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\begin{document}

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\section*{two-sided fibration}

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

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

[[!include category theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{twosided_fibrations}{}\section*{{Two-sided fibrations}}\label{twosided_fibrations}

\noindent\hyperlink{twosided_fibrations_2}{Two-sided fibrations}\dotfill \pageref*{twosided_fibrations_2} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_2monads}{In terms of 2-monads}\dotfill \pageref*{in_terms_of_2monads} \linebreak
\noindent\hyperlink{a_representable_definition}{A representable definition}\dotfill \pageref*{a_representable_definition} \linebreak
\noindent\hyperlink{as_iterated_fibrations}{As iterated fibrations}\dotfill \pageref*{as_iterated_fibrations} \linebreak
\noindent\hyperlink{twosided_discrete_fibrations}{Two-sided discrete fibrations}\dotfill \pageref*{twosided_discrete_fibrations} \linebreak
\noindent\hyperlink{definition_2}{Definition}\dotfill \pageref*{definition_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{profunctors_and_collages}{Profunctors and collages}\dotfill \pageref*{profunctors_and_collages} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{twosided_fibrations_2}{}\subsection*{{Two-sided fibrations}}\label{twosided_fibrations_2}

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

Recall that a functor $E \to B$ is called a [[fibration]] if its [[fibres]] $E_b$ vary (pseudo-)[[pseudofunctor|functorially]] in $b$. Taking \emph{fibre} to mean \emph{strict fibre} results in the notion of [[Grothendieck fibration]], while taking it to mean [[essential fibre]] gives the notion of [[Street fibration]].

Similarly, a \emph{two-sided fibration} $A \leftarrow E \to B$ is a pair of functors whose fibres $E(a,b)$ vary functorially in both $a$ and $b$ (contravariantly in one and covariantly in the other).

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

Let $K$ be a [[bicategory]] with finite [[2-limits]], and recall that [[fibration in a 2-category|fibrations]] in $K$ may be defined in any of several ways. Each of these has an analogous version for two-sided fibrations.

\hypertarget{in_terms_of_2monads}{}\paragraph*{{In terms of 2-monads}}\label{in_terms_of_2monads}

Recall that (cloven) fibrations $E \to B$ in $K$ are the (pseudo)algebras for a (pseudo) 2-monad $L$ on $K/B$. For a morphism $p \colon E \to A$ in $K$, $L p$ is given by composing the [[span]] $A \overset{p}{\leftarrow} E \to 1$ with the canonical span $\Phi A = A \overset{dom}{\leftarrow} A^{\mathbf{2}} \overset{cod}{\to} A$, so that $L p \colon E/p \to A$ is the canonical projection. This can equivalently be described as the [[comma object]] $(1_A/p)$. This 2-monad is [[lax-idempotent 2-monad|lax-idempotent]], so that $p\colon E\to B$ is a fibration if and only if the unit $p\to L p$ has a left adjoint with invertible counit.

More generally, the same construction gives a 2-monad $L$ on $Span K(B,A)$, whose algebras we call \textbf{left fibrations}. In [[Cat]], a span $C \overset{p}{\leftarrow} H \overset{q}{\to} D$ is a left fibration if $p$ is a cloven fibration whose chosen cartesian lifts are $q$-vertical. (Since we are working bicategorically, ``$q$-vertical'' means that they map to isomorphisms under $q$.)

Dually, there is a colax-idempotent 2-monad $R$ on each $Span K(B,A)$ whose algebras are called \textbf{right fibrations}, the special case of $Span Cat(D,1)$ yielding cloven opfibrations.

There is then a composite 2-monad $M$ that takes a span $E$ from $B$ to $A$ to $M E = \Phi A \circ E \circ \Phi B$, and $M$-algebras are called \textbf{two-sided fibrations}. Although $M$ is neither lax- nor colax-idempotent, it is still [[property-like 2-monad|property-like]].

\begin{prop}
\label{TwoSidedFibrationsInCat}\hypertarget{TwoSidedFibrationsInCat}{}
A two-sided Street fibration from $B$ to $A$ in $Cat$ is given by a span $p \colon E \to A$, $q \colon E \to B$ such that

\begin{enumerate}%
\item each $i \colon a \to p x$ in $A$ has a $p$-[[cartesian morphism|cartesian]] lift $\kappa_i \colon i^* x \to x$ in $E$ that is $q$-vertical (that is, $E$ is a left fibration)


\item each $j \colon q x \to b$ in $B$ has a $q$-opcartesian lift $\kappa^j \colon x \to j_! x$ in $E$ that is $p$-vertical ($E$ is a right fibration)


\item for every cartesian--opcartesian composite $i^* x \to x \to j_! x$ in $E$, the canonical morphism $j_! i^* x \to i^* j_! x$ is an [[isomorphism]].



\end{enumerate}
\end{prop}
\begin{proof}
By the usual theory of [[distributive laws]], an $M$-algebra $m \colon M E \to E$ gives rise to $L$- and $R$-algebras $m \cdot (\Phi A \circ \eta^R_E)$ and $m \cdot (\eta^L_E \circ \Phi B)$, and conversely an $L$-algebra $\ell$ and an $R$-algebra $r$ underlie an $M$-algebra if and only if there is an isomorphism $r \cdot (\ell \circ \Phi B) \cong \ell \cdot (\Phi A \circ r)$ that makes $r$ a morphism of $L$-algebras.

Now given $\ell$ and $r$, because $L$ is [[colax-idempotent]], there is a unique 2-cell $\bar r \colon r \cdot (\ell \circ \Phi B) \Rightarrow \ell \cdot (\Phi A \circ r)$ that makes $r$ a colax morphism of $L$-algebras. So we want to show that in the case of $Cat$, the components of this natural transformation are the canonical morphisms of (3).

The 2-cell $\bar r$ is given by $\ell \cdot (\Phi A \circ r) \cdot (\epsilon \circ \Phi B)$, where $\epsilon$ is the counit of the adjunction $\eta^L_E \dashv \ell$. Its components are thus given, for each $i \colon a \to p x$ in $A$ and $j \colon p x \to b$ in $B$, by first factoring $\kappa^j \kappa_i$ through the opcartesian $i^* x \to j_! i^* x$ and then factoring the result through the cartesian $i^* j_! x  \to j_! x$, to obtain exactly the canonical morphism $j_! i^* x \to i^* j_! x$.

\end{proof}
If $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ is a two-sided fibration, then the operation sending $(a,b)$ to the corresponding (essential) fiber of $(p,q)$ defines a pseudofunctor $A^{op}\times B \to Cat$. The third condition in Proposition \ref{TwoSidedFibrationsInCat} corresponds to the ``interchange'' equality $(\alpha,1)(1,\beta) = (1,\beta)(\alpha,1)$ in $A^{op}\times B$. We write $Fib(B,A)$ for the 2-category of two-sided fibrations from $B$ to $A$.

\hypertarget{a_representable_definition}{}\paragraph*{{A representable definition}}\label{a_representable_definition}

Another definition of internal fibration is that a (cloven) fibration in $K$ is a morphism $p\colon E\to B$ such that $K(X,p)\colon K(X,E)\to K(X,B)$ is a (cloven) fibration in $Cat$, for any $X\in K$, and for any $X\to Y$ the corresponding square is a morphism of fibrations in $Cat$. To adapt this definition to two-sided fibrations, we therefore need only to say what is a two-sided fibration in $Cat$. For this we can use the characterization of Proposition \ref{TwoSidedFibrationsInCat}.

\hypertarget{as_iterated_fibrations}{}\paragraph*{{As iterated fibrations}}\label{as_iterated_fibrations}

Let $Fib(A) = Fib_K(A)$ denote the 2-category of fibrations over $A\in K$. It is a well-known fact (apparently due to Benabou) that a morphism in $Fib(A)$ is a fibration in $Fib(A)$ if and only if its underlying morphism in $K$ is a fibration. See [[fibration in a 2-category]]. Thus, for any fibration $r\colon C\to A$, we have $Fib_{Fib_K(A)}(r) \simeq Fib_K(C)$.

Of course there is a dual result for opfibrations: for any opfibration $r\colon C\to A$ we have $Opf_{Opf_K(A)}(r) \simeq Opf_K(C)$. When we combine variance of iteration, however, we obtain two-sided fibrations.

\begin{utheorem}
A span $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ is a two-sided fibration from $B$ to $A$ if and only if 1. $p\colon E\to A$ is a fibration and 1. $(p,q)\colon E\to A\times B$ is an opfibration in $Fib(A)$.

\end{utheorem}
\begin{proof}
Recall that the projection $A\times B \to A$ is a fibration (and also an opfibration, although that is irrelevant here), and the cartesian 2-cells are precisely those whose component in $B$ is an isomorphism. Therefore, saying that $(p,q)$ is a \emph{morphism} in $Fib(A)$, i.e. that it preserves cartesian 2-cells, says precisely that $q$ takes $p$-cartesian 2-cells to isomorphisms.

Now $q$ is an opfibration in $K$ iff $E\to (q/1_B)$ has a left adjoint with invertible counit in $K/B$, and $(p,q)$ is an opfibration in $Fib(A)$ iff $E\to ((p,q)/1_{A\times B})$ has a left adjoint with invertible counit in $Fib(A)/(A\times B)$. Of crucial importance is that here $((p,q)/1_{A\times B})$ denotes the comma object calculated \emph{in the 2-category $Fib(A)$}, or equivalently in $K/A$ (since monadic forgetful functors create limits), and it is easy to check that this is in fact \emph{equivalent} to the comma object $(q/1_B)$ calculated in $K$.

Therefore, $(p,q)$ is an opfibration in $Fib(A)$ iff $q$ is an opfibration in $K$ and the left adjoint of $E\to (q/1_B)$ is a morphism in $Fib(A)$. It is then easy to check that this left adjoint is a morphism in $K/A$ iff $p$ inverts $q$-opcartesian arrows, and that it is a morphism of fibrations iff the final condition in Proposition \ref{TwoSidedFibrationsInCat} is satisfied.

\end{proof}
In particular, we have $Fib(B,A) \simeq Opf_{Fib(A)}(A\times B)$. By duality, $Fib(B,A) \simeq Fib_{Opf(B)}(A\times B)$, and therefore $Fib_{Opf(B)}(A\times B) \simeq Opf_{Fib(A)}(A\times B)$, a commutation result that is not immediately obvious.

This result appears in \hyperlink{BournPenon}{Bourn--Penon}; it was noticed independently and recorded here by [[Mike Shulman]].

\hypertarget{twosided_discrete_fibrations}{}\subsection*{{Two-sided discrete fibrations}}\label{twosided_discrete_fibrations}

\hypertarget{definition_2}{}\subsubsection*{{Definition}}\label{definition_2}

A two-sided fibration $A \leftarrow E \to B$ in $K$ is \textbf{discrete} if it is [[discrete object|discrete]] as an object of $K/A \times B$. Since discreteness is a limit construction, it is created by monadic forgetful functors; hence this is equivalent to being discrete as an object of the 2-category $Fib(A,B)$ of two-sided fibrations.

For Grothendieck fibrations in [[Cat]], this means the following.

\begin{udefn}
A \textbf{two-sided discrete fibration} is a [[span]] $q \colon E \to A$, $p \colon E \to B$ of [[categories]] and [[functors]] such that

\begin{enumerate}%
\item each $b \to p(e)$ in $B$ has a unique lift in $E$ that has codomain $e$ and is in the fiber over $q(e)$
\item each $q(e) \to a$ in $A$ has a unique lift in $E$ that has domain $e$ and is in the fiber over $p(e)$
\item for each $f\colon e \to e'$ in $E$, the codomain of the lift of $q(f)$ equals the domain of the lift of $p(f)$ and their composite is $f$.

\end{enumerate}
\end{udefn}
We write

\begin{displaymath}
DFib(A,B) \subset Span(A,B)
\end{displaymath}
for the full [[subcategory]] on the 2-category $Span K(A,B)$ of [[span]]s on the 2-sided discrete fibrations. Since a morphism of spans between discrete fibrations is automatically a morphism of fibrations, this is also the full sub-2-category of the 2-category of two-sided fibrations $Fib(A,B)$. And since they are discrete objects, this 2-category is actually (equivalent to) a 1-category.

\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\hypertarget{profunctors_and_collages}{}\paragraph*{{Profunctors and collages}}\label{profunctors_and_collages}

\begin{udef}
Given a profunctor $F : B^{op} \times A \to Set$, its [[collage]] is the category $K_F$ over the [[interval category]]

\begin{displaymath}
p : K_F \to \Delta[1]
\end{displaymath}
With $p^{-1}(0) = B$, $p^{-1}(1) = A$, $K_F(b,a) = F(b,a)$ and $K_F(a,b) = \emptyset$ for all $b \in B$, $a \in A$, where

\begin{itemize}%
\item the composite of $b \stackrel{e}{\to} a$ with $a \stackrel{f}{\to} a'$ is given by $F(b,f)(e)$;


\item the composite of $b \stackrel{g}{\to} b'$ with $b' \stackrel{e'}{\to} a'$ is given by $F(g, a')(e')$.



\end{itemize}
\end{udef}
\begin{uprop}
There is an [[equivalence of categories]]

\begin{displaymath}
[B^{op} \times A, Set]
    \stackrel{\simeq}{\to}
  DFib(A,B)
\end{displaymath}
\begin{displaymath}
F \mapsto E_F
  \,,
\end{displaymath}
[[pseudo-natural transformation|pseudo-natural]] in $A, B \in Cat$, between [[profunctor]]s in [[Set]] and discrete fibrations from $A$ to $B$, where $E_F$ is the category whose

\begin{itemize}%
\item objects are [[section]]s $\sigma : \Delta[1] \to K_F$ of the [[collage]] $p : K_F \to \Delta[1]$


\item morphisms are [[natural transformation]]s between such sections;


\item the two projections $A \leftarrow E_F \to B$ are the two functors induced by restriction along $\{0\} \to \Delta[1] \leftarrow \{1\}$.



\end{itemize}
\end{uprop}
\begin{proof}
First we write out $E_F$ in detail. In the following $b, b', \cdots \in B$ and $a,a', \dots \in A$.

The objects of $E_F$ are morphisms

\begin{displaymath}
\itexarray{
    b 
    \\
    {}^{\mathllap{e}}\downarrow
    \\
    a
  }
\end{displaymath}
in $K_F$, hence triples $(b \in B, a \in A, e \in F(b,a))$.

Morphisms are [[commuting diagram]]s

\begin{displaymath}
\itexarray{
    b &\stackrel{g}{\to}& b'
    \\
    {}^{\mathllap{e}}\downarrow && \downarrow^{\mathrlap{e'}}
    \\
    a &\stackrel{f}{\to}& a'
  }
\end{displaymath}
in $K_F$. We may identify these with pairs $((b \stackrel{g}{\to}b') \in B,(a \stackrel{f}{\to} a') \in A)$ such that

\begin{displaymath}
F(g,a')(e') = F(b,f)(e)
  \,.
\end{displaymath}
We check that this construction yields a two-sided fibration. The three conditions are

\begin{enumerate}%
\item For

\begin{displaymath}
\itexarray{
     b 
     \\
     {}^{\mathllap{e}}\downarrow
     \\
     a
  }
\end{displaymath}
an object of $E_F$ and $a \stackrel{f}{\to} a'$ a morphism in $A$, we have that

\begin{displaymath}
\itexarray{
     b &\stackrel{Id}{\to}& b
     \\
     {}^{\mathllap{e}}\downarrow 
     & & 
     \downarrow^{\mathrlap{f e}}
     \\
     a &\underset{f}{\to}& a'
  }
\end{displaymath}
is the unique lift to a morphism in $E$ that maps to $Id_b$.


\item Analogously, for

\begin{displaymath}
\itexarray{
     b' 
     \\
     {}^{\mathllap{e'}}\downarrow
     \\
     a'
  }
\end{displaymath}
an object of $E_F$ and $b \stackrel{g}{\to} b'$ a morphism in $B$, we have that

\begin{displaymath}
\itexarray{
     b &\stackrel{g}{\to}& b'
     \\
     {}^{\mathllap{e' g}}\downarrow 
     & & 
     \downarrow^{\mathrlap{e'}}
     \\
     a' &\underset{id}{\to}& a'
  }
\end{displaymath}
is the unique lift to a morphism in $E$ that maps to $Id_{a'}$.


\item For

\begin{displaymath}
\itexarray{
     b &\stackrel{g}{\to}& b'
     \\
     {}^{\mathllap{e}}\downarrow && 
     \downarrow^{\mathrlap{e'}}
     \\
     a &\underset{f}{\to}& a'
  }
\end{displaymath}
an arbitrary morphism in $E_F$, these two unique lifts of its $A$- and its $B$-projection, respectively, are

\begin{displaymath}
\itexarray{
     b &\stackrel{Id}{\to}& b
     \\
     {}^{\mathllap{e}}\downarrow 
     & & 
     \downarrow^{\mathrlap{f e}}
     \\
     a &\underset{f}{\to}& a'
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
     b &\stackrel{g}{\to}& b'
     \\
     {}^{\mathllap{e' g}}\downarrow & & 
     \downarrow^{\mathrlap{e'}}
     \\
     a' &\underset{Id}{\to}& a'
  }
  \,.
\end{displaymath}
The codomain and domain do match, since $f e = e' g$ by the existence of the original morphism, and their composite is the original morphism

\begin{displaymath}
\itexarray{
     b &\stackrel{Id}{\to}& b &\stackrel{g}{\to}& b
     \\
     {}^{\mathllap{e}}\downarrow 
     & & 
     {}^{\mathllap{f e}}\downarrow^{\mathrlap{e' g}}
      &&
      \downarrow^{\mathrlap{e'}}
     \\
     a &\underset{f}{\to}& a' &\stackrel{Id}{\to}& a'
  }    
  \,.
\end{displaymath}


\end{enumerate}
To see that this construction indeed yields an equivalence of categories, define a functor $(A\leftarrow E \to B) \mapsto (F_E : B^{op} \times A \to Set)$ by setting

\begin{itemize}%
\item $F_E(b,a) := E_{b,a}$;


\item for a morphism $b \stackrel{g}{\to} b'$ let $F_E(g,a') : F_E(b',a') \to F_E(b,a')$ be the function that sends $b' \stackrel{e'}{\to} a'$ to the domain of the unique lift of $b \stackrel{g}{\to} b'$ with this codomain and mapping to $Id_{a'}$;


\item for a morphism $a \stackrel{f}{\to} a'$ let $F_E(b,f) : F_E(b,a) \to F_E(b,a')$ be the function that sends $b \stackrel{e}{\to} a$ to the codomain of the unique lift of $a \stackrel{f}{\to} a'$ with this domain and mapping to $Id_{b}$;.



\end{itemize}
One checks that this yields an equivalence of categories.

\end{proof}
\begin{ulemma}
The category $E_F$ is equivalently characterized as being the [[comma category]] of the diagram $B \to K_F \leftarrow A$.

\end{ulemma}
Note that profunctors can also be characterized by their collages, these being the two-sided [[codiscrete cofibrations]]; and the collage corresponding to a two-sided fibration is its [[cocomma object]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Grothendieck fibration]],


\item [[Street fibration]],


\item [[discrete fibration]],


\item [[fibration in a 2-category]],


\item \textbf{two-sided fibration},


\item [[Conduche functor]]


\item [[Cartesian fibration]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion is originally discussed in

\begin{itemize}%
\item [[Ross Street]]. \emph{Fibrations and Yoneda's lemma in a 2-category}. In Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 104 133. Lecture Notes in Math., Vol. 420. Springer, Berlin, 1974.


\item [[Ross Street]], \emph{Fibrations in bicategories}. Cahiers Topologie G\'e{}om. Diff\'e{}rentielle, 21(2):111--160, 1980. (Corrections in 28(1):53--56, 1987)



\end{itemize}
Some further discussion of discrete fibrations can be found in

\begin{itemize}%
\item Dominique Bourn and Jacques Penon. \emph{2-cat\'e{}gories r\'e{}ductibles}. Preprint, University of Amiens Department of Mathematics, 1978. Reprinted as \emph{TAC Reprints} no. 19, 2010 (\href{http://www.tac.mta.ca/tac/reprints/articles/19/tr19abs.html}{link}).

\end{itemize}
Useful reviews are in

\begin{itemize}%
\item [[Emily Riehl]], \emph{Two-sided discrete fibrations in 2-categories and bicategories} 2010 (\href{http://www.math.harvard.edu/~eriehl/fibrations.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Fosco Loregian]] and [[Emily Riehl]], \emph{Categorical notions of fibration}, \href{https://arxiv.org/abs/1806.06129}{arxiv}

\end{itemize}
[[!redirects two-sided fibrations]] [[!redirects two-sided discrete fibration]] [[!redirects two-sided discrete fibrations]]

[[!redirects 2-sided fibration]] [[!redirects 2-sided fibrations]] [[!redirects 2-sided discrete fibration]] [[!redirects 2-sided discrete fibrations]]



\end{document}