# Example of splitting a fibred category.

Given a fibred category $$\mathcal{F} \to \mathcal{C}$$, we can choose a cleavage, which is a class of cartesian arrows $$K$$ in $$\mathcal{F}$$ s.t. for each arrow $$f:U\to V$$ in $$\mathcal{C}$$ and each object $$\eta$$ in $$\mathcal{F}(V)$$ there exists a unique arrow in $$K$$ with target $$\eta$$ mapping to $$f$$ in $$C$$. A split fibred category is a fibred category admit a splitting, i.e. there is a splitting cleavage which contains all identities and is closed under composition of arrows. Such a splitting doesn’t always exist.

For example, if we consider a group $$G$$ as a category with only one object, and arrows are multiplied by group elements, then a surjective group homomorphism $$G\to H$$ can be seen as a fibred category. A cleavage is a subset $$K$$ of $$G$$ that maps bijectively onto $$H$$. Such a cleavage splits iff $$K$$ is a subgroup of $$G$$. Thus we have a homomorphism $$H\to G$$ s.t. the composition $$H\to G\to H$$ is identity. Surely, such a homomorphism doesn’t always exist.

There is a statement that a fibred category is equivalent to a split fibred category by using 2-Yoneda lemma. If $$U$$ is an object of $$\mathcal{C}$$, we identify the functor $$h_U=Hom(-,U)$$ with comma category ($$\mathcal{C}/U$$). We can have a functor $$Hom(-,\mathcal{F})$$ sending $$U$$ into the category $$Hom(h_U,\mathcal{F})$$. Now we denote by $$\mathcal{F}’$$ the fibred category associated with this functor. Then we have a split fibred category equivalent to $$\mathcal{F}$$ by 2-Yoneda lemma.

My first question is how can we see the fibred category constructed above is split? Second, what does this construction yields in the case of surjective group homomorphism $$G\to H$$ as the example above? Is it $$Hom(H,G)$$? Thank in advance!