# DE RHAM

# COHOMOLOGY

**Definition** (Cohesive $\infty$-Topos).
An $\infty$-Topos which is local and $\infty$-connected is called cohesive.

# COHESION

Here is very short intro to de Rham cohesion built on top of $ʃ$ and $\flat$ modalities.

**Definition** (Cohesive $\infty$-Topos).
An $\infty$-Topos which is local and $\infty$-connected is called cohesive.

**Definition** (de Rham shape modality).
de Rham cohesive homotopy type of A is defined as a homotopy
cofiber of the unit of the shape modality:
$$
ʃ_{dR} A =_{def} \mathrm{cofib}\ \Big(A \to ʃ A \Big),
$$
or the (looping opetaion of) the cokernel of the unit of the shape modality.
It is also called de Rham shape modality.
$$
\begin{array}{ccc}
A & \mapright{} & {}1 \\
\mapdown{} & \square & \mapdown{} \\
ʃ A & \mapright{} & ʃ_{dR} A
\end{array}.
$$

**Definition** (de Rham flat modality).
de Rham complex with coefficients in A is defined as the homotopy fiber
of the counit of the flat modality:
$$
\flat_{dR} A =_{def} \mathrm{fib}\ \Big( \flat A \to A \Big),
$$
or the (delooping opetaion of) the cokernel of the unit of the shape modality.
It is also called de Rham flat modality.
$$
\begin{array}{ccc}
\flat_{dR} A & \mapright{} & \flat A \\
\mapdown{} & \square & \mapdown{} \\
{}1 & \mapright{} & A \\
\end{array}.
$$
The object A is called de Rham coefficient object of $pt_A : {}1 \rightarrow A$.

**Definition** (Loop Space Object).
Loop space objects are defined in any $\infty$-category $C$
with homotopy pullbacks: for any pointed object $X$ of $C$
with point ${}1 \rightarrow X$, its loop space object is the homotopy pullback
$\Omega(X)$ of this point along itself:
$$
\begin{array}{ccc}
\Omega(X) & \mapright{} & {}1 \\
\mapdown{} & \square & \mapdown{} \\
{}1 & \mapright{} & X \\
\end{array}.
$$

**Definition** (Delooping).
if $X$ is given and a homotopy pullback diagram
$$
\begin{array}{ccc}
X & \mapright{} & {}1 \\
\mapdown{} & \square & \mapdown{} \\
{}1 & \mapright{} & \mathbb{B}X \\
\end{array}.
$$
exists, with the point ${}1 \rightarrow \mathbb{B}X$
being essentially unique, by the above $X$ has been
realized as the loop space object of $\mathbb{B}X$.
$\mathbb{B}X$ is called delooping of X:
$$
X = \Omega\mathbb{B}X.
$$

**Theorem** (Milnor–Lurie).
There is an adjoint functor
$$
\mathrm{\infty\text{-Grp}}(\mathbb{H})
\mathrel{\mathop{\rightleftarrows}^{\mathrm{\Omega}}_{\mathrm{\\ \\ \mathbb{B}}}}
\mathbb{H}_{conn}
$$
between $\infty$-groups of $\mathbb{H}$ and uniquely pointed
connected objects $\mathbb{B}\mathrm{G}$ in $\mathbb{H}$ which are doneted $\mathbb{H}_{conn}$.
Where $\Omega$ is a looping and $\mathbb{B}$ is a delooping operations.

**Definition** (Maurer-Cartan form).
For $G \in \text{Group}(\mathbb{H})$ and $\infty$-group in the cohesive
$\infty$-topos $\mathbb{H}$ Maurer-Cartan form $\theta$ is defines as
$$
\theta_G =_{def} G \rightarrow \flat_{dR}\mathbb{B}G
$$
for the $G$-valued de Rham cocycle on $G$ induced by pullback pasting:
$$
\begin{array}{ccc}
G & \mapright{} & {}1 \\
\mapdown{\theta} & \square & \mapdown{} \\
\flat_{dR}\mathbb{B}G & \mapright{} & \flat \mathbb{B}G \\
\mapdown{} & \square & \mapdown{} \\
{}1 & \mapright{} & \mathbb{B}G \\
\end{array}.
$$

# LITERATURE

[1]. Urs Schreiber. Differential cohomology in a cohesive ∞-topos