nLab stratified homotopy hypothesis

Idea

Where the homotopy hypothesis is the assertion that ∞-groupoids are equivalent to topological spaces (considered modulo weak homotopy equivalence), in AFR15 the authors construct an equivalence for (∞,1)-categories. They do so in terms of what are called striation sheaves, which are sheaves on conically smooth stratified spaces satisfying a certain descent condition.

The construction relies on a fully faithful embedding of conically smooth stratified spaces, and conically smooth maps among them, into $(\infty, 1)$-categories via the exit-path functor, which maps stratified spaces to paths within them that once they leave a stratum do not re-enter it.

In Haine18 the author presents a completely self-contained proof of a precise form of the stratified homotopy hypothesis.

References

• David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv

• Peter J. Haine, On the homotopy theory of stratified spaces (arXiv:1811.01119)