structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
typical contexts
In a cohesive (∞,1)-topos $\mathbf{H}$, the canonical natural transformation
from the flat modality to the shape modality may be thought of as sending “points to the pieces in which they sit”.
Notice the existence of the following canonical natural transformations induced from the structure of a cohesive topos (a special case of the construction at unity of opposites).
Given a cohesive topos $\mathcal{E}$ with ($ʃ \dashv \flat$) its (shape modality $\dashv$ flat modality)-adjunction of def. \ref{AdjointTriple}, then the natural transformation
(given by the composition of the $\flat$-counit followed by the $ʃ$-unit) may be called the transformation from points to their pieces or the points-to-pieces-transformation, for short.
The $(f^\ast \dashv f_\ast)$-adjunct of the transformation from pieces to points, def. 1,
is (by the rule of forming right adjuncts by first applying the right adjoint functor and then precomposing with the unit and by the fact that the adjunct of a unit is the identity) the map
Observe that going backwards by applying $f^\ast$ to this and postcomposing with the $(f^\ast \dashv f_\ast)$-counit is equivalent to just applying $f^\ast$, since by idempotency of $\flat$ the counit is an isomorphism on the discrete object $f^\ast f_! X$. Therefore the points-to-pieces transformation and its adjunct are related by
Observe then finally that since $f^\ast$ is a full and faithful left and right adjoint, the points-to-pieces transform is an epimorphism/isomorphism/monomorphism precisely if its adjunct $f_\ast X \longrightarrow f_! X$ is, respectively.
For a cohesive 1-topos, if the pieces-to-points transform is an epimorphism then there is Aufhebung of the initial opposition $(\emptyset \dashv \ast)$ in that $\sharp \emptyset \simeq \emptyset$ (Lawvere-Menni 15, lemma 4.1). Conversely, if the base topos is a Boolean topos, then this Aufhebung implies that the pieces-to-points transform is epi (Lawvere-Menni 15, lemma 4.2).
Given an ∞-group $G$ in a cohesive (∞,1)-topos $\mathbf{H}$, with delooping $\mathbf{B}G$, then for any other object $X$ the ∞-groupoid $\mathbf{H}(X,\mathbf{B}G)$ is that of $G$-principal ∞-bundles with equivalences between them. Alternatively one may form the internal hom $[X,\mathbf{B}G]$. Applying the shape modality to this yields the $\infty$-groupoid $\mathbf{H}^\infty(X,\mathbf{B}G) \coloneqq ʃ [X,\mathbf{B}G]$ of $G$-principal $\infty$-bundles and concordances between them. Alternatively, the flat modality applied to the internal hom is again just the external hom $\flat [X,\mathbf{B}G] \simeq \mathbf{H}(X,\mathbf{B}G)$.
In conclusion, in this situation the points-to-pieces transform is the canonical map
from $G$-principal $\infty$-bundles with bundle equivalences between them, to $G$-principal $\infty$-bundles with concordances between them.
In a tangent cohesive (∞,1)-topos on stable homotopy types the points-to-pieces transform is one stage in a natural hexagonal long exact sequence, the differential cohomology diagram. See there for more.
Applied to stable homotopy types in $Stab(\mathbf{H}) \hookrightarrow T\mathbf{H}$ the tangent cohesive (∞,1)-topos which arise from a symmetric monoidal (∞,1)-category $V \in CMon_\infty(Cat_\infty(\mathbf{H}))$ internal to $\mathbf{H}$ under internal algebraic K-theory of a symmetric monoidal (∞,1)-category, the points-to-pieces transform interprets as the comparison map between algebraic and topological K-theory. See there for more
In infinitesimal cohesion the points-to-pieces transform in an equivalence.
(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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$