differential cohomology in an (∞,1)-topos – survey
internal homotopy ∞-groupoid?
(…)
Background fields in twisted differential nonabelian cohomology?
An Ehresmann $\infty$-connection is a reformulation of a cocycle in differential cohomology? in terms of abstract deRham classes on the total space of the underlying principal ∞-bundle. It generalizes the ordinary notion of an Ehresmann connection on an ordinray principal bundle.
More in detail, recall that, as discussed at differential cohomology?,
using the local model structure on simplicial presheaves $SPSh(C)^{loc}$ as a model for smooth ∞-stacks that compute smooth cohomology
in the presence of a comsimplicial $C$-object that induces a path ∞-groupoid a differential cocycle on an object $X$ with coefficients in an object $A$ is represented by a diagram
where $Y \to X$ is a hypercover, i.e. an acyclic fibration in $SPSh(C)^{loc}$.
Here the top horizontal morphism $g : Y \to A$ is the underlying $A$-valued cocycle that classifies an $A$-principal ∞-bundle $p : P \to X$.
We show below how the entire diagram above may be pulled back along $p$ to the total space of this principal ∞-bundle where it gives rise to $A$-valued differential form data on $P$ that satisfies two constraints. This datum and these constraints are analogous to and generalize the notion of an Ehresmann connection on an ordinary principal bundle.
It is exhibited by a diagram
Here $\Pi_vert(P)$ denotes the vertical path ∞-groupoid of $P$, described in detail below.
All three horizontal morphisms depicted trivialize when restricted along the canonical inclusion $Y \to \Pi(Y)$ and $P \to \Pi(P)$, respectively. This means these are cocycles in flat differential $A$-cohomology whose underlying $A$-class vanishes. This is the characterization of nonabelian deRham cohomology.
In a next step we may express the abstract (∞,1)-topos-theoretic nonabelian deRham cocycle appearing here in terms of Lie-∞-algebra connection data. This turns the Ehresmann ∞-connection into a
Let $A \in SPSh(C)$ be a pointed object with point $pt_A : {*} \to A$.
Recall from principal ∞-bundle that iven an $A$ cocycle $X \stackrel{\simeq}{\leftarrow} Y \stackrel{g}{\to} A$ the principal ∞-bundle $P \to X$ classified by it is the homotopy fiber of this morphism
If the cocycle $g : X \to A$ is modeled in the model structure on simplicial presheaves as a span $X \stackrel{\simeq}{\leftarrow} Y \stackrel{}{\to} A$ with $A$ fibrant then this $P$ here is modeled by the ordinary pullback of the object $\mathbf{E}_{pt}A$ that is described at list of notation and constructions in categories of fibrant objects i.e. by the ordinary pullback diagram
Given any morphism $p : P \to Y$ in $SPSh(C)$ we say that the the vertical path ∞-groupoid with respect to $p$ is the objec $\Pi_{vert}(P)$ given by the ordinary pullback
By the universal property of the pullback the commutativity of
induces a universal morphism
into the vertical path ∞-groupoid.
Given a strictly commuting diagram
in $SPSh(C)$ representing a diffential cocycle?, let $p : P \to Y$ be the morphism classified by $g : Y \to A$ as recalled above.
We may paste to the diagra the square that defines the vertical path ∞-groupoid of $p : P \to Y$ to obtain
All three horizontal morphisms of this pasted diagram trivialize when restricted to constant paths along $P \to \Pi_{vert}(P)$, along $P \to \Pi(P)$ and $Y \to \Pi(Y)$, respectively.
Therefore all three morphism represent cocycles in (nonabelian) deRham cohomology.
The bottom horizontal morphism trivializes by assumption that the diagram representes a differential cocycle.
The trivialization of the top horizontal morphism when restricted to $P$ reduces to the trivialization of any $A$-cocycle when pulled back along the total space of the bundle it classifies: we have a commuting diagram
By the very definition of $\mathbf{E}_{pt}A$ in terms of the pullback diagram this extends to a diagram
which exhibits the homotopy from the pulled back cocycle to the trivial cocycle.
Last revised on March 10, 2019 at 03:14:22. See the history of this page for a list of all contributions to it.