a Grpd-enriched category.

This means that for X,YPreSmooth1TypeX,Y \in PreSmooth1Type two pre-smooth groupoids, then the bare hom-groupoid

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

between them has

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

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

    such that for every smooth function ϕ: n 1 n 2\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 XX and YY:

    X( n 1) f( n) Y( n 1) X(ϕ) Y(ϕ) X( n 2) f( n) Y( n 2); \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:fgH \colon f \rightarrow g the “modifications” of these, hence collections H( n):f( n)g( n)H(\mathbb{R}^n) \colon f(\mathbb{R}^n)\to g(\mathbb{R}^n) of natural transformations, such that for every smooth function ϕ: n 1 n 2\phi \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} the whiskering of these satisfies

    X( n 1) X(ϕ) f( n 2) X( n 2) H( n 2) Y( n 2) g( n 1)= f( n 1) X( n 1) H( n 1) Y( n 1) g( n 1) Y(ϕ) Y( n 2) \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. 6, what they encode is immediate: this just says that smooth maps between pre-smooth groupoids take all probes of XX by abstract coordinate charts to probes of YY 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 2 is fully correct:

Proposition

Every Cartesian space n\mathbb{R}^n defines a pre-smooth groupoid ̲ n\underline{\mathbb{R}}^n, def. 5, by the assignment

̲ n: kC ( k, n)SetGrpd \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

()̲:CartSpPreSmooth1Type \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 XPreSmooth1TypeX\in PreSmooth1Type and any Cartesian space n\mathbb{R}^n, there is a natural equivalence of groupoids

PreSmooth1Type(̲ n,X)X( n) 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. 3 expresses just the intuition of remark 2 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 n\mathbb{R}^n with the pre-smooth groupoid that it represents.

Example

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

X:CartSp opSet X \colon CartSp^{op} \to Set

then its embedding into groupoids

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

is a pre-smooth groupoid.

More generally:

Example

Let

𝒢 ={𝒢 1tis𝒢 0} \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 𝒢 0,𝒢 1Smooth0Type\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 𝒢 1𝒢 0×𝒢 1𝒢 1\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 n\mathbb{R}^n then

𝒢( n) ={𝒢( n) 1t( n)i( n)s( n)𝒢( n) 0} \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

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

This exhibits sequence of full subcategory inclusions

GrpLieGrpdDiffGrpdGrpd(Smooth0Type)PreSmooth1Type 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 𝒢 \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

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

exhibiting the fact that smooth functions from an n\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 GG a Lie group, we write (BG) Grpd(Smooth0Type)(\mathbf{B}G)_\bullet \in Grpd(Smooth0Type) for the Lie groupoid which

(BG) =(Gi*) (\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 GG). The pre-smooth groupoid that this corresponds to under the embedding of example 2 has groupoids of probes of the form

nB(C ( n,G) disc) \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 GG-valued functions on n\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 XX a smooth manifold, let {U iX} iI\{U_i \to X\}_{i \in I} be an open cover of XX. Its Cech groupoid is the Lie groupoid (diffeological groupoid) C({U i}) C(\{U_i\})_\bullet whose

  • manifold of objects is C({U i}) 0iIU iC(\{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}) 1i,jIU i×XU jC(\{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 *C({U i}) \ast \to C(\{U_i\})_\bullet may be thought of as a pair (x,i)(x,i) of a point in the manifold XX and a label ii of a chart U iU_i that contains it, and there is is precisely one morphism between two such global point (x,i)(y,j)(x,i)\to (y,j) whenever x=yx = y in XX and both U iU_i as well as U jU_j contain xx, 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}) ={ (x,j) (x,i) (x,k)}. C(\{U_i\})_\bullet = \left\{ \array{ && (x,j) \\ & \nearrow && \searrow \\ (x,i) && \longrightarrow && (x,k) } \right\} \,.

Under the embedding of example 2 there is an evident morphism of pre-smooth groupoids

C({U i})X 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 XX.

((x,i)(x,j))(xid xx). ((x,i) \to (x,j)) \mapsto (x \stackrel{id_x}{\to} x) \,.
Proposition

Let XX a smooth manifold, {U iX}\{U_i \to X\} an open cover and C({U i})C(\{U_i\}) the corresponding Cech groupoid, def. 4. Let GG be a Lie group and BG\mathbf{B}G its groupoidal delooping according to example 3.

Then the hom-groupoid PreSmooth1Type(C({U i}),BG)PreSmooth1Type(C(\{U_i\}), \mathbf{B}G) of maps from C({U i})C(\{U_i\}) to BG\mathbf{B}G, def. 6, has

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

    g ij:U i×XU jG g_{i j} \;\colon\; U_i \underset{X}{\times} U_j \longrightarrow G

    satisfying on each triple intersection the cocycle condition

    g ijg jk=g ik g_{i j} g_{j k} = g_{i k}
  • as morphisms the coboundaries between such cocycles {g ij}\{g_{i j}\} and {g˜ jk}\{\tilde g_{j k}\}, hence collections of smooth functions

    h i:U iG h_{i} \colon U_i \longrightarrow G

    such that on each intersection of charts

    g ijh j=h ig˜ ij. 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:

π 0PreSmooth1Type(C({U i}),BG)H Cech 1(X,{U i};G). \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})BGC(\{U_i\})\longrightarrow \mathbf{B}G look as follows

( (x,j) (x,i) (x,k))( g ij(x) g jk(x) g ik(x) ). \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 *= 0\ast = \mathbb{R}^0, hence after forgetting the smooth structure, then it is a functor of groupoids as shown. Since BG( 0)\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)(x,i,j) in a double intersection of the cover to a group element g ij(x)Gg_{i j}(x)\in G. Hence the functor is given by functions

(g ij) i,jI:i,jU i×XU jG. ( 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 ij(x)g jk(x)=g ik(x) g_{i j}(x) g_{j k}(x) = g_{i k}(x)

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

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

(x,i) (x,j) h i(x) g ij(x) g˜ ij(x) h j(x) \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) iI:iU iG (h_i)_{i \in I} \colon \underset{i}{\coprod} U_i \longrightarrow G

such that

h i(x)g˜ ij(x)=g ij(x)h j(x). h_i(x) \tilde g_{i j}(x) = g_{i j}(x) h_j(x) \,.

This is precisely the formula characterizing (h i) iI(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. 3, 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 XX a pre-smooth groupoid, then given a probe of it of the form σ: nX\sigma \colon \mathbb{R}^n \to X and given a covering {U i n}\{U_i \to \mathbb{R}^n\} of the probe space by other probe spaces U i nU_i \simeq \mathbb{R}^n, then it should be possible to reconstruct σ\sigma from knowing its restrictions σ |U i:U iX\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})C(\{U_i\}), def. 4, encode. This is shown by prop. 4 for the case that XX is of the form (BG) (\mathbf{B}G)_\bullet, example 3, but this is already the archetypical case.

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

nX \mathbb{R}^n \longrightarrow X

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

C({U i})X 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}) nC(\{U_i\}) \to \mathbb{R}^n.

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

A pre-smooth groupoid XPreSmooth1TypeX \in PreSmooth1Type, def. 5, satisfies descent if for all nn \in \mathbb{N} and for all differentiably good open covers {U i n}\{U_i \to \mathbb{R}^n\} of the nn-dimensional abstract coordinate chart, the functor

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

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

Remark

The condition in def. 7 is called the stack condition, or the condition of descent, alluding to the fact that it says that XX “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 XX 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 XPreSmooth0TypePreSmooth1TypeX \in PreSmooth0Type \hookrightarrow PreSmooth1Type be a pre-smooth groupoid which is really just a pre-smooth set, hence a presheaf on CartSpCartSp that takes values in groupoids with only identity morphisms

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

Then XX is a smooth groupoid in the sense of def. 7 precisely if it is a smooth set, hence precisely if, as a presheaf, it satisfies the sheaf condition.

Example

In particular, for XSmoothMfdPreSmooth0TypePreSmooth1TypeX \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. 7 is a much stronger condition.

For instance the presheaves of the form (BG) =(G*)(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow} \ast) from example 3 satisfy descent on CartSpCartSp, but not all SmoothMfdSmoothMfd. 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 CartSpCartSp.

For example:

Proposition

For GG a Lie group, then the pre-smooth delooping groupoid (BG) (\mathbf{B}G)_\bullet of example 3 satisfies descent on CartSpCartSp, def. 7.

Proof

For {U i n}\{U_i \to \mathbb{R}^n\} a differentiably good open cover, then by prop. 4 the groupoid PreSmooth1Type(C({U i}),(BG) )PreSmooth1Type(C(\{U_i\}),(\mathbf{B}G)_\bullet) is the groupoid of Cech 1-cocycles and coboundaries with coeffcients in GG on n\mathbb{R}^n relative to the cover. But this is equivalently the groupoid of GG-principal bundles on n\mathbb{R}^n. Now because the underlying topological space of n\mathbb{R}^n is contractible, all GG-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 (G) disc)PreSmooth1Type( n,(BG) )PreSmooth1Type(C({U i}),(BG) ). 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 GG-principal bundles are precisely the smooth GG-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. 10. 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 XX 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 nn \in \mathbb{N}, and 0<r<10 \lt r \lt 1 \in \mathbb{R} let

nD r n n \mathbb{R}^n \simeq D^n_r \hookrightarrow \mathbb{R}^n

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

For XPreSmooth1TypeX \in PreSmooth1Type we write

(D n) *Xlim rX(D r n)Grpd (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 nn-stalk of XX. This extends to a functor

(D n) *:PreSmooth1TypeGrpd (D^n)^* \colon PreSmooth1Type \longrightarrow Grpd

This means that objects in (D n) *X(D^n)^\ast X are equivalence classes of pairs (r,x r)(r,x_r) where 0<r<10 \lt r \lt 1 and where x rX(D r n)x_r \in X(D^n_r), with two such pairs being equivalent (r 1,x r 1)(r 2,x r 2)(r_1, x_{r_1})\sim (r_2, x_{r_2}) precisely if there is r 0<r 1,r 2r_0 \lt r_1,r_2 such that x r 1x_{r_1} becomes equal to x r 2x_{r_2} after restriction to D r 0 nD^n_{r_0}.

Definition

A morphism f:XYf \colon X \longrightarrow Y of pre-smooth groupoids, def. 6, is called a local weak equivalence if for every nn \in \mathbb{N} the nn-stalk, def. 8, is an equivalence of groupoids

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

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

Proposition

For XX a smooth manifold and {U iX}\{U_i \to X\} an open cover for it, then the canonical morphism from the corresponding Cech groupoid to XX, def. 4, is a local weak equivalence in the sense of def. 9.

Proof

The nn-stalk of the smooth manifold XX regarded as a presheaf is the set of equivalence classes of maps from open pointed nn-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 XX (but beware that this terminology is typically used for something a little bit more restrictive, namely for the case that nn is the dimension of XX and that all maps from the disks into XX are required to be embeddings).

On the other hand the nn-stalk of C({U i})C(\{U_i\}) is the groupoid whose set of objects is the set of germs, in this sense, of the disjoint union iU i\underset{i}{\coprod} U_i, and whose set of morphisms is the set of germs of the disjoint union i,jU i×XU j\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)i,jU i×XU j(x,i,j) \in \underset{i,j}{\coprod} U_i \underset{X}{\times} U_j, then every germ of objects [g][g] around (x,i)(x,i) has a representative gg that factors through this double intersection charts: g:D r nU i×XU jiU ug \colon D^n_r \to U_i \underset{X}{\times} U_j \to \underset{i}{\coprod} U_u. And similarly for (x,j)(x,j).

This means that the groupoid of nn-stalks is a disjoint union of groupoids, one for each germ of XX, 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:YXp \colon Y \longrightarrow X of pre-smooth groupoids is called a split hypercover if

  1. YY is

    1. degreewise a coproduct of Cartesian spaces;

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

  2. pp is a weak equivalence, def. 9.

Proposition

For XX a smooth manifold and {U iX}\{U_i \to X\} an open cover, then the canonical projection C({U i})XC(\{U_i\}) \to X from the corresponding Cech groupoid, def. 4, is a split hypercover precisely if the cover is differentiably good.

Proof

For every cover the map is a weak equivalence, by prop. 7.

For the Cech groupoid the condition of cofibrancy in def. 10 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)(2,1)-category of smooth groupoids

The Grpd-enriched category of genuine smooth groupoids is that obtained from that of pre-smooth groupoids, def. 6 by “universally turning local weak equivalences, def. 9, into actual homotopy equivalences”. This is stated formally by def. 11 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. 9.

Definition

Write

Smooth1TypeL lwePreSmooth1Type 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. 9.

An object XSmooth1TypeX \in Smooth1Type we call a smooth groupoid or smooth homotopy 1-type.

Proposition

Let X,APreSmooth1TypeX,A \in PreSmooth1Type such that AA satisfies descent, def. 7. Let YXY \to X be a split hypercover of XX, def. 10.

Then there is an equivalence of groupoids

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

between the hom-groupoid of smooth groupoids from XX to AA, and that of pre-smooth groupoids, def. 6, from YY to AA.

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

PreSmooth1TypeL lwePreSmooth1Type=Smooth1Type. 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. 6 to the localized definition.

When a pre-smooth groupoid is given by an internal groupoid 𝒢 \mathcal{G}_\bullet in smooth sets via example 2, then we indicate its image under this functor by removing the subscript index, writing just 𝒢\mathcal{G}. This reflects the fact that in Smooth1TypeSmooth1Type 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 BG\mathbf{B}G for the image of (BG) (\mathbf{B}G)_\bullet in Smooth1TypeSmooth1Type.

Proposition

Let XX be a smooth manifold, regarded as a smooth 0-groupoid, and let GG be a Lie group, with smooth delooping groupoid BG\mathbf{B}G (example 3, remark 7).

Then

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

is equivalently the groupoid of GG-principal bundles on XX.

Proof

By prop. 6 the object (BG) (\mathbf{B}G)_\bullet satisfies descent on CartSp. Choose {U iX}\{U_i \to X\} a differentiably good open cover. By prop. 8 the correspoding Cech groupoid projection C({U i})XC(\{U_i\}) \to X is a split hypercover, def. 10. Hence, by prop. 9, there is an equivalence of groupoids

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

The groupoid on the right is, by prop. 4, the groupoid of Cech 1-cocycles and coboundaries with values in GG relative to a good open cover. This is equivalently the groupoid of GG-principal bundles.

Remark

The content of prop. 10 is in common jargon that: BGSmoothGrpd\mathbf{B}G \in SmoothGrpd is the moduli stack of GG-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 CC is to remember the rule which for each test space UU in CC produces the \infty-groupoid of UU-parameterized families of kk-morphisms in KK. For instance for an ∞-Lie groupoid we could test with each Cartesian space U= nU = \mathbb{R}^n and find the \infty-groupoids K(U)K(U) of smooth nn-parameter families of kk-morphisms in KK.

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

K:C opKanCplxsSet 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][C^{op}, sSet] on the opposite category of the category of test objects CC serves as a model for the (∞,1)-category of \infty-groupoids with CC-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 1K 2K_1 \stackrel{\simeq}{\to} K_2.

For common choices of CC there is a well-understood way to define the weak equivalences Wmor[C op,sSet]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 XYX \to Y between \infty-groupoids with CC-structure is not just a morphism XYX\to Y in [C op,sSet][C^{op}, sSet], but is a span of such ordinary morphisms

    X^ Y X \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 XX to YY.

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

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

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

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

Good covers and split hypercovers
Proposition

For XX a smooth manifold let {U iX} i\{U_i \hookrightarrow X\}_i be an open cover. This is a differentiably good open cover, def. \ref{DifferentiallyGoodOpenCover}, precisely if the corresponding Cech nerve C({U i})XC(\{U_i\}) \to X in sPh(CartSp) proj,locsPh(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 k\mathbb{R}^k is a smooth function γ: k\gamma \colon \mathbb{R} \to \mathbb{R}^k such

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

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

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

Write

[, k]//Diff()Smooth0Type [\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 P 1( k)B\mathbf{P}_1(\mathbb{R}^k) \to \mathbf{B}\mathbb{R} from the smooth path groupoid of k\mathbb{R}^k is

  • a homomorphism of smooth space

    S:[, k] S \colon [\mathbb{R}, \mathbb{R}^k] \to \mathbb{R}
  • such that

    1. composition of paths γ 1,γ 2\gamma_1, \gamma_2 is sent to addition in \mathbb{R}

      S(γ 2γ 1)=S(γ 1)+S(γ 2). 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

[P 1(),B] ()()DK(C ()dΩ 1(). [\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(𝔤)CE(\mathfrak{g}) of a finite dimensional Lie algebra 𝔤\mathfrak{g} is the semifree graded-commutative dg-algebra whose underlying graded algebra is the Grassmann algebra

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

(with the nnth skew-symmetrized power in degree nn)

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

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

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

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

Remark

If in the situation of prop. 14 we choose a dual basis {t a}\{t^a\} of 𝔤 *\mathfrak{g}^* and let {C a bc}\{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

dt a=12C a bct bt c, 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. 14 yields a fully faithful functor

CE():LieAlgdgAlg op 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 𝔤\mathfrak{g}, its Weil algebra W(𝔤)W(\mathfrak{g}) is the semi-free dga whose underlying graded-commutative algebra is the exterior algebra

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

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

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

of two graded derivations of degree +1 defined by

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

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

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

    with d\mathbf{d}:

    d CE(𝔤)dω:=dd CE(𝔤)ω d_{CE(\mathfrak{g})} \mathbf{d} \omega := - \mathbf{d} d_{CE(\mathfrak{g})} \omega

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

Proposition

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

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

hence there is a natural bijection

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

The form AA is flat in that its curvature differential 2-form F AF_A vanishes, precisely if this morphism factors through the CE-algebra.

Remark

With a choice of basis as in remark 9, then the content of prop. 14 is seen in components as follows:

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

t a, A a Ω 1( n) r a F a Ω 2( n). \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

t a A a d W(𝔤) d dR 12C a bct bt c+r a (12C a bcA bA c+F a) = d dRA a. \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 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():L AlgdgAlg op CE(-) \;\colon\; L_\infty Alg \hookrightarrow dgAlg^{op}
(𝔤,[],[,],[,,],)( 𝔤 *,d CE=[] *+[,] *+[,,] *+) (\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 𝔤\wedge^\bullet \mathfrak{g} on a \mathbb{N}-graded vector space 𝔤\mathfrak{g} of finite type.

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

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

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

(𝔞 X,[],[,],[,,],)( C (X) 𝔞 *,d CE=[] *+[,] *+[,,] *+) (\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 (X)C^\infty(X) of an \mathbb{N}-graded projective module 𝔞\mathfrak{a} of finite type over C (X)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 XX a smooth manifold and 𝔤\mathfrak{g} an ∞-Lie algebra or more generally an ∞-Lie algebroid, a ∞-Lie algebroid valued differential forms on XX is a morphism of dg-algebras

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

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

A:TXinn(𝔤) 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

Ω (X)AW(𝔤)F () 1𝔤 *:F A. \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(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

(F A=0)( CE(𝔤) A flat Ω (X) A W(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 AA flat.

Higher dimensional paths in an \infty-Lie algebroid
Example

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

CE(TX)=(Ω (X),d dR). CE(T X) = (\Omega^\bullet(X), d_{dR}) \,.
Definition

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

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

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

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

Ω (Δ k)CE(𝔞):Σ * \Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^*

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

See also at differential forms on simplices.

Remark

A kk-path in 𝔞\mathfrak{a}, def. 17, is equivalently

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

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

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

Definition

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

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

More generally, for any UU \in CartSp a smooth differential form ω\omega on U×Δ kU \times\Delta^k is said to have sitting instants if there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that for all points u:*Uu : * \to U the pullback along (u,Id):Δ kU×Δ k(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k is a form with sitting instants on ϵ\epsilon-neighbourhoods of faces.

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

We write Ω si,vert (U×Δ k)\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×Δ kUp : U \times \Delta^k \to U, hence the coequalizer

Ω (U)0p *Ω si (U×Δ k)Ω si,vert (U×Δ k). \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 kk perpendicular directions to a vertex.

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

    A smooth function f:U×Δ 1f : U \times \Delta^1 \to \mathbb{R} is in Ω vert 0(U×Δ 1)\Omega^0_{\mathrm{vert}}(U \times \Delta^1) if there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that for each uUu \in U the function f(u,):Δ 1[0,1]f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R} is constant on [0,ϵ)(1ϵ,1)[0,\epsilon) \coprod (1-\epsilon,1).

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

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

    ϕ:Δ kX \phi \colon \Delta^k \to X

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

    Then the pullback form ϕ *ωΩ (Δ k)\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 Δ k\Delta^k. Notably for ω jΩ (Δ k1)\omega_j \in \Omega^\bullet(\Delta^{k-1}) a collection of forms with sitting instants on the (k1)(k-1)-cells of a horn Λ i k\Lambda^k_i that coincide on adjacent boundaries, and for

p:Δ kΛ i k1 p \colon \Delta^k \to \Lambda^{k-1}_i

a standard piecewise smooth retract, the pullbacks

p *ω i p^* \omega_i

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

Remark

That ωΩ (Δ k)\omega \in \Omega^\bullet(\Delta^k) having sitting instants does not imply that there is a neighbourhood of the boundary of Δ k\Delta^k on which ω\omega is entirely constant. It is important for the following constructions that in the vicinity of the boundary ω\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 𝔤\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 𝔤\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 𝔞\mathfrak{a} be an ∞-Lie algebroid (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra).

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

exp(𝔞) bare (Hom LieAlgd(TΔ 2,𝔞)Hom LieAlgd(TΔ 1,𝔞)Hom LieAlgd(TΔ 0,𝔤)) =(Hom dgAlg(CE(𝔞),Ω (Δ 2))Hom dgAlg(CE(𝔞),Ω (Δ 1))Hom dgAlg(CE(𝔞),Ω (Δ 0))). \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(𝔞)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[CartSp_{smooth}^{op}, sSet]_{proj,loc} over the site CartSp smooth{}_{smooth}.

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

Definition

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

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

by

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

for all UU \in CartSp and [k]Δ[k] \in \Delta.

Remark

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

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

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

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

Proof

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

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

f:Δ kKΛ i kK. f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,.

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

The resulting assignment

(CE(𝔤)AΩ si,vert (U×Λ i k))(CE(𝔤)AΩ si,vert (U×Λ i k)f *Ω si,vert (U×Δ n)) (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 UU \in CartSp.

Definition

Write cosk n+1exp(a)\mathbf{cosk}_{n+1} \exp(a) for the simplicial presheaf obtained by postcomposing exp(𝔞):CartSp opsSet\exp(\mathfrak{a}) : CartSp^{op} \to sSet with the (n+1)(n+1)-coskeleton functor cosk n+1:sSettr nsSet n+1cosk n+1sSet\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 𝔤L \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 GG integrating 𝔤\mathfrak{g}.

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

BG c:UN(C (U,G)*)*. \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 Pexp():Ω 1([0,1],𝔤)GP \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G yields a weak equivalence (in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj})

Pexp():cosk 3exp(𝔤)cosk 2exp(𝔤)BG c. 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 XX a simply connected smooth manifold and x 0Xx_0 \in X a basepoint, there is a canonical bijection

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

between the set of Lie-algebra valued 1-forms on XX whose curvature 2-form vanishes, and the set of smooth functions XGX\to G that take x 0x_0 to the neutral element eGe \in G.

Proof

The bijection is given as follows. For AΩ flat 1(X,𝔤)A \in \Omega^1_{flat}(X,\mathfrak{g}) a flat 1-form, the corresponding function f A:XGf_A : X \to G sends xXx \in X to the parallel transport along any path x 0xx_0 \to x from the base point to xx

f A:xtra A(x 0x). f_A : x \mapsto tra_A(x_0 \to x) \,.

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

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

A=f *θ. A = f^* \theta \,.

From this we obtain

Proof of the proposition

The \infty-groupoid cosk 2exp(𝔤)\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 Δ 1G\Delta^1 \to G (with sitting instants), where two of these are taken to be equivalent if there is a smooth homotopy D 2GD^2 \to G (with sitting instant) between them.

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

Remark

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

Integrating to line/circle Lie nn-groups
Definition

For n,n1n \in \mathbb{N}, n \geq 1 write b n1b^{n-1} \mathbb{R} for the L-∞-algebra whose Chevalley-Eilenberg algebra is given by a single generator in degree nn and vanishing differential. We may call this the line Lie nn-algebra.

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

Observation

The discrete ∞-groupoid underlying exp(b n1)\exp(b^{n-1} \mathbb{R}) is given by the Kan complex that in degree kk has the set of closed differential nn-forms (with sitting instants) on the kk-simplex

exp(b n1) disc:[k]Ω si,cl n(Δ k) \exp(b^{n-1} \mathbb{R})_{disc} : [k] \mapsto \Omega^n_{si,cl}(\Delta^k)
Proposition

The \infty-Lie integration of b n1b^{n-1} \mathbb{R} is the line Lie n-group B n\mathbf{B}^{n} \mathbb{R}.

Moreover, with B n chn[CartSp smooth op,sSet]\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 nn on C (,)C^\infty(-, \mathbb{R}) the equivalence is induced by the fiber integration of differential nn-forms over the nn-simplex:

Δ :exp(b n1)B n chn. \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

Δ :(ωΩ si,vert,cl n(U×Δ k)) Δ kωC (U,) \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 n1)B n chn\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn} on CartSp smooth{}_{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 nn. Let λΩ si,vert,cl n(Δ n+1)\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 nn-faces of the (n+1)(n+1)-simplex. Followed by the integral over Δ n\Delta^n this is the piecewise integral of λ\lambda over the boundary of the nn-simplex. Since λ\lambda has sitting instants, there is 0<ϵ0 \lt \epsilon \in \mathbb{R} such that there are no contributions to this integral in an ϵ\epsilon-neighbourhood of the (n1)(n-1)-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the (n+1)(n+1)-simplex, as indicated in the following diagram

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Layer 1

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

It remains to show that Δ :exp(b n1)B n chn\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 UCartSpU \in CartSp. This amounts to the statement that

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

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

To demonstrate this, we want to work with forms on the (k+1)(k+1)-ball instead of the (k+1)(k+1)-simplex. To achieve this, choose again 0<ϵ0 \lt \epsilon \in \mathbb{R} and construct the diffeomorphic image of S k×[1,1ϵ]S^k \times [1,1-\epsilon] inside the (k+1)(k+1)-simplex as indicated in the above diagram: outside an ϵ\epsilon-neighbourhood of the corners the image is a rectangular ϵ\epsilon-thickening of the faces of the simplex. Inside the ϵ\epsilon-neighbourhoods of the corners it bends smoothly. By the Steenrod-Wockel approximation theorem the diffeomorphism from this ϵ\epsilon-thickening of the smoothed boundary of the simplex to S k×[1ϵ,1]S^k \times [1-\epsilon,1] extends to a smooth function from the (k+1)(k+1)-simplex to the (k+1)(k+1)-ball.

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

It is now sufficient to show: a smooth family of smooth nn-forms ωΩ vert,cl n(U×S k)\omega \in \Omega^n_{vert,cl}(U \times S^k) extends to a smooth family of closed nn-forms ω^Ω vert,cl n(U×B k+1)\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<kn \lt k and for k=nk = n precisely if its smooth family of integrals vanishes, S kω=0C (U,)\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 nn-form ω\omega on S kS^k is exact precisely if n<kn \lt k or if n=kn = k and its integral vanishes. In that case there is an (n1)(n-1)-form AA with ω=dA\omega = d A. Choosing any smoothing function f:[0,1][0,1]f : [0,1] \to [0,1] (smooth, surjective, non,decreasing and constant in a neighbourhood of the boundary) we obtain an nn-form fAf \wedge A on (0,1]×S k(0,1] \times S^k, vertically constant in a neighbourhood of the ends of the interval, equal to AA at the top and vanishing at the bottom. Pushed forward along the canonical (0,1]×S kD k+1(0,1] \times S^k \to D^{k+1} this defines a form on the (k+1)(k+1)-ball, that we denote by the same symbol fAf \wedge A. Then the form ω^:=d(fA)\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 (n1)(n-1)-form AA in a way depending smoothly on the the nn-form ω\omega.

One way of achieving this is using Hodge theory. Fix a Riemannian metric on S nS^n, and let Δ\Delta be the corresponding Laplace operator, and π\pi the projection on the space of harmonic forms. Then the central result of Hodge theory for compact Riemannian manifolds states that the operator π\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 *GP := d^* G expressed in terms of the adjoint d *d^* of the de Rham differential and of the Green operator GG of Δ\Delta. Since the kk-form ω\omega is exact its projection on harmonic forms vanishes. Therefore

ω =(Idπ)ω =d(Pω)+P(dω) =d(Pω). \begin{aligned} \omega & = (Id-\pi)\omega \\ & = d (P\omega)+P (d\omega) \\ & = d (P\omega). \end{aligned}

Hence A:=PωA := P\omega is a solution of the differential equation dA=ωd A=\omega depending smoothly on ω\omega.

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

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

Then cosk 3exp(𝔤 μ)\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu) is equivalent to the 2-groupoid BString\mathbf{B}String

  • with a single object;

  • whose morphisms are based paths in GG;

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

    • Σ:D * 2G\Sigma : D^2_* \to G is a smooth based map (where we use a homeomorphism D 2Δ 2D^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 * 2D^2_* is the 0-vertex of Δ 2\Delta^2)

    • and cU(1)c \in U(1), and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball ϕ:D 3G\phi : D^3 \to G filling them the labels c 1,c 2U(1)c_1, c_2 \in U(1) differ by the integral D 3ϕ *μ(θ)mod\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z},,

where θ\theta is the Maurer-Cartan form, μ(θ)=θ[θθ]\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)

Semantic Layer

\infty-Toposes

Slice \infty-toposes
Local, \infty-connected and cohesive \infty-toposes

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& &#643; &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Some terminology:

Definition

For XHX \in \mathbf{H} locally \infty-connected, we say Π(X)\mathbf{\Pi}(X) is the shape of XX. If H\mathbf{H} is in addition cohesive we also say that Π(X)\mathbf{\Pi}(X) is the geometric realization of XX or the fundamental ∞-groupoid of XX.

If Π(X)*\mathbf{\Pi}(X) \simeq * we say that XX is geometrically contractible.

The \infty-topos of smooth \infty-groupoids

Local \infty-Connectedness

(…)

Locality

(…)

Cohesion

(…)

Syntactic Layer

Identity types

Homotopy type theory

Cohesive modality II: flat types

flat type A\flat A

maps XBGX \to \flat \mathbf{B}G are flat GG-connections

BG:TypeUnderlyingBundle BG:BGBG \mathbf{B}G \colon Type \;\vdash\; UnderlyingBundle_{\mathbf{B}G} \colon \flat \mathbf{B}G \to \mathbf{B}G
Revised on January 26, 2016 05:31:25 by Urs Schreiber (194.210.233.114)