(Cohesive -Topos).
An -Topos which is local and -connected is called cohesive.
Here is very short intro to de Rham cohesion built on
top of and modalities.

(Cohesive -Topos).
An -Topos which is local and -connected is called cohesive.

(de Rham shape modality).
de Rham cohesive homotopy type of A is defined as a homotopy
cofiber of the unit of the shape modality:

or the (looping opetaion of) the cokernel of the unit of the shape modality.
It is also called de Rham shape modality.

(de Rham flat modality).
de Rham complex with coefficients in A is defined as the homotopy fiber
of the counit of the flat modality:

or the (delooping opetaion of) the cokernel of the unit of the shape modality.
It is also called de Rham flat modality.

The object A is called de Rham coefficient object of .

(Loop Space Object).
Loop space objects are defined in any -category
with homotopy pullbacks: for any pointed object of
with point , its loop space object is the homotopy pullback
of this point along itself:

(Delooping).
if is given and a homotopy pullback diagram

exists, with the point
being essentially unique, by the above has been
realized as the loop space object of .
is called delooping of X:

(Milnor–Lurie).
There is an adjoint functor

between -groups of and uniquely pointed
connected objects in which are doneted .
Where is a looping and is a delooping operations.

(Maurer-Cartan form).
For and -group in the cohesive
-topos Maurer-Cartan form is defines as

for the -valued de Rham cocycle on induced by pullback pasting: