infinitesimal cohesive (infinity,1)-topos

**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 formal moduli problems/L-∞ algebras which in turn are infinitesimally cohesive over geometrically discrete ∞-groupoids.

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

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 January 9, 2014 21:32:57
by Urs Schreiber
(89.204.137.152)