# nLab points-to-pieces transform

Contents

### Context

#### Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

## Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

## Structures with infinitesimal cohesion

infinitesimal cohesion?

# Contents

## Idea

In a cohesive (∞,1)-topos $\mathbf{H}$, the canonical natural transformation

$\flat \to \Pi$

from the flat modality to the shape modality may be thought of as sending “points to the pieces in which they sit”.

## Definition and basic properties

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).

###### Definition

Given a cohesive topos $\mathcal{E}$ with ($ʃ \dashv \flat$) its (shape modality $\dashv$ flat modality)-adjunction, then the natural transformation

$\flat X \longrightarrow X \longrightarrow ʃ X$

(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.

If this is an epimorphism for all $X$, we say that pieces have points or that the Nullstellensatz is verified.

###### Remark

The $(f^\ast \dashv f_\ast)$-adjunct of the transformation from pieces to points, def. ,

$\flat X \longrightarrow X \longrightarrow ʃ X$

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

$(f_\ast X \longrightarrow f_! X) \coloneqq \left( f_\ast X \longrightarrow f_\ast f^\ast f_! X \stackrel{\simeq}{\longrightarrow} f_!X \right) \,.$

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

$\left( \flat X \longrightarrow X \longrightarrow ʃ X \right) = f^\ast \left( f_\ast X \longrightarrow f_! X \right).$

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.

$\flat X \overset{\epsilon^{\flat}}{\longrightarrow} X \overset{\eta^\sharp}{\longrightarrow} \sharp X$

is a monomorphism, we say that discrete objects are concrete.

## Relation to points to co-pieces

###### Proposition

(pieces have points iff discrete objects are concrete)

For a cohesive topos $\mathbf{H}$, the the following two conditions are equivalent:

1. pieces have points, i.e. $\flat X \to X \to ʃ X$ is an epimorphism for all $X \in \mathbf{H}$;

2. discrete objects are concrete, i.e. $\flat X \overset{ \eta^{\sharp}_{\flat X} }{\longrightarrow} \sharp \flat X$ is a monomorphism.

See at cohesive topos this prop..

## Relation to Aufhebung of the initial opposition

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, see also Shulman 15, section 3). Conversely, if the base topos is a Boolean topos, then this Aufhebung implies that the pieces-to-points transform is an epimorphism (Lawvere-Menni 15, lemma 4.2).

## Examples

### Bundle equivalence and concordance

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

$\mathbf{H}(X,\mathbf{B}G) \longrightarrow \mathbf{H}^\infty(X,\mathbf{B}G)$

from $G$-principal $\infty$-bundles with bundle equivalences between them, to $G$-principal $\infty$-bundles with concordances between them.

### In global equivariant homotopy theory

In global equivariant homotopy theory an incarnation of the points to pieces transform is the comparison map from homotopy quotients to ordinary quotients

$X//G \longrightarrow X/G$

which in terms of the Borel construction is induced by the map $E G \to \ast$

$E G \times_G X \longrightarrow \ast \times_G X = X/G \,.$

### In tangent cohesion: the differential cohomology diagram

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.

### Comparison map between algebraic and topological K-theory

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

In infinitesimal cohesion the points-to-pieces transform is an equivalence.

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$