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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2012
   • (edited Dec 2nd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have started making notes at _[[differential cohesion]]_ on the axiomatic formulation of * _[cotangent bundles](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CotangentBundles)_ * _[critical loci](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CriticalLocus)_ . So far just the bare basics. To be expanded... The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type $X$, the inclusion of the [formally étale maps](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#FormallySmoothEtaleUnramified) into $X$ into the full slice over $X$ is not only reflective but also co-reflective (since the formally étale maps are the [[Pi-closed morphisms|Pi_inf-closed morphisms]] with the infinitesimal path groupoid functor / de Rham space functor $\Pi_{inf}$ being a left adjoint). This means that for $G$ any differential cohesive $\infty$-group with the corresponding de Rham coefficient object $\flat_{dR}\mathbf{B}G$ (the universal moduli for flat $\mathfrak{g}$-valued differential forms), the sheaf of flat $\mathfrak{g}$-valued forms over any $X$ is given by the sections of the coreflection of the product projection $X \times \flat_{dR}\mathbf{B}G \to X$ into the formally étale morphisms into $X$.

   I have started making notes at differential cohesion on the axiomatic formulation of

   • cotangent bundles

   • critical loci .

   So far just the bare basics. To be expanded…

   The basic observation (easy in itself, but fundamental for the concept formation) is that for any differential cohesive homotopy type $X$, the inclusion of the formally étale maps into $X$ into the full slice over $X$ is not only reflective but also co-reflective (since the formally étale maps are the Pi_inf-closed morphisms with the infinitesimal path groupoid functor / de Rham space functor $\Pi_{inf}$ being a left adjoint).

   This means that for $G$ any differential cohesive $\infty$-group with the corresponding de Rham coefficient object $\flat_{dR}\mathbf{B}G$ (the universal moduli for flat $\mathfrak{g}$-valued differential forms), the sheaf of flat $\mathfrak{g}$-valued forms over any $X$ is given by the sections of the coreflection of the product projection $X \times \flat_{dR}\mathbf{B}G \to X$ into the formally étale morphisms into $X$.