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.

Idea

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.

Definition

Remark

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

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

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

Remark

The underlying adjoint triple $\Pi \dashv Disc \dashv \Gamma$ 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.

Examples

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

$$\array{ cohesion && SmoothSuper\infty Grpds &\stackrel{\overset{\Pi^s}{\longrightarrow}}{\stackrel{\overset{Disc^s}{\leftrightarrow}}{\stackrel{\overset{\Gamma^s}{\longrightarrow}}{\underset{coDisc^s}{\leftarrow}}}}& Super \infty Grpds \\ &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} && {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}} \\ cohesion && Smooth \infty Grpds &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}& \infty Grpds \\ \\ && diff.\;cohesion && inf.\;cohesion }$$

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.

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

See also at differential cohesion and idelic structure.

Goodwillie-tangent cohesion

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

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

Related concepts

• bireflective subcategory

• classical modality

• linear homotopy type theory

• differential cohesion

• differential cohesion and idelic structure

References

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

• William Lawvere, Some Thoughts on the Future of Category Theory (1991)

The above examples of infinitesimal cohesion appear in

• Urs Schreiber, differential cohomology in a cohesive topos (2013)

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)