nLab points-to-pieces transform



Cohesive \infty-Toposes

Discrete and concrete objects



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

Π \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).


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

XXʃX \flat X \longrightarrow X \longrightarrow ʃ X

(given by the composition of the \flat-counit followed by the ʃʃ-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 XX, we say that pieces have points or that the Nullstellensatz is verified.


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

XXʃX \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 *Xf !X)(f *Xf *f *f !Xf !X). (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 *f^\ast to this and postcomposing with the (f *f *)(f^\ast \dashv f_\ast)-counit is equivalent to just applying f *f^\ast, since by idempotency of \flat the counit is an isomorphism on the discrete object f *f !Xf^\ast f_! X. Therefore the points-to-pieces transformation and its adjunct are related by

(XXʃX)=f *(f *Xf !X). \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 *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 *Xf !Xf_\ast X \longrightarrow f_! X is, respectively.

If this adjunct

Xϵ Xη X \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


(pieces have points iff discrete objects are concrete)

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

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

  2. discrete objects are concrete, i.e. Xη X X\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).


Bundle equivalence and concordance

Given an ∞-group GG in a cohesive (∞,1)-topos H\mathbf{H}, with delooping BG\mathbf{B}G, then for any other object XX the ∞-groupoid H(X,BG)\mathbf{H}(X,\mathbf{B}G) is that of GG-principal ∞-bundles with equivalences between them. Alternatively one may form the internal hom [X,BG][X,\mathbf{B}G]. Applying the shape modality to this yields the \infty-groupoid H (X,BG)ʃ[X,BG]\mathbf{H}^\infty(X,\mathbf{B}G) \coloneqq ʃ [X,\mathbf{B}G] of GG-principal \infty-bundles and concordances between them. Alternatively, the flat modality applied to the internal hom is again just the external hom [X,BG]H(X,BG)\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

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

from GG-principal \infty-bundles with bundle equivalences between them, to GG-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//GX/G X//G \longrightarrow X/G

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

EG× GX*× GX=X/G. 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(H)THStab(\mathbf{H}) \hookrightarrow T\mathbf{H} the tangent cohesive (∞,1)-topos which arise from a symmetric monoidal (∞,1)-category VCMon (Cat (H))V \in CMon_\infty(Cat_\infty(\mathbf{H})) internal to H\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.


infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Last revised on March 15, 2019 at 13:13:16. See the history of this page for a list of all contributions to it.