metadata
pipeline_tag: text-classification
widget:
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Dijkstra's algorithm is an algorithm for finding the paths between nodes
in a weighted graph
example_title: Djikstra
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Proposition 1. The sum of the differential weight d on the Dowker complex
D(X, Y, R) is the number of elements of Y .
example_title: Proposition
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Proof. This follows from showing that t is order-reversing in the
following way: if σ ⊆ τ, then t(σ) ≥ t(τ).
example_title: Proof 1
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Theorem 1. Given the Dowker complex D(X, Y, R) and differential weight d,
one can reconstruct R up to a bijection on Y .
example_title: Theorem
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The Dowker complex D(X,Y,R) is a functor between an appropriately con-
structed category of relations and the category of abstract simplicial
complexes. We prove this fact in Theorem 3 along with a few other
observations.
example_title: Basic
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Lemma 1. Let f : X → Y be a simplicial map. For every pair of simplices σ,
τ of X satisfying σ ⊆ τ, their images in Y satisfy f(σ) ⊆ f(τ).
example_title: Theorem
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Since f is a simplicial map, then f(σ) is a simplex of Y and so is f(τ).
If σ ⊆ τ, then every vertex v of σ is also a vertex of τ. By the
definition of simplicial maps, f(v) is a vertex of both f(σ) and f(τ).
Conversely, every vertex of f(σ) is the image of some vertex w of σ. □
example_title: Proof without keyword
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The Dowker complex defined in Definition 2 is a covariant functor D : Rel
→ Asc.
example_title: th without keyword