# Deriving local data for 2-transition function of a 2-bundle (getting stuck)

Let $P \xrightarrow{p} B$ be a 2-$\mathcal{G}$-bundle where $P$ and $B$ are two 2-spaces (categories internalized in $\mathrm{Diff_{\infty}}$ thus forming a 2-category $\mathrm{2Cat}$ with 2-spaces as objects, functors as morphisms and natural transformations as 2-morspisms), $\mathcal{G}$ is a 2-group (a category internalized in $\mathrm{Grp}$) and $p$ the projection map (a functor in $\mathrm{2Cat}$), coherence laws (the categorical analogous to $g_{ij}g_{jk}=g_{ik}$) only happen up to natural isomorphisms represented by (components) $f_{ijk}$ on every triple intersection $U_{ijk}$:

$g^{2}_{ik} \circ f_{ijk} = f_{ijk} \circ (g^{2}_{ij}.g^{2}_{jk})$

(Where $g^2_{..}$ are morphisms of $\mathcal{G}$)

From this last property, John Baez and Urs Schreiber derived the following matching conditions (in Higher Gauge Theory: 2-Connections on 2-Bundles pp. 26-27):

$t(f^{2}_{ijk})t(g^{2}_{ik})g^{1}_{ik}=t(g^{2}_{ij})g^{1}_{ij}t(g^{2}_{jk})g^{1}_{jk}$

where $g^{1}_{..}$ are objects of $\mathcal{G}$, and the natural isomorphisms components have been written in their crossed module form $f_{ijk} = (f^{1}_{ijk},f^{2}_{ijk})$ where $f^{1}_{ijk}$ is the source label and $f^{2}_{ijk}$ is the morphism label. $t$ is the target map.

It is this last relation that seems obscure to me as one can rewrite it (in crossed modul terms):

$(g^{1}_{ik}, f^{2}_{ijk}g^{2}_{ik}) = (g^{1}_{ij},g^{2}_{ij})(g^{1}_{jk},g^{2}_{jk})$ which seems for me different from what the coherence laws impose.

My question is can we derive this last relation from that of the coherence laws? If yes, can you give me a hint?

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