Return to Answer

February 26, 2020 Theorem 5.4.7 of the NAT book is a generalisation of a result of JHC Whitehead on free crossed modules, in the form of a complete description of the crossed module $\pi_2(X \cup_f CA,X,x) \to \pi_1(X,x)$ in terms of the morphism $f_*:\pi_1(A,a) \to \pi_1(X, x)$. The proof in the NAT book uses cubical methods. (Whitehead's case was $A$ is a wedge of circles.) Thus we need to look at comparisons of a reasonably sophisticated level of applications.

In all these cases the use of connections in cubical methods is crucial. There is more discussion on this mathoverflow. For us connections arose in order to define commutative cubes in higher cubical categories: compare this paper.

In all these cases the use of connections in cubical methods is crucial. There is more discussion on this mathoverflow. For us connections arose in order to define commutative cubes in higher cubical categories: compare this paper.

Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15

See also some beamer presentations available on my preprint page.

composition http://groupoids.org.uk/images/composition2.jpg

See also this 2014 presentation The intuition for cubical methods in algebraic topology.

November 29, 2016 There is more information in this paper on Modelling and Computing Homotopy Types: I which can serve as an introduction to the NAT book.

Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15

See also some beamer presentations available on my preprint page.

See also this 2014 presentation The intuition for cubical methods in algebraic topology.

November 29, 2016 There is more information in this paper on Modelling and Computing Homotopy Types: I which can serve as an introduction to the NAT book.