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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{fibration in a 2-category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{2category_theory}{}\paragraph*{{2-category theory}}\label{2category_theory} [[!include 2-category theory - contents]] \hypertarget{fibrations_in_a_2category}{}\section*{{Fibrations in a 2-category}}\label{fibrations_in_a_2category} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{the_2fibration_of_fibrations}{The 2-fibration of fibrations}\dotfill \pageref*{the_2fibration_of_fibrations} \linebreak \noindent\hyperlink{iterated_fibrations}{Iterated fibrations}\dotfill \pageref*{iterated_fibrations} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Recall the notion of a [[Grothendieck fibration]]: a [[functor]] $p \colon E \to B$ whose fibres $E_b$ are (contravariantly) functorial in $b \in B$. This idea may be generalized to work in any suitable [[2-category]], although if the 2-category is not [[strict 2-category|strict]], then one has to generalize instead the non-strict notion of [[Street fibration]]. The generalized definition can be given in any of several equivalent ways, in such a way that When $K$ is [[Cat]], strict fibrations in this sense are precisely [[Grothendieck fibration|Grothendieck fibrations]], while non-strict ones are precisely [[Street fibrations]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Fix a [[2-limit|finitely complete]] (non-strict) 2-category $K$. Recall that for any two morphisms $f: A \to C$ and $g: B \to C$, $f/g$ denotes their [[comma object]]. Let $f/_{\cong} g$ be their [[2-pullback]]. A morphism $p \colon E \to B$ in $K$ is a (non-strict) \textbf{fibration} when the following equivalent conditions hold: \begin{itemize}% \item $p_* = K(X,p) \colon K(X,E) \to K(X,B)$ is a Street fibration in $Cat$ for each $X \in K$, and for all $f \colon Y \to X$ in $K$ \begin{displaymath} \itexarray{ K(X,E) & \overset{f^*}{\to} & K(Y,E) \\ \mathllap{p_*} \downarrow & & \downarrow \mathrlap{p_*} \\ K(X,B) & \overset{f^*}{\to} & K(Y,B) } \end{displaymath} is a [[morphism of fibrations]]. \item For every morphism $f: X \to B$, the canonical map $i: f/_{\cong} p \to f/p$ has a [[right adjoint]] in the [[slice 2-category]] $K / X$. \item The canonical map $i \colon p \to B/p$ has a right adjoint in $K / B$. \item $p \colon E \to B$ is an algebra for the [[2-monad]] $L$ on $K/B$ given by $L p = B/p$. \item The induced [[F-functor]] $\Sigma_p : K\swarrow E \to K\swarrow B$ between the [[lax slice 2-category|oplax slice]] [[F-categories]] has a [[lax F-adjunction|right semi-lax F-adjoint]]. (See \hyperlink{JohnstonePP}{Johnstone}.) \end{itemize} If $K$ is a [[strict 2-category]] with finite strict 2-limits, then we say that $p \colon E \to B$ is a \textbf{strict fibration} when the corresponding conditions hold where ``Street fibration'' is replaced by ``Grothendieck fibration'', slice 2-categories are replaced by strict-slice 2-categories, comma objects are replaced by strict comma objects, and 2-pullbacks are replaced by strict 2-pullbacks. In this case the 2-monad $L$ is in fact a strict 2-monad, but we do \emph{not} require $p$ to be a \emph{strict} algebra, only a [[pseudoalgebra]]; strict algebras for $L$ would instead be [[split fibrations]]. Note also that the \emph{first} definition makes perfect sense regardless of whether $K$ has any limits, although this is not true of the others. \hypertarget{details}{}\subsection*{{Details}}\label{details} By way of spelling out the first definition, we may define a 2-cell $\eta \colon e \to e' \colon X \to E$ to be \textbf{$p$-cartesian} if $f^*\eta = \eta f$ is $p_*$-cartesian for every $f \colon Y \to X$. Since this definition already incorporates stability under pullback, we can then say that $p$ is a fibration in $K$ if for every 2-cell $\beta \colon b \to p e$ there is a cartesian $\hat\beta \colon e' \to e$ such that $p e' = b$ and $p \hat\beta = \beta$. The third definition is perhaps the simplest. Of course, it is implied by the second (take $f = 1$), but the converse is also true by the pasting lemma for [[comma object|comma]] and [[pullback]] squares. We can show that the first and third definitions are equivalent by using the representability of fibrations and adjunctions, plus the following lemma, whose proof can be found at [[Street fibration]]. \begin{ulemma} A functor $p \colon E \to B$ is a [[cleavage|cloven]] Street fibration if and only if the canonical functor $i \colon B \to E/p$ has a right adjoint $r$ in $Cat / B$. \end{ulemma} It follows that a morphism $p \colon E \to B$ in any 2-category $K$ is representably a fibration (i.e. satisfies the first definition) if and only if the adjunction $i \dashv r$ exists in $K/B$ (i.e. it satisfies the third condition). Now we connect the first three conditions with the fourth. Because $K$ is finitely complete, we may form the [[tricategory]] $Span(K)$ of [[spans]] in $K$. In particular, $K/B$ is equivalent to the hom-2-category $Span(K)(B,1)$. Now [[comma object|recall]] that $B/p$ can be expressed as a pullback or span composite \begin{displaymath} \itexarray{ & & & & B/p & & & & \\ & & & \swarrow & & \searrow & & & \\ & & B^{\mathbf{2}} & & & & E & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & \\ B & & & & B & & & & 1 } \end{displaymath} Write $\Phi B = B^{\mathbf{2}}\rightrightarrows B$. The functor $L \colon p \mapsto B/p$ is then given by composition: $L p =\Phi B \circ p$. To show that $p$ is a fibration iff it is an $L$-algebra, it suffices to show \textbf{Lemma.} \emph{$\Phi B$ is a [[colax-idempotent monad]] in $Span(K)$ with unit $i \colon B \to B^{\mathbf{2}} = B/B$.} \begin{proof} Thus for each $n \geq 1$, there is a span $B^{[n]} \colon B ⇸ B$, which is canonically equivalent to the $(n-1)$-fold composite $B^{[2]} \circ B^{[2]} \circ \cdots \circ B^{[2]} \colon B ⇸ B$, because cotensor preserves finite limits. For the same reason, $B^- \colon Cat^{op}_{fp} \to K$ almost restricts to a monoidal 2-functor $\mathbf{\Delta}^{op} \to Span(K)(B,B)$, except that $[0]$ does not yield a span from $B$ to $B$. Precomposing $B^-$ with the functor $\mathbf{\Delta}^{co} \to \mathbf{\Delta}^{op}$ that sends $[n]$ to $[n+1]$, while $\delta_i \mapsto \sigma_i$ and $\sigma_i \mapsto \delta_{i+1}$ does yield a monoidal 2-functor $\mathbf{\Delta}^{co} \to Span(K)(B,B)$, which by the universal property of $\mathbf{\Delta}$ corresponds to a unique colax-idempotent monad in $Span(K)(B,B)$. In detail, the monoid object $[0] \to [1] \leftarrow [2]$ in $\mathbf{\Delta}$ is sent to the monad $\Phi B = B^{[2]}$ in $Span(K)$, with structure maps $i = (\sigma^1_0)^* \colon B \to \Phi B$ and $c = (\delta^2_1)^* \colon \Phi B \circ \Phi B \to \Phi B$. Moreover, because of the adjunction $\delta^2_1 \dashv \sigma^2_0 = \sigma^1_0 \oplus [1]$ in $\mathbf{\Delta}$, we have $i \circ \Phi B \dashv c$ in $Span(K)(B,B)$ with identity unit. \end{proof} It follows that $L = \Phi B \circ -$ is a monad, and is colax-idempotent because, for a span $H \colon B ⇸ A$, we have $\eta_H = i \circ H$, $\mu_H = c \circ H$, and \begin{displaymath} \eta_{L H} = i \circ \Phi B \circ H \dashv c \circ H \end{displaymath} with identity unit. It is clear, too, that the morphism $i \colon p \to B/p$ as above is given by $\eta_{p^\circ} = i \circ p^\circ$, where $p^\circ$ denotes the obvious span $B ⇸ 1$. Thus, by the definition of a [[colax-idempotent monad]], a morphism $p \colon E \to B$ carries the structure of an $L$-algebra if and only if the unit $i = \eta_{p^\circ} \colon p \to B/p$ has a right adjoint. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{the_2fibration_of_fibrations}{}\subsubsection*{{The 2-fibration of fibrations}}\label{the_2fibration_of_fibrations} For a [[strict 2-category]] $K$, let $Fib_K$ be the 2-category of strict fibrations in $K$, morphisms of fibrations, and 2-cells between them. Then the codomain functor $cod : Fib_K \to K$ is a strict [[2-fibration]]. (This is arguably an instance of the [[microcosm principle]].) Similarly, for any [[bicategory]] $K$, the bicategory $Fib_K$ of weak fibrations in $K$, morphisms of fibrations, and 2-cells admits a weak 2-fibration $cod : Fib_K \to K$. \hypertarget{iterated_fibrations}{}\subsubsection*{{Iterated fibrations}}\label{iterated_fibrations} It is easy to show that a composite of fibrations is a fibration. Moreover, if $Fib(X)= Fib_K(X)$ denotes the 2-category of fibrations over $X\in K$, then we have: \begin{theorem} \label{}\hypertarget{}{} A morphism in $Fib(X)$ is a fibration in the 2-category $Fib(X)$ iff its underlying morphism in $K$ is a fibration. \end{theorem} This is a standard result, at least in the case $K=Cat$, and is apparently due to Benabou. References include (\hyperlink{Benabou85}{Bénabou 1985}), (\hyperlink{Hermida99}{Hermida 1999}) and (\hyperlink{Jacobs}{Jacobs 1999, Chapter 9}). Therefore, for any fibration $A\to X$ in $K$ we have $Fib_K(A) \simeq Fib_{Fib_K(X)}(A\to X)$, and similarly for opfibrations. This is a fibrational 2-categorical analogue of the standard equivalence $K/A \simeq (K/X)/(A\to X)$ for ordinary [[slice categories]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Grothendieck fibration]], [[Street fibration]], [[discrete fibration]], [[two-sided fibration]] \item [[Cartesian fibration]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Ross Street]] \emph{Fibrations in bicategories}, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], 21 no. 2 (1980), p. 111--160 (\href{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_2_111_0}{numdam}). \item [[Mark Weber]] \emph{Yoneda structure from 2-toposes}, Applied Categorical Structures \textbf{15} (2007) pp259-323. doi:\href{https://doi.org/10.1007/s10485-007-9079-2}{10.1007/s10485-007-9079-2} (\href{https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes}{author's page}) \item [[Peter Johnstone]], \emph{Fibrations and partial products in a 2-category}, Applied Categorical Structures \textbf{1} (1993) pp141-179 doi:\href{https://doi.org/10.1007/BF00880041}{10.1007/BF00880041} \item [[Jean Bénabou]] \emph{Fibered categories and the foundations of naive category theory}, Journal of Symbolic Logic, \textbf{50}(1) (1985) pp10-37. doi:\href{https://doi.org/10.2307/2273784}{10.2307/2273784}, (\href{http://www.jstor.org/stable/2273784}{JSTOR}) \item [[Bart Jacobs]], \emph{Categorical Logic and Type Theory}, Studies in Logic and the Foundations of Mathematics, 141. North-Holland Publishing Co., Amsterdam, 1999. xviii+760 pp. ISBN: 0-444-50170-3 (\href{https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf}{pdf}) (\href{http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html}{contents}) \item [[Claudio Hermida]], \emph{Some properties of Fib as a fibred 2-category}, Journal of Pure and Applied Algebra \textbf{134} Issue 1 (1999) pp83-109 doi:\href{https://doi.org/10.1016/S0022-4049%2897%2900129-1}{10.1016/S0022-4049(97)00129-1}, (\href{https://core.ac.uk/display/82656524}{Core}) \end{itemize} [[!redirects fibrations in a 2-category]] [[!redirects fibrations in 2-categories]] [[!redirects internal fibration]] [[!redirects internal fibrations]] [[!redirects opfibration in a 2-category]] [[!redirects opfibrations in a 2-category]] [[!redirects opfibrations in 2-categories]] [[!redirects internal opfibration]] [[!redirects internal opfibrations]] \end{document}