2015/02/05 by Ayala, David, Francis, John, Rozenblyum, Nick · 1 citation
#32S60 #55N40 #57N80 #57P05 #57R40 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1502.01713
We show that conically smooth stratified spaces embed fully faithfully into ∞-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each ∞-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include ℝ1-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in \mathbbA1-homotopy theory. In this way, we identify ∞-categories as striation sheaves, which are those sheaves on conically smooth stratified spaces satisfying the indicated descent. We use this identification to construct by hand two remarkable examples of ∞-categories: B\sf un, an ∞-category classifying constructible bundles; and E\sf xit, the absolute exit-path ∞-category. These constructions are deeply premised on stratified geometry, the key geometric input being a characterization of conically smooth stratified maps between cones and the existence of pullbacks for constructible bundles.