This entry is about a variant of the concept of cohesive (∞,1)-topos. The definition here expresses an intuition not unrelated to that at infinitesimally cohesive (∞,1)-presheaf on E-∞ rings but the definitions are unrelated and apply in somewhat disjoint contexts.
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A cohesive (∞,1)-topos is infinitesimal cohesive if all its objects behave like built from infinitesimally thickened geometrically discrete ∞-groupoids in that they all have “precisely one point in each cohesive piece”.
(There is an evident version of an infinitesimally cohesive 1-topos. In (Lawvere 07, def. 1) such is referred to as a “quality type”. A hint of this seems to be also in (Lawvere 91, p. 9)).
A cohesive (∞,1)-topos $\mathbf{H}$ with its shape modality $\dashv$ flat modality $\dashv$ sharp modality denoted $ʃ \dashv \flat \dashv \sharp$ is infinitesimal cohesive if the canonical points-to-pieces transform is an equivalence
The underlying adjoint triple $\Pi \dashv Disc \dashv \Gamma$ in the case of infinitesimal cohesion is an ambidextrous adjunction. Such a localization is called a “quintessential localization” in (Johnstine 96).
Given an (∞,1)-site with a zero object, then the (∞,1)-presheaf (∞,1)-topos over it is infinitesimally cohesive. This class of examples contains the following ones.
super ∞-groupoids are infinitesimally cohesive over geometrically discrete ∞-groupoids, while smooth super ∞-groupoids are cohesive over super ∞-groupoids and differentially cohesive over smooth ∞-groupoids
synthetic differential ∞-groupoids are cohesive over generalized formal moduli problems/L-∞ algebras (generalized meaning without the condition of vanishing on the point and of without the condition of being infinitesimally cohesive sheaves in Lurie's sense) which in turn are infinitesimally cohesive over geometrically discrete ∞-groupoids.
See also at differential cohesion and idelic structure.
A tangent (∞,1)-topos $T \mathbf{H}$ is infinitesimally cohesive over $\mathbf{H}$:
The notion of infinitesimal cohesion appears under the name “quality type” in def. 1 of
An earlier hint of the same notion seems to be that on the bottom of p. 9 in
The above examples of infinitesimal cohesion appear in
Localization by an ambidextrous adjunction is also discussed in