TITLE: How to construct a free 2-group on a groupoid?
QUESTION [8 upvotes]: Let G
  be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
  equipped with a functor $i:G\longrightarrow\mathcal{F}\left(G\right)$
  such that for any 2-group $\mathcal{G}$
  and any functor $F:G\longrightarrow\mathcal{G}$
  there is a monoidal functor $F':\mathcal{F}\left(G\right)\longrightarrow\mathcal{G}$
  and an isomorphism $\alpha:F'\circ i\Longrightarrow F$
  such that the functor $F':\mathcal{F}\left(G\right)\longrightarrow\mathcal{G}$
  is unique up to coherent isomorphism. Here is my attempt at a description of $\mathcal{F}\left(G\right)$
 :
The objects are defined inductively by requiring that $\mathcal{F}\left(G\right)$
  contain an object 1
 , the objects of G
  and for each object $g\in G$
  an object $\overline{g}\in G$
  such that for any two objects $x,y\in\mathcal{F}\left(G\right)$
  there is an object $x\otimes y$
  in $\mathcal{F}\left(G\right)$
 . 
The 'prearrows' in $\mathcal{F}\left(G\right)$
  are built as follows. We require that every arrow from G
  be a prearrow in $\mathcal{F}\left(G\right)$
 . For each pair of prearrows $a:x\longrightarrow z$
  and $b:y\longrightarrow w$
  in $\mathcal{F}\left(G\right)$
  there is a prearrow $a\otimes b:x\otimes y\longrightarrow z\otimes w$
 . In addition we adjoin a prearrow $e_{g}:g\otimes\overline{g}\longrightarrow1$
  for each $g\in G$
  and we adjoin prearrows $\alpha_{x,y,z}:\left(x\otimes y\right)\otimes z\longrightarrow x\otimes\left(y\otimes z\right)$
 , $\lambda_{x}:1\otimes x\longrightarrow x$
 , and $\rho_{x}:x\otimes1\longrightarrow x$
  for each $x,y,z\in\mathcal{F}\left(G\right)$
 . The arrows are then given by equivalence classes of prearrows generated by the requirement that $\alpha,\rho,\lambda$
  and e
  be isomorphisms, the naturality conditions on $\alpha,\rho$
  and $\lambda$
 , the condition that $\otimes$
  is a functor, and the axioms of a monoidal category.
My question is, does this construction work? If not, can anybody give an indication of a construction that does? Also, to define the free symmetric 2-group on a groupoid, does it suffice to add prearrows $\gamma_{x,y}:x\otimes y\longrightarrow y\otimes x$
  for each $x,y\in\mathcal{F}\left(G\right)$
  and generate the equivalence relation from the axioms of a symmetric monoidal category?

REPLY [3 votes]: The notion of free crossed module was a major feature of JHC Whitehead's 1949 paper "Combinatorial homotopy II", in which he proved, using methods of transversality and knot theory, that the crossed module 
$$\pi_2(X \cup \{e^2_\lambda\},X,x) \to \pi_1(X,x) $$
is free on the characteristic maps of the $2$-cells. This theorem is sometimes mentioned but rarely proved in topology texts. Work of Philip Higgins and I showed how this theorem was a special case of a $2$-dim van Kampen type theorem, i.e. a colimit theorem, and the full story of this is given in Part I of the book partially titled Nonabelian Algebraic Topology, EMS 2011 (NAT). Thus the more general theorem determines
$$\pi_2(X \cup _f CA,A,x) \to \pi_1(X,x)$$
for $A$ connected, in terms of the  induced morphism $f_*: \pi_1(A,a) \to \pi_1(X,x)$, so that Whitehead's theorem is the case $A$ is a wedge of circles. 
Feb 14, 2016
This question could also be looked at in the light of the forgetful functor $$\Phi: (\text{2-groups)} \to (\text{groupoids})$$
which is a bifibration with a left adjoint say $D$, and a right adjoint $I$. As developed in Appendix B3 (Theorem B.3.2) of the book NAT, the cofibration (cocartesian) property of $\Phi$ is given by a pushout of 2-groupoids
$$ \begin{matrix} D\Phi(K) &\xrightarrow{D(F)}& D(H) \\
\downarrow&& \downarrow \\
K & \to &F_*(K).\end{matrix} $$
The notion of ``free" 2-groupoid on $F: G \to H$ is the special case when $K=I(G)$. 
This is not stated in that Appendix but is the construction used in Part 1. For example if $F:P \to Q$ is a morphism of groups then the free crossed module on $F$ is the induced crossed module  $F_*(P \to P)$, where the identity crossed module is $I(P)$. This generalises to free crossed modules over groupoids. 
February 25,2016 I think Fernando is right and I apologise for my misapprehension. To explain it, I am commonly looking for some algebraic explanations of how low dimensional identifications in spaces influence high dimensional homotopy invariants. The prototype was that groupoids enable the computation of 1-types by a van Kampen type theorem because, it seems, groupoids have structure in dimensions 0 and 1. Crossed modules (over groupoids, and so equivalent to 2-groupoids) have structure in dimensions 0,1,2, and give computational models of 2-types in an analogous way. So I am used to the "inducing" process coming from a bifibration of algebraic models from dimension $n$ to dimension $n-1$ and contributing to "free" structures in dimension $n$.