nLab fundamental category



Category theory

Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




For directed spaces

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

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-11 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.

For simplicial sets

The left adjoint of the nerve functor N:CatSSetN:Cat \to SSet, which takes a simplicial set to a category, is sometimes called the fundamental category functor. One notation for it is τ 1\tau_1. Explicitly, for a simplicial set XX, τ 1(X)\tau_1(X) is the category freely generated by the directed graph whose vertices are 0-simplices of XX and whose edges are 1-simplices (the source and target are defined by the face maps), modulo the relations s 0(x)id xs^0(x) \sim id_x for xX 0x \in X_0 and d 1(x)d 0(x)d 2(x)d^1(x) \sim d^0(x) \circ d^2(x) for xX 2x \in X_2. Here s is^i and d id^i denote the degeneracy and face maps, respectively.

If XX is a quasicategory, then its fundamental category is equivalent to its homotopy category.

QuasiCat sSet Ho τ 1 Cat. \array{ QuasiCat &&\hookrightarrow&& sSet \\ & {}_{\mathllap{Ho}}\searrow && \swarrow_{\mathrlap{\tau_1}} \\ && Cat } \,.


  • 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.