homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
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In generalization to how a topological space has a fundamental groupoid whose morphisms are homotopy-classes of paths in and whose composition operation is the concatenation of paths, a directed space has a fundamental category whose morphisms are directed paths in .
A stratified space has a ‘fundamental -category with duals’, which generalizes the fundamental -groupoid of a plain old space. When a path crosses a codimension- 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 and paper it inspired, J. Woolf Transversal homotopy theory.
The left adjoint of the nerve functor , which takes a simplicial set to a category, is sometimes called the fundamental category functor. One notation for it is . Explicitly, for a simplicial set , is the category freely generated by the directed graph whose vertices are 0-simplices of and whose edges are 1-simplices (the source and target are defined by the face maps), modulo the relations for and for . Here and denote the degeneracy and face maps, respectively.
If is a quasicategory, then its fundamental category is equivalent to its homotopy category.
fundamental category, fundamental (∞,1)-category
Marco Grandis, section 3 of:: Directed Algebraic Topology, New Mathematical Monographs 13, Cambridge Univ. Press (2009) [doi:10.1017/CBO9780511657474, pdf]
Marco Grandis, Directed algebraic topology, categories and higher categories [pdf]
J. Woolf, Transversal homotopy theory, Theory and applications of categories 24 7 (2010) 148-178 [arXiv:0910.3322]
J. Woolf, The fundamental category of a stratified space, Journal of Homotopy and Related Structures 4 1 (2009) 359-387 [arXiv:0811.2580]
Last revised on June 17, 2025 at 17:28:21. See the history of this page for a list of all contributions to it.