infinitesimal cohesive (infinity,1)-topos
Synthetic differential geometry
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)).
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
Formal moduli problems/ strong homotopy Lie algebras
synthetic differential ∞-groupoids are cohesive over formal moduli problems/L-∞ algebras which in turn are infinitesimally cohesive over geometrically discrete ∞-groupoids.
A tangent (∞,1)-topos is infinitesimally cohesive over :
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