points-to-pieces transform


Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory



Presentation over a site

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

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?


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 (Unknown characterUnknown characterʃ \dashv \flat) its (shape modality \dashv flat modality)-adjunction of def. \ref{AdjointTriple}, then the natural transformation

XXUnknown characterUnknown characterX \flat X \longrightarrow X \longrightarrow ʃ X

(given by the composition of the \flat-counit followed by the Unknown characterUnknown characterʃ-unit) may be called the transformation from points to their pieces or the points-to-pieces-transformation, for short.


The (f *f *)(f^\ast \dashv f_\ast)-adjunct of the transformation from pieces to points, def. 1,

XXUnknown characterUnknown characterX \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

(XXUnknown characterUnknown characterX)=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.

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). 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).


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)Unknown characterUnknown character[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 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 in an equivalence.


tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive Unknown characterUnknown character discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 }


  • 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)

Revised on July 18, 2015 06:19:24 by Urs Schreiber (