structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
Given a cohesive (∞,1)-topos $\mathbf{H}$, then then shape $ʃ X$ of any object $X$ behaves like the path ∞-groupoid of $X$. For some $\mathbf{H}$ this is true verbatim, in that the shape operation is represented by a geometric singular simplicial complex
where on the right we have a homotopy colimit over internal homs from a cohesive incarnation of the n-simplex into $X$.
This is true for instance for the case
This is due to (BEBP, see Pavlov, theorem 0.2). For $X$ a stable homotopy type in the tangent (∞,1)-topos $T Smooth\infty Grpd$ this was observed in Bunke-Nikolaus-Voelkl 13
Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)
Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves (arXiv:1912.10544)
Dmitri Pavlov, Structured Brown representability via concordance (pdf)
Last revised on July 23, 2020 at 04:33:45. See the history of this page for a list of all contributions to it.