This page discusses a cubical version of simplicial sets. Simplicial sets are described on the page here.
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The delta category Δ has:
The vertices of the simplicies are ordered by labeling them with natural numbers. |
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Δop has the same objects as Δ but the morphisms are reversed. The morphisms are now the face maps and degeneracy maps. |
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Face MapsThese maps go from dimension n+1 dimension n. That is they send a (hyper)cube to one of its faces.
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Degeneracy MapsThese maps go in the opposite direction and increase the dimension. That is they add an index. |
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| Order preserving map from simplex to complex: |  |
Gluing
We have a contravarient functor which picks out faces but the faces are not all separate so how do we encode this additional structure?
The edges are connected when they share a common sub-face.
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In order to understand this we need the concepts of fibration and lifting as described on the page here. |
Paths
What happens when there are a sequence of edges following each other? (the target of the first is equal to the source of the second and so on). Can the edge in the interval Is this the same thing as requiring composition? In cubical type theory equalities are represented by a path, in a given space, between equal terms. |
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match the whole path? |
In order to understand this we need the concepts of cofibration and extentions as described on the page here. |
Fibration and Co-fibration
Homotopy has the concept of:
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a fibration which has the lifting property.
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a co-fibration which has the extension property -extension is dual to lift.
Fibration |
Co-fibration (Extension Property) |
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| Homotopy | Fibration |
Co-fibration (see page here) |
Combinatorics |
Kan fibration (see page here) |
Kan extension (see page here) |
Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.
Further Information
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