nLab infinitesimal cohesive (infinity,1)-topos


This entry is about a variant of the concept of cohesive (∞,1)-topos. The definition here expresses an intuition not unrelated to that at infinitesimally cohesive (∞,1)-presheaf on E-∞ rings but the definitions are unrelated and apply in somewhat disjoint contexts.


Cohesive \infty-Toposes

Synthetic differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



A cohesive (∞,1)-topos is infinitesimal cohesive if all its objects behave as though built from infinitesimally thickened geometrically discrete ∞-groupoids in that they all have “precisely one point in each cohesive piece”.

(There is an evident version of an infinitesimally cohesive 1-topos. In (Lawvere 07, def. 1) such is referred to as a “quality type”. A hint of this seems to be also in (Lawvere 91, p. 9)).

Infinitesimal cohesion may also be defined relative to any (∞,1)-topos.



A cohesive (∞,1)-topos H\mathbf{H} with its shape modality \dashv flat modality \dashv sharp modality denoted ʃʃ \dashv \flat \dashv \sharp is infinitesimal cohesive if the canonical points-to-pieces transform is an equivalence

ʃ \flat \stackrel{\simeq}{\longrightarrow} ʃ

in which case all three modal operators collapse to a single “classical modality\natural of a bireflective subcategory.


The underlying adjoint triple ΠDiscΓ\Pi \dashv Disc \dashv \Gamma in the case of infinitesimal cohesion is an ambidextrous adjunction. Such a localization is called a “quintessential localization” by Johnstone (1996), see at bireflective subcategory.


Given an (∞,1)-site with a zero object, then the (∞,1)-presheaf (∞,1)-topos over it is infinitesimally cohesive. This class of examples contains the following ones.

Super \infty-groupoids

super ∞-groupoids are infinitesimally cohesive over geometrically discrete ∞-groupoids, while smooth super ∞-groupoids are cohesive over super ∞-groupoids and differentially cohesive over smooth ∞-groupoids

cohesion SmoothSuperGrpds coDisc sΓ sDisc sΠ s SuperGrpds Disc inf Π inf Disc inf Π inf cohesion SmoothGrpds coDiscΓDiscΠ Grpds diff.cohesion inf.cohesion \array{ cohesion && SmoothSuper\infty Grpds &\stackrel{\overset{\Pi^s}{\longrightarrow}}{\stackrel{\overset{Disc^s}{\leftrightarrow}}{\stackrel{\overset{\Gamma^s}{\longrightarrow}}{\underset{coDisc^s}{\leftarrow}}}}& Super \infty Grpds \\ &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} && {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} \\ cohesion && Smooth \infty Grpds &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}& \infty Grpds \\ \\ && diff.\;cohesion && inf.\;cohesion }

Formal moduli problems/ strong homotopy Lie algebras

synthetic differential ∞-groupoids are cohesive over generalized formal moduli problems/L-∞ algebras (generalized meaning without the condition of vanishing on the point and of without the condition of being infinitesimally cohesive sheaves in Lurie's sense) which in turn are infinitesimally cohesive over geometrically discrete ∞-groupoids.

cohesion SynthDiffGrpds coDisc iΓ iDisc iΠ i L Alg gen op Disc inf Π inf Disc inf Π inf cohesion SmoothGrpds coDiscΓDiscΠ Grpds diff.cohesion inf.cohesion \array{ cohesion && SynthDiff\infty Grpds &\stackrel{\overset{\Pi^i}{\longrightarrow}}{\stackrel{\overset{Disc^i}{\leftrightarrow}}{\stackrel{\overset{\Gamma^i}{\longrightarrow}}{\underset{coDisc^i}{\leftarrow}}}}& L_\infty Alg^{op}_{gen} \\ &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} && {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} \\ cohesion && Smooth \infty Grpds &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}& \infty Grpds \\ && diff.\;cohesion && inf.\;cohesion }

See also at differential cohesion and idelic structure.

Goodwillie-tangent cohesion

A tangent (∞,1)-topos THT \mathbf{H} is infinitesimally cohesive over H\mathbf{H}:

Stab(H) coDisc seqΓ seqDisc seqLΠ seq Stab(Grpd)Spectra T *H T *Grpd incl incl H Ω totd TH coDisc seqΓ seqDisc seqLΠ seq TGrpd base 0 base 0 H coDiscΓDiscΠ Grpd. \array{ && Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \simeq Spectra \\ && \simeq && \simeq \\ && T_\ast \mathbf{H} && T_\ast \infty Grpd \\ && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}& T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} \\ && \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd } \,.


The notion of infinitesimal cohesion appears under the name “quality type” in def. 1 of

  • William Lawvere, Axiomatic cohesion, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

An earlier hint of the same notion seems to be that on the bottom of p. 9 in

The above examples of infinitesimal cohesion appear in

Localization by an ambidextrous adjunction is also discussed in

  • Peter Johnstone, Remarks on quintessential and persistent localizations, Theory and Applications of Categories, Vol. 2, No. 8, 1996, pp. 90–99 (TAC)

Last revised on March 29, 2023 at 12:11:38. See the history of this page for a list of all contributions to it.