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