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 $X$ 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 $X$.
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 and paper it inspired, J. Woolf Transversal homotopy theory.
The left adjoint of the nerve functor $N:Cat \to SSet$, which takes a simplicial set to a category, is sometimes called the fundamental category functor. One notation for it is $\tau_1$. Explicitly, for a simplicial set $X$, $\tau_1(X)$ is the category freely generated by the directed graph whose vertices are 0-simplices of $X$ and whose edges are 1-simplices (the source and target are defined by the face maps), modulo the relations $s^0(x) \sim id_x$ for $x \in X_0$ and $d^1(x) \sim d^0(x) \circ d^2(x)$ for $x \in X_2$. Here $s^i$ and $d^i$ denote the degeneracy and face maps, respectively.
If $X$ is a quasicategory, then its fundamental category is equivalent to its homotopy category.
fundamental category, fundamental (∞,1)-category
Marco Grandis, Directed algebraic topology, categories and higher categories (pdf)
Andre Joyal, Myles Tierney, Notes on simplicial homotopy theory, 2008, citeseerx
J. Woolf, Transversal homotopy theory, Theory and applications of categories, Vol 24, Issue 7, pp 148-178, 2010. (arXiv:0910.3322)
J. Woolf, The fundamental category of a stratified space, Journal of Homotopy and Related Structures, Vol 4, Issue 1 pp 359-387, 2009. (arXiv:0811.2580)
Last revised on December 27, 2021 at 15:49:25. See the history of this page for a list of all contributions to it.