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\begin{document}

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\section*{cleavage}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{cloven_fibrations_vs_pseudofunctors}{Cloven fibrations vs pseudofunctors}\dotfill \pageref*{cloven_fibrations_vs_pseudofunctors} \linebreak
\noindent\hyperlink{nonstrict_cleavages}{Non-strict cleavages}\dotfill \pageref*{nonstrict_cleavages} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Being a [[Grothendieck fibration]] is a [[property-like structure]] on a [[functor]], like the existence of [[limits]] in a [[category]]: it is defined by the existence of certain objects (in this case, [[cartesian morphisms]]) which, when they exist, are unique up to unique [[isomorphism]]. Any property-like structure can be ``algebraicized'' by requiring a specific \emph{choice} of the objects that are required to exist; a \emph{cleavage} is this ``algebraicization'' of being a fibration.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $p\colon E\to B$ be a functor. A \textbf{cleavage} of $p$ is a choice, for each $e\in E$ and $u\colon b\to p(e)$ in $B$, of a single [[cartesian arrow]] $f:e'\to e$ such that $p(f) = u$. Evidently if there exists a cleavage for $p$, then $p$ is a [[Grothendieck fibration]]. If $p$ is equipped with a cleavage, it said to be \textbf{cloven}. Conversely, if we assume the [[axiom of choice]], then every fibered category has a cleavage.

If the cartesian arrow $f$ is an identity whenever $u$ is, the cleavage is said to be \textbf{normal}, and if each composite $f_2 f_1$ is the specified lifting of $u_2 u_1$, the cleavage is said to be a \textbf{splitting}. Any cleavage can be modified to become normal, but not necessarily to become split. (Any fibration is, however, \emph{equivalent} to a split fibration.)

\hypertarget{cloven_fibrations_vs_pseudofunctors}{}\subsection*{{Cloven fibrations vs pseudofunctors}}\label{cloven_fibrations_vs_pseudofunctors}

Given a cleavage of $p$ and an arrow $u:b'\to b$ in $B$, there is a functor $u^*\colon p^{-1}(b)\to p^{-1}(b')$ which to every object $e\in p^{-1}(b)$ assigns the domain of the specified arrow in the cleavage which is above $u$ and whose codomain is $e$. This correspondence extends to a functor, thanks to the universal property of the cartesian arrows.

The functor $u^*$ may be called either the \emph{inverse image functor} along $u$, or the \emph{direct image functor} of $u$, depending on the context; see the remarks on notation at \href{http://ncatlab.org/nlab/show/domain+opfibration#notation}{domain opfibration}. It depends on the choice of cleavage as well as on $u$, although different cleavages produce canonically naturally isomorphic functors. If one doesn't choose a cleavage, then $u^*$ can still be defined as an [[anafunctor]].

The direct image functors corresponding to varying morphisms in $B$ together form a [[pseudofunctor]] $B^{op}\to Cat$. The inverse [[Grothendieck construction]] produces a cloven fibration from a pseudofunctor, setting up an equivalence of 2-categories (in fact, a [[strict 2-equivalence of strict 2-categories]]) between cloven fibrations and pseudofunctors $B^{op} \to Cat$. If we either assume the axiom of choice, or allow the morphisms in [[Cat]] to be anafunctors, then this extends to an equivalence between pseudofunctors and not-necessarily-cloven fibrations.

\hypertarget{nonstrict_cleavages}{}\subsection*{{Non-strict cleavages}}\label{nonstrict_cleavages}

There is a corresponding notion of cleavage for a [[Street fibration]], namely a choice of, for each $e\in E$ and $u\colon b\to p(e)$ in $B$, a cartesian arrow $f:e'\to e$ in $E$ and an isomorphism $v\colon p(e') \xrightarrow{\cong} b$ such that $u\circ v = p(f)$. Such a cleavage induces, for every $u\colon b'\to b$, a functor between the [[essential fibers]] of $b$ and $b'$, and thereby a pseudofunctor and another equivalence of 2-categories (though not a strict 2-equivalence).

[[!redirects pseudo-splitting]] [[!redirects split fibration]] [[!redirects cloven fibration]] [[!redirects split opfibration]] [[!redirects cloven opfibration]] [[!redirects pseudo-splittings]] [[!redirects split fibrations]] [[!redirects cloven fibrations]] [[!redirects split opfibrations]] [[!redirects cloven opfibrations]] [[!redirects cleavages]]



\end{document}