DE RHAM COHESION

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

$$\def\mapright#1{\xrightarrow{{#1}}} \def\mapdown#1{\Big\downarrow\rlap{\raise2pt{\scriptstyle{#1}}}} \def\mapdiagl#1{\vcenter{\searrow}\rlap{\raise2pt{\scriptstyle{#1}}}} \def\mapdiagr#1{\vcenter{\swarrow}\rlap{\raise2pt{\scriptstyle{#1}}}} $$

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