structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
For $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd$ a cohesive (∞,1)-topos, we call the action of the shape modality
the geometric realization functor. For $X \in \mathbf{H}$ any object, hence any cohesive ∞-groupoid, $\vert \Pi(X)\vert$ is its geometric realization.
Notice that $\Pi(X)$ is the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos and $\vert - \vert :$ ∞Grpd $\to$ Top is the “homotopy hypothesis” equivalence of (∞,1)-categories.
See at cohesive (∞,1)-topos -- structures the section Geometric homotopy and Galois theory.
In $\mathbf{H} =$ ETop∞Grpd the geometric realization of cohesive $\infty$-groupoids subsumes the geometric realization of simplicial topological spaces (see there for details).
of categories, of simplicial topological spaces, of cohesive ∞-groupoids