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 $\mathbb R^2$.