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I think I’m confused. We work with a distinction between path groupoids and fundamental groupoids and their higher versions, the former requiring considerations regarding smoothness, thin homotopies, etc. But we then have an entry shape via cohesive path ∞-groupoid and yet at shape modality it says that this modality
sends an object to something that may be regarded equivalently as its geometric realization or its fundamental ∞-groupoid
Is ’shape via cohesive path ∞-groupoid’ right? Why not fundamental ∞-groupoid?
But then there is a “smooth shape modality” mentioned at shape modality.
A probably related confusion is when connections are flat. So a connection on a bundle for a smooth space is defined via the path groupoid, but is flat if it factors through the fundamental groupoid. This explains by the shape-flat adjunction, why $\flat A$ is acting as coefficients for flat connections.
But then it’s back here to speaking of shape as the path ∞-groupoid to define flat ∞-connections. So what’s a (non-flat) ∞-connection defined on?
This is the very issue that, way back, made me pass from the idea of parallel transport along “smooth path n-groupoids” as $n\to \infty$ to the concept of cohesion in $\infty$-toposes: It turns out that as $n \to \infty$ then the smooth path $n$-groupoid of a smooth space becomes equivalent to its fundamental $\infty$-groupoid.
This statement in full beauty was finally proven by Dmitri Pavlov only recently (and is still not on the arXiv, it seems). In the stable case, i.e. for sheaves of spectra, and evaluated just on compact test spaces, this is prop. 7.6 in Bunke-Nikolaus-Völkl 13
Ah, interesting. But then on that basis, all connections factor through the fundamental $\infty$-groupoid, and so are flat? But there are not-flat such connections, no? connection on a smooth principal infinity-bundle.
@David
Flatness happens “at the top dimension”, so never, in the infinity case.
Hmm, looking at the page you linked to, it gives a definition of flat that uses CE algebras, not factoring through the fundamental infinity-groupoid. See definitely one can have flat things, they just aren’t forced by homotopy invariance at the top level.
Yes, right. The higher parallel transport in terms of smooth $n$-functors out of smooth path $n$-groupoids
$\mathbf{\Pi}_n(X) \longrightarrow \mathbf{B}G$comes out as “fake flat”, namely with all $k \leq n$-form curvature components vanishing. This reflects nothing but the fact that in this case between any two small enough “parallel” $(k-1)$-paths there is a $k$-path that makes their paralell transport equal up to homotopy. Now as $n \to \infty$ this means that all curvature components whatsoever have to vanish.
To nevertheless have non-flat transport in terms of the path $\infty$-groupoid, one observes that for $G$ a smooth $n$-group, then $INN(G)$ its “inner automorphism $(n+1)$-group” is the coefficient for the curvature of non-flat $G$ parallel transport, and that the corresponding “curvature transport” with coefficients in $INN(G)$ is completely flat, this flatness of the curvatures being the Bianchi identities on the original connection.
(That’s the reason why we have an article Roberts-Schreiber 07 on the “inner automorphism 3-group of a 2-group”! :-)
Now however $INN(G)$ is contractible, so flat $INN(G)$-transport alone is not the answer either, because it is always gauge equivalent to the trivial transport. But this is only because $INN(G)$ admits more gauge transformations than are present in the $G$-transport that we actually wanted to describe. So one has to combine both and consider anafunctor diagrams of the form
$\array{ X &\overset{bundle}{\longrightarrow}& \mathbf{B}G \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\overset{ \text{connection and} \atop \text{curvature} }{\longrightarrow}& \mathbf{B} INN(G) }$Here the top horizontal morphism is the Cech cocycle for the $G$-principal bundle, while the bottom morphism is the non-flat (and non-fake flat) parallell transport on it. A gauge transformation is now a homotopy between such morphisms in the arow category, and this constrains the homotopy on the “curvatur transport” bottom arrow to be compatible with the homotopy of the Cech cocycle top arrow, and that constrains the former such as to retain the genuine connection information.
Or almost. In fact one will want to impose one more condition, namely that the curvature invariants of the connection do descend to globally defined differential forms. For instance in the case that $G = \mathbf{B}^n U(1)$ then we want not just
$\array{ X &\overset{bundle}{\longrightarrow}& \mathbf{B}\mathbf{B}^n U(1) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\overset{ \text{connection and} \atop \text{curvature} }{\longrightarrow}& \mathbf{B} \mathbf{B}^n(U(1) \to U(1)) }$but in fact
$\array{ X &\overset{ \text{bundle} }{\longrightarrow}& \mathbf{B}\mathbf{B}^n U(1) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\overset{ \text{connection and} \atop \text{curvature} }{\longrightarrow}& \mathbf{B} \mathbf{B}^n(U(1) \to U(1)) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\underset{\text{curvature}}{\longrightarrow}& \mathbf{B} \mathbf{B}^{n+1}U(1) }$where now the bottom morphism picks out the curvature invariants, which in this abelian case just happen to be the plain $(n+2)$-form curvature itself. For non-abelian connections it is instead the result of evaluating the non-abelian curvature forms in all invariant polynomials, or at least in those for which one wants the connection to have a Chern-Weil homomorphism that refines these curvature invariants to higher Chern-Simons gerbes.
As $X$ here varies over smooth manifolds or just Cartesian spaces, the $\infty$-groupoids of diagrams as above form the $\infty$-stack $\mathbf{B}G_{conn}$ of non-flat $G$-principal $\infty$-connections
Finally one wants to construct all this with just an $L_\infty$-algebra $\mathfrak{g}$ specified. And this is accomplished by considering (some discrete quotient of some truncation of) the simplicial sheaf
$\exp(\mathfrak{g})_{conn} \;\colon\; U \mapsto \left( [k] \mapsto \left( \array{ \Omega^\bullet_{vert}(U \times \Delta^k) &\overset{A}{\longleftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\overset{(A,F)}{\longleftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\underset{\langle F \wedge \cdots \wedge F\rangle}{\longleftarrow}& inv(\mathfrak{g}) } \right) \right)$where on the right we have the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of $\mathfrak{g}$, as well as its Weil algebra $W(\mathfrak{g})$ and its algebra of invariant polynomials $inv(\mathfrak{g})$, respectively.
(This finally became Forenza-Schreiber-Stasheff 10.)
This story is also discussed in the “Introduction” section of dcct, in the subsection “Introduction – Geometry” and there in the subsections “Principal connections”, “Characteristic classes”, “Lie algebras” and “Chern-Weil homomorphism”.
Thanks! But I have the feeling that there’s still something fundamental I’m missing in what led me to pose my question in #1, which means I’m not able to foresee when differential aspects will appear when thinking about the shape-flat-sharp modalities.
I guess a chunk of my intuition for these modalities is coming from the adjoint quadruple between $Top$ and $Set$, so no whiff of anything differential.Then when Science of Logic (relying on the Wayback Machine as the nLab isn’t working right now) notes that the smooth cylinder and smooth circle are not diffeomorphic but are the same under shape, that’s fitting with the intuition of underlying topological spaces up to homotopy, and agrees with $\int X$ as
the fundamental infinity-groupoid of $X$.
Now to muddy the waters, we’ve been discussing the equivalence of the latter with the smooth path-infinity groupoid. Does that have something to do with why something differential may appear when dealing with the shape-flat-sharp oppositions, e.g., why the exact hexagon is about differential cohomology theories?
Science of Logic characterises these modalities thus:
Hence this unity of opposites is geometric quality. In standard models this geometric quality is for instance topology or smooth structure or formal smooth structure or supergeometric structure.
cohesive (infinity,1)-topos says that the cohesion may be continuous or smooth. What tells me which it is?
Is the problem that I’m thinking bottom up, i.e., along the lines that the minimal kind of geometry which models the shape-flat-sharp stage is merely topological-continuous, so I don’t expect anything differential there? Is it instead that I should think of the shape-flat-sharp stage from the perspective of settings which already involve smoothness?
Maybe I should look for the easiest non-trivial cohesive topos over $\infty Grpd$, such as the Sierpinski $(\infty,1)$-topos. $\flat$ sends $P \to X$ to $P \to P$. The cofiber (say for stable objects) has something to do with differential forms?
Of course, there was plenty of relevant conversation a year ago in fundamental theorem of calculus claim from hpg.
It seems you are realizing some of the magic that goes into cohesion. One is this fact we mentioned that the smooth path $n$-groupoid become a discrete homotopy type when $n = \infty$.
Another fun fact to notice is this: We can define flat $G$-valued differential forms even for $G$ a topological group, in the following sense.
One way to see, for $G$ a Lie group, that $\flat_{dR} \mathbf{B}G \coloneqq hofib (\flat \mathbf{B}G \to \mathbf{B}G)$ is equivalently the sheaf $\Omega^1_{flat}(-,\mathfrak{g})$ of flat Lie algebra valued form is to first observe that $\flat \mathbf{B}G$ is presented by the sheaf of groupoids whose objects are flat $G$-valued forms, and whose morphisms are gauge transformations, then observe that with this presentation the map $\flat \mathbf{B}G \to \mathbf{B}G$ is presented by a fibration, so that its homotopy fiber is its ordinary fiber.
But of course there is another, equivalent, way to compute this homotopy fiber, which never mentions presentations by sheaves of differential forms: since $\flat \mathbf{B}G \simeq \mathbf{B} \flat G$ we discover that
$\Omega^1_{flat}(-,\mathfrak{g}) \simeq G/\flat G \,,$where the right hand side is presented by the sheaf that assigns to a Cartesian space $U$ the smooth functions $U \to G$ modulo the relation which identifies two such smooth functions if they differ by multiplication (from the right, say) with an element in $G$, hence with a constant smooth function.
This equivalence is another secret incarnation of the parallel transport: Given a flat $\mathfrak{g}$-valued differential form on $U$, then for every choice of basepoint in $U$ we get a smooth function from $U$ to $G$ which sends every other point to the parallel transport of that differential from from that basepoint to that other point, along any smooth path connecting them. If we choose another basepoint, then this function changes by global multiplication with the parallel transport along any path connecting the two basepoints. This is why parallel transport of flat $\mathfrak{g}$-valued forms without choice of basepoint constitutes $G/\flat G$.
But now the expression $G/\flat G$ makes sense even if there is no concept of differential forms! For a topological group $G$ it makes sense: it’s now the sheaf of continuous $G$-valued functions modulo global (constant) multiplication by any group element.
Moreover, the synthetic development of the theory from axiomatic cohesion shows that no matter what the underlying model is, the object $G/\flat G$ will always behave in key aspects as a sheaf of flat Lie algebra valued differential forms is supposed to behave, even if actual differential forms don’t exist in the model.
Thanks! Something to digest after (during?) the three hours of meetings I have to fill up my afternoon.
Just a remark about that magical convergence of path and fundamental $n$-groupoids as one passes along $n \to \infty$.
Recalling some fun we had a decade or so ago with Tangle hypotheses and fundamental $n$-categories with duals of stratified spaces (entering and exiting rather than the one way process that Ayala et al. study), John said
Well, this is the mystery that makes the Tangle Hypothesis so interesting! You start with an algebraically natural sort of $n$-category, namely $k$-tuply monoidal $n$-categories with duals, and you discover that the free such gadget on one object describes $n$-dimensional surfaces in $\mathbb{R}^{n+k}$!
Eugenia Cheng’s paper – An ω-category with all duals is an ω-groupoid – came out at about that time, and discusses such Tangle hypotheses in the limit as $n \to \infty$.
For all $k\geq 0$ there should be a $k$-monoidal ω-category whose $m$-cells are $m$-manifolds embedded in $(m+k)$-cubes. This should have all duals as above and hence by the Main Theorem should in fact be an ω-groupoid. Likewise for oriented manifolds.
Is there some relation from this kind of result to the path $\infty$-groupoid surprise? Presumably the latter involves maps out of some nice entity which captures smooth paths, surfaces, etc. Then if the latter turns out to be some kind of free $\infty$-groupoid on one object, maps out of this in turn give something like a fundamental $\infty$-groupoid?
By the way, we have a rather neglected smooth structure of the path groupoid.
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