A crossed module (over groups) $\mathcal{M} = (H,G,\partial)$ is a homomorphism $\partial\colon H \to G$ (called the boundary) together with an action $\alpha\colon (g,h) \mapsto {}^gh$ of $G$ on $H$ such that the following two axioms are satisfied:
- $\partial({}^gh) = g\partial(h)g^{-1}$
- ${}^{\partial(h)}h' = hh'h^{-1}$
for all $h,h' \in H$ and $g \in G$.
Given two crossed modules $\mathcal{M} = (H,G,\partial)$ and $\mathcal{N} = (H',G',\partial')$, a morphism $$(\mu,\nu)\colon \mathcal{M} \to \mathcal{N}$$ is a pair of group homomorphisms such that $\mu$ and $\nu$ interacts well with the boundaries of $\mathcal{M}$ and $\mathcal{N}$. Moreover, the action is preserved, in the sense that $$\mu({}^gh) = {}^{\nu(g)}\mu(h)$$ for all $h \in H$ and $g \in G$.
With the objects and morphisms defined above, we form the category $\mathsf{CrossedMod}$ of crossed modules.
In Crossed modules and homology [p.44, 1], the authors defined a free functor $$F\colon (\mathsf{Set} \downarrow U) \to \mathsf{CrossedMod},$$ where $U\colon \mathsf{Grp} \to \mathsf{Set}$ is the forgetful functor and $(\mathsf{Set} \downarrow U)$ is the comma category of objects $U$-under $\mathsf{Set}$. They have called $F(f\colon X \to U(G))$ the free crossed module over $f\colon X \to U(G)$.
I am not totally convinced that this indeed forms a crossed module. Are there any other free functors that can be constructed to the category of crossed modules?
I am interested to see how these free crossed modules are useful and specifically where they are useful.
[1] http://www.sciencedirect.com/science/journal/00224049/95/1
I found a partial answer to my question a while ago and so I thought I should put it here. I shall construct free crossed modules that are associated with group presentations. The references that I have used to understand this are:
H. J. Baues, Combinatorial homotopy 4-dimensional complexes, Walter de Gruyter Expos. Math. 2 (1991), 94,95.
A. J. Sieradski, Algebraic topology for two-dimensional complexes, Two-dimensional homotopy and combinatorial group theory 197 (1993), 51–96, London Mathematical Society Lecture Note Series, Cambridge University Press.
Let $X$ be a set and let $F[X]$ be the free group on the set $X$. Let $R \subseteq F[X]$ be a subset of $F[X]$. The elements of $R$ are called relators. Then, $\mathcal{P} = \left< X\;|\;R \right>$ is a presentation of a group. Now, let $E(\mathcal{P})$ denote the free group on the set $F[X] \times R$ of ordered pairs $(w,r)$, where $w \in F[X]$ and $r \in R$.
There is a group action $$F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$$ given by $$(v, (w,r)) \mapsto {}^v(w,r) = (vw,r)$$ with $v,w \in F[X]$, $r \in R$ and a group homomorphism $\delta\colon E(\mathcal{P}) \to F[X]$ defined on the basis elements by $\delta(w,r) = wrw^{-1}$.
Notice that $\mathrm{im}(\delta) = N(R)$, the normal closure of $R$ in $F[X]$. The subgroup $I(\mathcal{P}) = \mathrm{ker}(\delta\colon E(\mathcal{P}) \to F[X])$ of $E(\mathcal{P})$ is called the group of identities for the presentation $\mathcal{P}$.
It is easy to see that the homomorphism $\delta\colon E(\mathcal{P}) \to F[X]$ is a pre-crossed module (just the first axiom is satisfied). That is, for all $v,w \in E(\mathcal{P})$ and $r \in R$, we have: \begin{align*} \delta({}^v(w,r)) &= \delta((vw,r))\\ &= vwr(vw)^{-1}, \text{ by defintion of $\delta$}\\ &= v\delta(w,r)v^{-1}. \end{align*}
The action $F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$ of $\delta(w,r) = wrw^{-1}$ in $F[X]$ on $(v,s)$ in $E(\mathcal{P})$ gives us ${}^{(wrw^{-1})}(v,s) =(wrw^{-1}v,s)$ but we notice that although $(wrw^{-1}v,s)$ does not equal the conjugate $(w,r)(v,s)(w,r)^{-1}$ of $(v,s)$ by $(w,r)$ in $E(\mathcal{P})$, they have the same boundary in $F[X]$: \begin{align*} \delta((wrw^{-1}v,s)) &= (wrw^{-1}v)s(wrw^{-1}v)^{-1}\\ &= \delta((w,r)(v,s)(w,r)^{-1}). \end{align*}
And thus, this implies that $$(w,r)(v,s)(w,r)^{-1}(wrw^{-1}v,s)^{-1}$$ in $E(\mathcal{P})$ is in kernel of $\delta$.
Elements in $E(\mathcal{P})$ of the form $$(w,r)(v,s)(w,r)^{-1}\left(^{\delta((w,r))}(v,s)\right)^{-1}$$ with $(w,r),(v,s) \in E(\mathcal{P})$ are called Peiffer commutators. It is obvious that the Peiffer commutators lie in $\mathrm{ker}(\delta) = I(\mathcal{P})$. Hence, the normal closure $P(\mathcal{P}) \subseteq I(\mathcal{P})$ in $E(\mathcal{P})$ is called the Peiffer group for the presentation $\mathcal{P}$.
The action $F[X] \times E(\mathcal{P}) \to E(\mathcal{P})$ and homomorphism $\delta$ induce an action of $F[X]$ on the quotient group $C(\mathcal{P}) = E(\mathcal{P})/P(\mathcal{P})$ $$F[X] \times C(\mathcal{P}) \to C(\mathcal{P})$$ and a boundary homomorphism $$\partial\colon C(\mathcal{P}) \to F[X]$$ given on the generators by $\partial((w,r)P(\mathcal{P})) = wrw^{-1}$.
Therefore, $\mathscr{F} = (C(\mathcal{P}), F[X], \partial)$ constitute a free crossed module associated with the group presentation $\mathcal{P}$. Finally, we see that $\mathrm{ker}(\partial) = I(\mathcal{P})/P(\mathcal{P})$ and $\mathrm{im}(\partial) = N(R)$.