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.

**structures in a cohesive (∞,1)-topos**

**infinitesimal cohesion?**

A cohesive (∞,1)-topos is *infinitesimal cohesive* if all its objects behave like 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)).

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

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

The underlying adjoint triple $\Pi \dashv Disc \dashv \Gamma$ in the case of infinitesimal cohesion is an ambidextrous adjunction.

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

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

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*.

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

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

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

Revised on November 11, 2014 08:11:19
by Urs Schreiber
(86.212.18.175)