I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.

My attention was drawn to the Section A.3 on "The Seifert-van Kampen Theorem" p. 845.

It starts by stating the classical theorem determining the fundamental group of pointed space which is a union of two open sets with path-connected intersection. (The most general theorem of this type is for the fundamental groupoid on a set $A$ of base points for a space $X$ which is the union of a family $\mathcal U$ of open sets and such that $A$ meets each path-component of all 1-,2-,3-fold intersections of the sets of $\mathcal U$).

It states: " In this section, we will prove a generalization of the Seifert-van Kampen theorem, which describes the entire weak homotopy type of $X$ in terms of any sufficiently nice covering of X by open sets: Theorem A.3.1." However this Theorem makes no mention of groups or connectivity conditions.

So my question is: How does one deduce the SvKT as stated there, or its more general version, from Theorem A.3.1?

Theorem A.3.1 itself seems closely related to classical theorems on excision, showing the singular complex is chain homotopy equivalent to the singular complex of $\mathcal U$-small simplices. (I don't have the earliest reference for this, but I like the proof by R. Sch"on from Proc. AMS 59 (1976).)

A particular point for the deduction of the most general version of the SvKT is: why the number 3? One explanation is that it has to do with the Lebesgue dimension of $\mathbb R^2$.