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.
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 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.
See also at differential cohesion and idelic structure.
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
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)
Revised on November 26, 2014 20:31:21
by Urs Schreiber