nLab fundamental 2-groupoid

Idea

Given a well-bahaved topological space $X$, its fundamental 2-groupoid is the 2-groupoid whose

• objects are the points (elements) of $X$;

• 1-morphisms are continuous paths $[0,1] \to X$ (or thin homotopy classes of such, to yield a strict 2-groupoid);

• 2-morphisms are homotopies between such paths, fixing their endpoints;

• composition is given by concatenation of paths and homotopies.

This may equivalently be understood as:

• the 2-groupoidal-categorification of the notion of the fundamental groupoid;

• the homotopy 2-category of the fundamental $\infty$-groupoid;

At least if $X$ admits the structure of a CW-complex, its fundamental 2-groupoid captures the underlying 2-truncated weak homotopy type (the homotopy 2-type).

Related concepts

• path 2-groupoid

References

Original discussion of the fundamental 2-groupoid of Hausdorff topological spaces as a strict 2-groupoid (Grpd-enriched category):

• Keith A. Hardie, Klaus H. Kamps, Rudger Kieboom, A homotopy 2-groupoid of a Hausdorff space. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures 8 (2000) 209-234 [doi:10.1023/A:1008758412196]

and as a weak 2-groupoid:

• Keith A. Hardie, Klaus H. Kamps, Rudger Kieboom, A Homotopy Bigroupoid of a Topological Space, Applied Categorical Structures 9 (2001) 311-327 [doi:10.1023/A:1011270417127]