Schreiber
path-structured smooth (∞,1)-toposes

Intro

We describe a general setup for differential nonabelian cohomology in the context of smooth spaces modeled by smooth ∞-stacks.

A smooth topos is a context in which (synthetic) differential geometry exists. An (∞,1)-topos is a context in which higher groupoids exist. Merging these two concepts yields the notion of a smooth (∞,1)-topos: a context in which ∞-Lie groupoids exist.

A lined topos is a context in which each space has a notion of path. A path-structured smooth $(\mathrm{\infty }\mathrm{,1})$-topos is a context in which each ∞-Lie groupoid comes with its smooth path ∞-groupoid, naturally.

Path-structured and smooth $(\mathrm{\infty }\mathrm{,1})$-toposes are the context in which differential nonabelian cohomology exists.

Motivation

For

• $X$ a smooth manifold

• and $\Sigma$ an $(n\mid 1)$-dimensional supermanifold

one expects that

$n$-dimensional quantum field theories describing the quantum dynamics of $(n-1)$-dimensional spinorial objects in $X$ may be obtained by push-forward operations of cocycles in differential cohomology along $[\Sigma ,X]\to *$: the path integral.

To make sense of this we need to understand two things:

• the context in which this can be made formal sense of –

  • physically this is the kinematics of $\Sigma$;

  • mathematically this in encoded in cohomology.

• the entities that we are pushing forward –

  • physically this is the dynamics of $\Sigma$: the structure encoding the forces subject to which $\Sigma$ propagates in $X$;

  • mathematically this in encoded in differential cohomology.

As we shall describe now.

Nonabelian cohomology - kinematics

Let $X\to BO(k)$ be a representative of the classifying map of the $O(n)$-principal bundle to which is associated the tangent bundle of $X$.

For the path integral to make sense (at all) a certain obstruction called a quantum anomaly has to vanish, which imposes the following constraints on the kinematics:

• for $n=0$, we need to require an orientation on $X$: a lift

  $$\begin{array}{ccccccc}& & B\mathrm{SO}(k)& & & & \mathrm{cokill}{\pi }_{0+1}\\ & \nearrow & \downarrow \\ X& \to & BO(k)\end{array}$$\array{ && B SO(k) &&&& cokill \pi_{0+1} \\ & \nearrow & \downarrow \\ X &\to& B O(k) }

• for $n=1$, we need to require a Spin structure on $X$: a lift

  $$\begin{array}{ccccccc}& & B\mathrm{Spin}(k)& & & & \mathrm{cokill}{\pi }_{1+1}\\ & \nearrow & \downarrow \\ X& \to & B\mathrm{SO}(k)\end{array}$$\array{ && B Spin(k) &&&& cokill \pi_{1+1} \\ & \nearrow & \downarrow \\ X &\to& B SO(k) }

• for $n=2$, we need to require a String structure on $X$: a lift

  $$\begin{array}{ccccccc}& & B\mathrm{String}(k)& & & & \mathrm{cokill}{\pi }_{3+1}\\ & \nearrow & \downarrow \\ X& \to & B\mathrm{Spin}(k)\end{array}$$\array{ && B String(k)&&&& cokill \pi_{3+1} \\ & \nearrow & \downarrow \\ X &\to& B Spin(k) }

• for $n=6$, we think we need to require a Fivebrane structure on $X$: a lift

  $$\begin{array}{ccccccc}& & B\mathrm{Fivebrane}(k)& & & & \mathrm{cokill}{\pi }_{7+1}\\ & \nearrow & \downarrow \\ X& \to & B\mathrm{String}(k)\end{array}$$\array{ && B Fivebrane(k) &&&& cokill \pi_{7+1} \\ & \nearrow & \downarrow \\ X &\to& B String(k) }

Here

$$\cdots B\mathrm{Fivebrane}(k)\to B\mathrm{String}(k)\to B\mathrm{Spin}(k)\to B\mathrm{SO}(k)\to BO(k)$$\cdots B Fivebrane(k) \to B String(k) \to B Spin(k) \to B SO(k) \to B O(k)

is the Whitehead tower of $BO(n)$: each step towards the left gives the universal next higher connected cover of the previous space.

Each morphism $X\to BG$ here may be thought of as a cocycle in nonabelian cohomology: the coefficient object $BG$ is a homotopy r-type (for some $0\le r\le \mathrm{\infty }$) in the (∞,1)-topos Top.

For describing dynamics this needs to be refined to a richer (∞,1)-topos.

Differential cohomology - dynamics

The dynamics of the objects $\Sigma$ – the forces they experience while propagating in $X$ – are encoded in refinements of the cocycles $X\to BG$ to differential cocycles.

Well known is this:

If $G$ is a Lie group then a differential refinement of a topological $G$-cocycle $X\to BG$ is a connection on the corresponding smooth $G$-principal bundle.

This story is well-known for the $n\le 1$-case of the spin group.

But: the topological groups $\mathrm{String}(n)$, $\mathrm{Fivebrane}(n)$, etc. do not admit refinements to (finite dimensional) Lie groups.

Question:

What is a differential refinement of a $\mathrm{String}(n)$-cocycle, or a $\mathrm{Fivebrane}(n)$-cocycle, etc. ?

What is a $\mathrm{String}(n)$- $\mathrm{Fivebrane}(n)$-principal ∞-bundle with connection?

Context for answer:

We need to

1. refine the (∞,1)-topos Top to a smooth (∞,1)-topos $H$ where $B\mathrm{String}(n)$, etc. exist as smooth objects;

2. such that in $H$ each object $X$ has a fundamental path ∞-groupoid $\Pi (X)$ of smooth trajectories / paths in $X$, generalizing the fundamental ∞-groupoid construction in Top.

Answer:

In such a context, define:

1. a flat differential $\mathrm{String}(n)$-cocycle refining a $\mathrm{String}(n)$-cocycle $X\to B\mathrm{String}(n)$ is an extension

   $$\begin{array}{ccc}X& \to & B\mathrm{String}\\ \downarrow & \nearrow \\ \Pi (X)\end{array};$$\array{ X &\to& \mathbf{B}String \\ \downarrow & \nearrow{} \\ \Pi(X) } \,;

2. a general differential $\mathrm{String}(n)$-cocycle is a cocycle in the twisted cohomology of $\Pi (X)$, with the twist given by a nonabelian Chern character morphism

   $$\mathrm{ch}:H(X,A)\to \prod _i{H}_{\mathrm{dR}}^{n_i}(X)$$ch : H(X,A) \to \prod_i H_{dR}^{n_i}(X)

   that determines the curvature characteristic classes.

In the following we indicate the realization of the context needed for this answer. The answer itself is described at theory of differential nonabelian cohomology.

Path structure on a smooth $(\mathrm{\infty }\mathrm{,1})$-topos

We describe a general class of realizations of the above desiderata.

bare

Choose a site of smooth test spaces such as $C=$

1. Diff : manifolds and smooth maps between them

2. or SmoothLoci : manifolds and infinitesimal spaces

3. or SuperSmoothLoci: supermanifolds and infinitesimal supermanifolds

The sheaf topos $𝒯=\mathrm{Sh}(C)$ characterized as a reflective subcategory of presheaves on $C$

$$\begin{array}{cccc}𝒯:=& \mathrm{Sh}(C)& \stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }& \mathrm{PSh}(C)\end{array}$$\array{ \mathcal{T} := & Sh(C) &\stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow}& PSh(C) }

contains as objects generalized smooth spaces modeled on objects in $C$ – such as diffeological spaces. For instance

• the manifold $X$ is in there;

• but also the mapping space $[\Sigma ,X]$.

Analogously – passing to higher geometry – the (∞,1)-sheaf (∞,1)-topos $H:={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$

contains as objects smooth ∞-groupoids modeled on objects in $C$: the ∞-stacks on $C$.

An ∞-stack (∞,1)-topos is conveniently presented by 1-categorical models for ∞-stack (∞,1)-toposes:

$$\begin{array}{cccccc}H:=& {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)& \stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }& {\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)& & \text{intrinsic def of ∞-stack (∞,1)-topos}\\ & \uparrow \simeq & & \uparrow \simeq & & \text{Lurie's theorem}\\ & (\mathrm{SPSh}(C{)}_{\mathrm{loc}}{)}^{\circ }& \stackrel{\stackrel{\text{left Bousfield loc.}}{\leftarrow }}{\to }& (\mathrm{SPSh}(C{)}_{\mathrm{glob}}{)}^{\circ }& & \text{model by simplicial presheaves}\end{array}$$\array{ \mathbf{H} := & Sh_{(\infty,1)}(C) & \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} & PSh_{(\infty,1)}(C) && \text{intrinsic def of ∞-stack (∞,1)-topos} \\ & \uparrow\mathrlap{\simeq} && \uparrow\mathrlap{\simeq} && \text{Lurie's theorem} \\ & (SPSh(C)_{loc})^\circ &\stackrel{\stackrel{\text{left Bousfield loc.}}{\leftarrow}}{\to}& (SPSh(C)_{glob})^\circ && \text{model by simplicial presheaves} }

The SSet-enriched category $\mathrm{SPSh}(C):=[C^{\mathrm{op}},\mathrm{SSet}]$ equipped with the local model structure on simplicial presheaves presents $H$.

Proposition

(smooth incarnation of coefficient objects)

The topological spaces $B\mathrm{String}$ and $B\mathrm{Fivebrane}(n)$ do naturally refine to objects $B\mathrm{String}(n)$, $B\mathrm{Fivebrane}(n)$ in $H$, i.e. they do naturally admit the structure of ∞-Lie groupoids.

Proof

See String 2-group and Fivebrane 6-group.

This takes care of the kinematics.

lined

To encode the dynamics, we need a notion of process : an object that models the notion of trajectory. We need a line. A worldline.

So refine $𝒯$ to a lined topos $(𝒯,R)$: a topos equipped with an internal $Z$-algebra object $R$. Using the cospan

$$\begin{array}{ccc}& & R\\ & {}^0\nearrow & & {\nwarrow }^1\\ *& & & & *\end{array}$$\array{ && R \\ & {}^{\mathllap{0}}\nearrow && \nwarrow^{\mathrlap{1}} \\ {*} &&&& {*} }

this is canonically a cartesian interval object in $𝒯$: a collared model for the interval $[\mathrm{0,1}]$.

Proposition

($\mathrm{\infty }$-groupoids of trajectories/paths)

Identifying inside each $R^k$ the boundary of a $k$-simplex yields a cosimplicial object

$${\Delta }_R:\Delta \to 𝒯$$\Delta_{R} : \Delta \to \mathcal{T}

of collared simplices in $𝒯$

This induces for each $X\in 𝒯$ a simplicial object

$$X^{{\Delta }_R^{•}}\in \mathrm{SSh}(C)\subset \mathrm{SPSh}(C)$$X^{\Delta^\bullet_R} \in SSh(C) \subset SPSh(C)

which presents in $H$ the smooth path ∞-groupoid of $X$.

Proof

The cosimplicial object ${\Delta }_R$ is described in detail at interval object.

As objects we simply have ${\Delta }_R^n=R^n$, but the coface maps ${\delta }_i:R^{n-1}\to R^n$ – built from $\mathrm{0,1}:*\to R$ and from the diagonal $R\to R\times R$ – identify an $n$-simplex inside that cube, rendering the $R^n$ a collared $n$-simplex.

Proposition

(path $\mathrm{\infty }$-groupoids)

There is functorially for each $U\in C$ a cofibrant replacement

$$\begin{array}{ccc}U& \hookrightarrow & \Pi (U)\\ & \searrow & {\downarrow }^{\simeq }\\ & & U^{{\Delta }_R^{•}}\end{array}$$\array{ U &\hookrightarrow& \Pi(U) \\ & \searrow & \downarrow^{\mathrlap{\simeq}} \\ && U^{\Delta^\bullet_R} }

which Yoneda extends to a Quillen adjunction $(\Pi ⊣(-{)}_{\mathrm{flat}})$

$$\Pi (-):\mathrm{SPSh}(C{)}_{\mathrm{glob}}:\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{glob}}:(-{)}_{\mathrm{flat}}.$$\Pi(-) : SPSh(C)_{glob} : \stackrel{\leftarrow}{\to} SPSh(C)_{glob} : (-)_{flat} \,.

If all representables $U$ are contractible with respect to the line object $R$ in that $U^{{\Delta }_R^{•}}\to *$ is a weak equivalence in $\mathrm{SPSh}(C{)}_{\mathrm{glob}}$, then this induces a Quillen adjunction

$$\Pi (-):\mathrm{SPSh}(C{)}_{\mathrm{loc}}:\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{loc}}:(-{)}_{\mathrm{flat}}$$\Pi(-) : SPSh(C)_{loc} : \stackrel{\leftarrow}{\to} SPSh(C)_{loc} : (-)_{flat}

and hence an (∞,1)-functor

$$\Pi (-):H:\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}.$$\Pi(-) : \mathbf{H} : \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat} \,.

Proof

Details are at path ∞-groupoid.

Remark

(parallel transport and local systems)

A morphism

$$\mathrm{tra}:\Pi (X)\to A$$tra : \Pi(X) \to A

may be thought of as a smooth $A$-valued $\mathrm{\infty }$-local system: the value of $\mathrm{tra}$ on a $k$-morphism $\Sigma$ in $\Pi (X)$ is the parallel transport along $\Sigma$.

$$\mathrm{tra}:\begin{array}{ccc}& & y\\ & {}^{{\gamma }_1}\nearrow & \Downarrow & {\searrow }^{{\gamma }_2}\\ x& & \underset{{\gamma }_3}{\to }& & z\end{array}\mapsto \begin{array}{ccc}& & E_y\\ & {}^{\mathrm{tra}({\gamma }_1)}\nearrow & \Downarrow & {\searrow }^{\mathrm{tra}({\gamma }_2)}\\ E_x& & \underset{\mathrm{tra}({\gamma }_3)}{\to }& & E_z\end{array}.$$tra \;\;\; : \;\;\; \array{ && y \\ & {}^{\mathllap{\gamma_1}}\nearrow &\Downarrow& \searrow^{\mathrlap{\gamma_2}} \\ x &&\underset{\gamma_3}{\to}&& z } \;\;\;\; \;\;\;\; \mapsto \;\;\;\; \;\;\;\; \array{ && E_y \\ & {}^{\mathllap{tra(\gamma_1)}}\nearrow &\Downarrow& \searrow^{\mathrlap{tra(\gamma_2)}} \\ E_x &&\underset{tra(\gamma_3)}{\to}&& E_z } \,.

infinitesimal

The functor $\Pi (-):H\to H$ induces a global differential structure on objects of $H$ in that it provides a notion parallel transport along smooth paths by morphisms $\mathrm{tra}:\Pi (X)\to A$. The familiar notion of differential structure is more local: every such ∞-Lie groupoid valued parallel transport should determine and be determined by its restriction along ${\Pi }^{\inf }(X)\hookrightarrow \Pi (X)$ to infinitesimal paths, where it identifies with ∞-Lie algebroid valued differential forms.

To model this, we assume now that the lined topos $(𝒯,R)$ is actually a smooth topos: that there is in a sensible way a subobject

$$D=\{x\in R\mid x^2=0\}\hookrightarrow R$$D = \{x \in R | x^2 = 0\} \hookrightarrow R

of generalized elements infinitesimally close to the origin of $R$: the infinitesimal interval object.

Using standard Models for Smooth Infinitesimal Analysis, this is the case for $C=$

1. smooth loci

2. super smooth loci.

Definition

(infinitesimal singular simplicial complex)

For each smooth locus $U\in 𝒯$ there naturally is the simplicial object

$$U^{({\Delta }_{\inf }^{•})}\in \mathrm{SSh}(C)$$U^{(\Delta^\bullet_{inf})} \in SSh(C)

of infinitesimal $k$-simplices in $U$: the infinitesimal singular simplicial complex of $U$.

Proposition

(infinitesimal path $\mathrm{\infty }$-groupoid)

There is functorially for each $U\in C$ a cofibrant replacement

$$\begin{array}{ccc}U& \hookrightarrow & {\Pi }^{\inf }(U)\\ & \searrow & {\downarrow }^{\simeq }\\ & & U^{({\Delta }_{\inf }^{•})}\end{array}$$\array{ U &\hookrightarrow& \Pi^{inf}(U) \\ & \searrow & \downarrow^{\mathrlap{\simeq}} \\ && U^{(\Delta^\bullet_{inf})} }

which Yoneda extends to a Quillen adjunction

$${\Pi }^{\inf }(-):\mathrm{SPSh}(C{)}_{\mathrm{glob}}:\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{glob}}:(-{)}_{\mathrm{flat}}^{\inf }$$\Pi^{inf}(-) : SPSh(C)_{glob} : \stackrel{\leftarrow}{\to} SPSh(C)_{glob} : (-)_{flat}^{inf}

where ${\Pi }^{\inf }$ preserves all weak equivalences between simplicial presheaves that are degreewise coproducts of representables.

Proof

Details are at infinitesimal path ∞-groupoid.

To get a feeling for what ${\Pi }^{\inf }(X)$ is like, it helps to dualize it.

Definition

(Chevalley-Eilenberg algebra)

For every simplicial object $A\in [{\Delta }^{\mathrm{op}},𝒯]\subset \mathrm{SPSh}(C)$ we have the cosimplicial algebra $𝒯(A_{•},R)$. Under the (monoidal) Dold-Kan correspondence $C^{•}:[\Delta ,\mathrm{Alg}]\stackrel{\simeq }{\to }\mathrm{dgAlg}$ this is turned into its normalized Moore cochain complex

$$\mathrm{CE}(A):=C^{•}(𝒯(A_{•},R)).$$CE(A) := C^\bullet(\mathcal{T}(A_\bullet,R)) \,.

This we call the Chevalley-Eilenberg algebra of $A$.

Proposition

(functoriality of CE)

This construction extends to a functor that is the left adjoint of a Quillen adjunction

$$\mathrm{CE}:\mathrm{SPSh}(C)\to {\mathrm{dgAlg}}^{\mathrm{op}}\stackrel{U}{\to }{\mathrm{Ch}}^{•}{}^{\mathrm{op}}.$$CE : SPSh(C) \to dgAlg^{op} \stackrel{U}{\to} Ch^\bullet^{op} \,.

Proof

This is discussed at Chevalley-Eilenberg algebra.

Proposition

(simplicial deRham complex)

Let $X$ be an ordinary manifold. Then

$$\mathrm{CE}(X^{{\Delta }_{\inf }^{•}})={\Omega }^{•}(X)$$CE(X^{\Delta_{inf}^\bullet}) = \Omega^\bullet(X)

is isomorphic in $\mathrm{dgAlg}$ to the deRham complex of $X$.

More generally, let $X=(X_{•})$ be a simplicial manifold. Then

$$\mathrm{CE}({\Pi }^{\inf }(X))\simeq {\Omega }^{•}(X_{•})$$CE(\Pi^{inf}(X)) \simeq \Omega^\bullet(X_\bullet)

is isomorphic in the homotopy category $\mathrm{Ho}(\mathrm{dgAlg})$ to the simplicial deRham complex of $X_{•}$.

Proof

The proof is described at simplicial deRham complex. The first statement is implicit in Anders Kock’s work on combinatorial differential forms. The second follows with Eilenberg-Zilber theory.

Remark

($\mathrm{\infty }$-Lie algebroids and integration)

We have a notion of ∞-Lie differentiation and integration. and integration of ∞-Lie algebroid valued differential forms.

For $U$ representable, the restriction ${\Pi }^{\inf }(U)\hookrightarrow \Pi (U)\to A$ of a parallel transport only hits infinitesimal k-morphisms in $A$.

The collection of these is the ∞-Lie algebroid $𝔞\hookrightarrow A$ of $A$. A morphism

$$\omega :{\Pi }^{\inf }(U)\to 𝔞$$\omega : \Pi^{inf}(U) \to \mathfrak{a}

is a collection of flat ∞-Lie algebroid valued differential forms. Dually this is a morphism of dg-algebras

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(𝔞):\omega .$$\Omega^\bullet(X) \leftarrow CE(\mathfrak{a}) : \omega \,.

An integration $\int \omega$ of these is an extension

$$\begin{array}{ccc}{\Pi }^{\inf }(X)& \stackrel{\omega }{\to }& 𝔞\\ \downarrow & & \downarrow \\ \Pi (X)& \stackrel{\int \omega }{\to }& A\end{array}$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(X) &\stackrel{\int \omega}{\to}& A }

super

To fully capture the motivating physics application we also need to model the supermanifold $\Sigma$ in $H$.

For that it is sufficient to promote the $k$-algebra object $R$ to a $k$-superalgebra – a ${\mathbb{Z}}_2$-graded algebra object – and thereby promote the smooth topos $(𝒯,R)$ to a super smooth topos.

This is naturally obtained for the choice of site $C=$ super smooth loci.

The corresponding (∞,1)-topos of super ∞-Lie groupoids is then a super smooth (∞,1)-topos

$$H_{\mathrm{super}}:=(\mathrm{SSh}(C{)}_{\mathrm{loc}}{)}^{\circ }.$$\mathbf{H}_{super} := (SSh(C)_{loc})^\circ \,.

Example

(supergravity field as cocycle in $H_{\mathrm{super}}$)

The D'Auria-Fre formulation of supergravity implicitly says that the fields of supergravity are differential cocycles in $H_{\mathrm{super}}$.

Description

This is described at D'Auria-Fre formulation of supergravity.

Thanks. This entry has received helpful edits by Toby Bartels and David Roberts. Thanks!