nLab
fundamental category

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 and paper it inspired, J. Woolf Transversal homotopy theory.

For simplicial sets

The left adjoint of the nerve functor $N:\mathrm{Cat}\to \mathrm{SSet}$, which takes a simplicial set to a category, is sometimes called the fundamental category functor. One notation for it is ${\tau }_1$. If $X$ is a quasicategory, then its fundamental category is equivalent to its homotopy category.

$$\begin{array}{ccccc}\mathrm{QuasiCat}& & \hookrightarrow & & \mathrm{sSet}\\ & {}_{\mathrm{Ho}}\searrow & & {\swarrow }_{{\tau }_1}\\ & & \mathrm{Cat}\end{array}.$$\array{ QuasiCat &&\hookrightarrow&& sSet \\ & {}_{\mathllap{Ho}}\searrow && \swarrow_{\mathrlap{\tau_1}} \\ && Cat } \,.

Related concepts

• fundamental groupoid, fundamental ∞-groupoid

• fundamental category, fundamental (∞,1)-category

• directed homotopy theory

References

• Marco Grandis, Directed algebraic topology, categories and higher categories (pdf)