Fix n∈Z>0 and let X=Ran⩽n(M)×R>0 for M a compact, connected PL manifold embedded in RN. Take ˜h:X→(B,⩽) the conical stratifying map from a previous post (``Conical stratifications via semialgebraic sets," 2018-04-16) compatible with the natural stratification h:X→SC. The goal of this post is to construct a functor F:SingB(X)→N(SC) from the ∞-category of entry paths that encodes the structure of X.
Recall that a simplicial set is a functor, an element of Fun(Δop,Set). A simplicial set S is defined by its collection of n-simplices Sn, its face maps si:Sn−1→Sn, and degeneracy maps di:Sn+1→Sk, for all i=0,…,n. For the first simplicial set of interest in this post, we have
SingB(X)n=HomBTop(|Δn|,X),(si:[n]→[n−1])↦((|Δn−1|→X)↦(|Δn|→X)collapses ith with (i+1)th vertex, then maps as source)(di:[n]→[n+1])↦((|Δn+1|→X)↦(|Δn|→X)maps as ith face of source map)
We write HomBTop for the subset of HomTop that respects the stratification B in the context of entry paths. For the second simplicial set, the nerve, we have
N(SC)n={(S0f1→⋯fn→Sn) : Si∈SC, fi are simplicial maps},(si:[n]→[n−1])↦((S0f1→⋯fn−1→Sn−1)↦(S0f1→⋯fi→Siid→Sifi+1→⋯fn−1→Sn−1)),(di:[n]→[n+1])↦(i=0:(S0⋯Sn+1)↦(S1f2→⋯fn+1→Sn+1)0<i<n:(S0⋯Sn+1)↦(S0f1→⋯fi−1→Si−1fi+1∘fi→Si+1fi+2→⋯fn+1→Sn+1)i=n:(S0⋯Sn+1)↦(S0f1→⋯fn→Sn)).
Define F on k-simplices as F(γ:|Δk|→Ran⩽n(M)×R>0)=(˜h(γ(1,0,…,0))(˜h∘γ∘sk∘⋯∘s2)(|Δ1|)→⋯(˜h∘γ∘sk−2∘⋯∘s0)(|Δ1|)→˜h(γ(0,…,0,1))). A morphism in SingB(X) is a composition of face maps si and degeneracy maps di, so F must satisfy the commutative diagrams
for all si, di. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.
Commutativity of the diagram on the left is immediate, as considering a simplex |Δn−1| as the ith face of a larger simplex |Δn| is the same as adding a step that is the identity map in the Hamiltonian path of vertices of |Δn−1|. Similarly, observing that the image of the shortest path vi−1→vi→vi+1 in |Δn+1|, for vi=(0,…,0,1,0,…,0) the ith standard basis vector, induced by an element γ:|Δn+1|→X in SingB(X)n+1, is homotopic to the image of the shortest path vi−1→vi+1 shows that the diagram on the right commutes. Since F is a natural transformation between the two functors SingB(X) and N(SC), it is a functor on the functors as simplicial sets.
Remark: The particular choice of X did not seem to play a large role in the arguments above. However, the stratifying map ˜h:X→B has image sitting inside SC, the nerve of which is the target of F, and every morphism in SingB(X) can be interpreted as a relation in B⊆SC (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor SingA(X)→N(A′) may exist for a stratified space X→A⊆A′.
Recall that a simplicial set is a functor, an element of Fun(Δop,Set). A simplicial set S is defined by its collection of n-simplices Sn, its face maps si:Sn−1→Sn, and degeneracy maps di:Sn+1→Sk, for all i=0,…,n. For the first simplicial set of interest in this post, we have
SingB(X)n=HomBTop(|Δn|,X),(si:[n]→[n−1])↦((|Δn−1|→X)↦(|Δn|→X)collapses ith with (i+1)th vertex, then maps as source)(di:[n]→[n+1])↦((|Δn+1|→X)↦(|Δn|→X)maps as ith face of source map)
We write HomBTop for the subset of HomTop that respects the stratification B in the context of entry paths. For the second simplicial set, the nerve, we have
N(SC)n={(S0f1→⋯fn→Sn) : Si∈SC, fi are simplicial maps},(si:[n]→[n−1])↦((S0f1→⋯fn−1→Sn−1)↦(S0f1→⋯fi→Siid→Sifi+1→⋯fn−1→Sn−1)),(di:[n]→[n+1])↦(i=0:(S0⋯Sn+1)↦(S1f2→⋯fn+1→Sn+1)0<i<n:(S0⋯Sn+1)↦(S0f1→⋯fi−1→Si−1fi+1∘fi→Si+1fi+2→⋯fn+1→Sn+1)i=n:(S0⋯Sn+1)↦(S0f1→⋯fn→Sn)).
Define F on k-simplices as F(γ:|Δk|→Ran⩽n(M)×R>0)=(˜h(γ(1,0,…,0))(˜h∘γ∘sk∘⋯∘s2)(|Δ1|)→⋯(˜h∘γ∘sk−2∘⋯∘s0)(|Δ1|)→˜h(γ(0,…,0,1))). A morphism in SingB(X) is a composition of face maps si and degeneracy maps di, so F must satisfy the commutative diagrams
for all si, di. Since the maps are unwieldy when in coordinates, we opt for heuristic arguments, neglecting to trace out notation-heavy diagrams.
Commutativity of the diagram on the left is immediate, as considering a simplex |Δn−1| as the ith face of a larger simplex |Δn| is the same as adding a step that is the identity map in the Hamiltonian path of vertices of |Δn−1|. Similarly, observing that the image of the shortest path vi−1→vi→vi+1 in |Δn+1|, for vi=(0,…,0,1,0,…,0) the ith standard basis vector, induced by an element γ:|Δn+1|→X in SingB(X)n+1, is homotopic to the image of the shortest path vi−1→vi+1 shows that the diagram on the right commutes. Since F is a natural transformation between the two functors SingB(X) and N(SC), it is a functor on the functors as simplicial sets.
Remark: The particular choice of X did not seem to play a large role in the arguments above. However, the stratifying map ˜h:X→B has image sitting inside SC, the nerve of which is the target of F, and every morphism in SingB(X) can be interpreted as a relation in B⊆SC (both were necessary for the commutativity of the diagrams). Hence it is not unreasonable to expect a similar functor SingA(X)→N(A′) may exist for a stratified space X→A⊆A′.
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