nLab
fundamental category

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Category theory

category theory

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Homotopy theory

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Paths and cylinders

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Idea

For directed spaces

In generalization to how a topological space X has a fundamental groupoid whose morphisms are homotopy-classes of paths in X and whose composition operation is the concatenation of paths, a directed space has a fundamental category whose morphisms are directed paths in X.

For stratified spaces

A stratified space? has a ‘fundamental n-category with duals’, which generalizes the fundamental n-groupoid of a plain old space. When a path crosses a codimension-1 stratum, “something interesting happens” – i.e., a catastrophe. So, we say such a path gives a noninvertible morphism. The idea is that going along such a path and then going back is not “the same” as having stayed put. So, going back along such a path is not its inverse, just its dual?.

See Café discussion

References

  • Marco Grandis, Directed algebraic topology, categories and higher categorries (pdf)
Revised on December 6, 2010 23:14:15 by Urs Schreiber (131.211.233.37)
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