nLab points-to-pieces transform

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.

If this adjunct

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

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 $E G \to \ast$

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

(see at global equivariant homotopy theory this prop.)

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.

Related concepts

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

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

References

• William Lawvere, Matías Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, Theory and Applications of Categories, Vol. 30, 2015, No. 26, pp 909-932. (TAC)

• Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)