Linked spaces and exit paths
Ödül Tetik
TL;DR
This work constructs a combinatorial exit-path ∞-category $\mathbf{EX}$ for linked spaces, providing a fully faithful embedding of the ∞-category of linked spaces into $\mathbf{Cat}_{\infty}$ and extending to linked ∞-categories. For depth-1 conically smooth spaces, the EX construction recovers the Lurie–MacPherson exit-path model, yielding a direct, self-contained handle on stratified homotopy types via spans $M\leftarrow L \rightarrow N$ with $\pi$ a fibration and $\iota$ a cofibration. The paper establishes explicit foundations for modeling exit paths combinatorially, derives counterexamples to previously proposed finiteness criteria for characterizing exit-path EPCs, and sets up a program to classify conically smooth bundles over depth-1 posets through span maps into $\mathrm{BO}(n,m)$ and $\mathrm{BG}(n,m)$, with a plan for a tangential theory in a sequel. Overall, it provides a robust, explicit bridge from non-stratified span data to stratified homotopy types, with immediate consequences for bundle theory and higher-categorical approaches to stratified geometry.
Abstract
Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrowπ L\xrightarrowιN$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.