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Sunday, April 22, 2018

A functor from entry paths to the nerve of simplicial complexes

Fix nZ>0 and let X=Rann(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:XSC. 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:Sn1Sn, and degeneracy maps di:Sn+1Sk, 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][n1])((|Δn1|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={(S0f1fnSn) : SiSC, fi are simplicial maps},(si:[n][n1])((S0f1fn1Sn1)(S0f1fiSiidSifi+1fn1Sn1)),(di:[n][n+1])(i=0:(S0Sn+1)(S1f2fn+1Sn+1)0<i<n:(S0Sn+1)(S0f1fi1Si1fi+1fiSi+1fi+2fn+1Sn+1)i=n:(S0Sn+1)(S0f1fnSn)).
 Define F on k-simplices as F(γ:|Δk|Rann(M)×R>0)=(˜h(γ(1,0,,0))(˜hγsks2)(|Δ1|)(˜hγsk2s0)(|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 |Δn1| 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 |Δn1|. Similarly, observing that the image of the shortest path vi1vivi+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 vi1vi+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:XB 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 BSC (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 XAA.

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