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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*{Grothendieck fibration} \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{fibrations_versus_pseudofunctors}{Fibrations versus pseudofunctors}\dotfill \pageref*{fibrations_versus_pseudofunctors} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{variations}{Variations}\dotfill \pageref*{variations} \linebreak \noindent\hyperlink{discrete_and_groupoidal_fibrations}{Discrete and groupoidal fibrations}\dotfill \pageref*{discrete_and_groupoidal_fibrations} \linebreak \noindent\hyperlink{opfibrations_and_bifibrations}{Opfibrations and bifibrations}\dotfill \pageref*{opfibrations_and_bifibrations} \linebreak \noindent\hyperlink{StreetFibration}{Version respecting the Principle of equivalence}\dotfill \pageref*{StreetFibration} \linebreak \noindent\hyperlink{internal_version}{Internal version}\dotfill \pageref*{internal_version} \linebreak \noindent\hyperlink{twosided_version}{Two-sided version}\dotfill \pageref*{twosided_version} \linebreak \noindent\hyperlink{multicategory_version}{Multicategory version}\dotfill \pageref*{multicategory_version} \linebreak \noindent\hyperlink{higher_categorical_versions}{Higher categorical versions}\dotfill \pageref*{higher_categorical_versions} \linebreak \noindent\hyperlink{higher_multicategory_version}{Higher multicategory version}\dotfill \pageref*{higher_multicategory_version} \linebreak \noindent\hyperlink{alternate_definitions}{Alternate definitions}\dotfill \pageref*{alternate_definitions} \linebreak \noindent\hyperlink{in_terms_of_adjoints}{In terms of adjoints}\dotfill \pageref*{in_terms_of_adjoints} \linebreak \noindent\hyperlink{discussions}{Discussions}\dotfill \pageref*{discussions} \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} A \textbf{[[Alexander Grothendieck|Grothendieck]] fibration} (also called a \textbf{fibered category} or just a \textbf{fibration}) is a [[functor]] $p:E\to B$ such that the [[fiber]]s $E_b = p^{-1}(b)$ depend (contravariantly) [[pseudofunctor|pseudofunctorially]] on $b\in B$. One also says that $E$ is a \textbf{fibered category} over $B$. Dually, in a (Grothendieck) \textbf{opfibration} the dependence is covariant. There is an [[equivalence of categories|equivalence]] of [[strict 2-category|2-categories]] \begin{displaymath} Fib(B) \stackrel{\simeq}{\leftrightarrow} [B^{op}, Cat] : \int \end{displaymath} between the 2-category of fibrations over $B$ and the 2-category $[B^{op},Cat]$ of contravariant [[pseudofunctor]]s from $B$ to [[Cat]], also called $B$-[[indexed categories]]. The construction $\int : [B^{op}, Cat] \to Fib(B) : F \mapsto \int F$ of a fibration from a pseudofunctor is sometimes called the \emph{[[Grothendieck construction]]}, although fortunately (or unfortunately) Grothendieck performed many constructions. Less ambiguous terms for $\int F$ are the [[category of elements]] and the [[oplax colimit]] of $F$. Those fibrations corresponding to pseudofunctors that factor through [[Grpd]] are called \textbf{[[categories fibered in groupoids]]}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Let $\phi:e'\to e$ be an arrow in $E$. We say that $\phi$ is \textbf{[[cartesian morphism|cartesian]]} if for any arrow $\psi:e''\to e$ in $E$ and $g:p(e'')\to p(e')$ in $B$ such that $p(\phi)\circ g = p(\psi)$, there exists a unique $\chi:e''\to e'$ such that $\psi = \phi\circ \chi$ and $p(\chi) =g$. In other words, for any $\psi$, any filling of the image of the following diagram under $p$ can be lifted up to a filling in $E$: \begin{displaymath} \itexarray{ e''& \dashrightarrow & e'\\ & _{\mathllap{\psi}}\searrow & \downarrow^\mathrlap{\phi}\\ & & e } \end{displaymath} We say that $p:E\to B$ is a \textbf{fibration} if for any $e\in E$ and $f:b\to p(e)$, there is a cartesian arrow $\phi:e'\to e$ with $p(\phi)=f$. Such an arrow is called a ``cartesian lifting'' of $f$ to $e$, and a choice of cartesian lifting for every $e$ and $f$ is called a [[cleavage]]. Thus, assuming the [[axiom of choice]], a functor is a fibration iff it admits some cleavage. As a side note, we say that $\phi$ is [[prefibered category|weakly cartesian]] if it has the property described above only when $g$ is an identity. One can prove that $p$ is a fibration if and only if firstly, it has the above property with ``cartesian'' replaced by ``weakly cartesian,'' and secondly, the composite of weakly cartesian arrows is weakly cartesian. In a fibration, every weakly cartesian lifting $\phi$ of $f$ to $e$ is in fact cartesian (as one can show by combining the universal properties of $\phi$ and of a given cartesian lifting of $f$ to $e$), but this is not true in general. Some sources say ``cartesian'' and ``strongly cartesian'' instead of ``weakly cartesian'' and ``cartesian,'' respectively. If weakly cartesian liftings exist but weakly cartesian arrows are \emph{not} necessarily closed under composition, one sometimes speaks of a [[prefibered category]]. A square \begin{displaymath} \itexarray{E' & \to & E \\ \downarrow && \downarrow \\ B' &\to & B} \end{displaymath} is a \textbf{cartesian morphism} or \emph{morphism of fibrations} if the top arrow takes cartesian arrows to cartesian arrows. Most frequently when considering morphisms of fibrations, the bottom arrow $B'\to B$ is an identity. A 2-cell between morphisms of fibrations is a pair of 2-cells, one lying over the other. If the bottom arrow is an identity, this means that the top 2-cell is ``vertical'' (its components lie in fibers). \hypertarget{fibrations_versus_pseudofunctors}{}\subsection*{{Fibrations versus pseudofunctors}}\label{fibrations_versus_pseudofunctors} Given a fibration $p:E\to B$, we obtain a pseudofunctor $B^{op}\to Cat$ by sending each $b\in B$ to the category $E_b = p^{-1}(b)$ of objects mapping onto $b$ and morphisms mapping onto $1_b$. To obtain the action on morphisms, given an $f:a\to b$ in $B$ and an object $e\in E_b$, we choose a cartesian arrow $\phi:e'\to e$ over $f$ and call its [[source]] $f^*(e)$. The universal factorization property of cartesian arrows then makes $f^*$ into a functor $E_b \to E_a$, and it is easy to verify that it is a pseudofunctor. The functor in the other direction is called the [[Grothendieck construction]]. This yields a [[strict 2-equivalence of 2-categories]] between \begin{itemize}% \item fibrations over $B$, morphisms of fibrations over $Id_B$, and 2-cells over $id_{Id_B}$, and \item [[pseudofunctors]] of the form $B^{op}\to Cat$, [[pseudonatural transformations]], and [[modifications]]. \end{itemize} In fact, this is an instance of the general theory of representability for [[generalized multicategories]]. There is a monad $T$ whose pseudoalgebras are pseudofunctors $B^{op}\to Cat$, and whose ``generalized multicategories'' are functors $E\to B$. Such a multicategory is ``representable'' precisely when it is a fibration, and moreover there is an induced monad $\hat{T}$ on $Cat/B$ which is [[lax-idempotent 2-monad|colax-idempotent]] and whose pseudoalgebras are precisely the fibrations. This correspondence also generalizes to the correspondence between arbitrary functors over $B$ and [[displayed categories]] over $B$, i.e. normal lax functors $B\to Prof$. \hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks} \begin{itemize}% \item Fibrations are a ``nonalgebraic'' structure, since the base change functors $f^*$ are determined by universal properties, hence uniquely up to isomorphism. By contrast, pseudofunctors are an ``algebraic'' structure, since the functors $f^*$ are specified, together with the necessary coherence data and axioms; the latter come for free in a fibration because of the universal property. \item A [[stack]], being a particular type of pseudofunctor, can also be described as a particular sort of fibration. This was the original application for which Grothendieck introduced the notion. \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item Let $Ring$ be the category of [[ring|rings]], and $Mod$ the category of pairs $(R,M)$ where $R$ is a ring and $M$ is a (left) $R$-module. Then the evident forgetful functor $Mod\to Ring$ is a fibration and an opfibration. The functors $f^*$ are given by restriction of scalars, $f_!$ is extension of scalars, and the right adjoint $f_*$ is coextension of scalars. \item Let $C$ be any category with pullbacks and $\mathbf{2}$ the free-living arrow, so that $C^{\mathbf{2}}$ is the category of arrows and commutative squares in $C$. Then the ``codomain'' functor $C^{\mathbf{2}} \to C$ is a fibration and opfibration. The cartesian arrows are precisely the pullback squares, and the functors $f_!$ are just given by composition. The right adjoints $f_*$ exist iff $C$ is [[locally cartesian closed category|locally cartesian closed]]. The term ``cartesian'' is motivated by this example, which is usually called the \textbf{[[codomain fibration]]} over $C$. \item Let $C$ be any category and let $Fam(C)$ be the category of set-indexed families of objects of $C$. The forgetful functor $Fam(C)\to Set$ taking a family to its indexing set is a fibration; the functors $f^*$ are given by reindexing. They have left adjoints iff $C$ has small coproducts, and right adjoints iff $C$ has small products. \end{itemize} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \begin{itemize}% \item It is easy to verify that fibrations are closed under [[pullback]] in [[Cat]], and that the composite of fibrations is a fibration. This latter property is notably difficult to even express in the language of pseudofunctors. \item Every fibration (or opfibration) is a [[Conduché functor]], and therefore an [[exponentiable morphism]] in [[Cat]]. \item Every fibration or opfibration is an [[isofibration]]. In particular, [[strict 2-limit|strict 2-pullbacks]] of fibrations are also [[2-pullbacks]]. \item If $p\colon A\to B$ is a fibration, then [[limits]] in $A$ can be constructed out of limits in $B$ and in the fiber categories. Specifically, given a [[diagram]] $f\colon I\to A$, let $L$ be the limit of $p f\colon I\to B$, with projections $\phi_i\colon L \to p(f(i))$. Then for each $i\in I$, let $g(i) = \phi_i^*(f(i)) \in p^{-1}(L)$; these objects form a diagram $g\colon I\to p^{-1}(L)$ whose limit is the limit of $f$. Dually, if $p\colon A\to B$ is an opfibration, then [[colimits]] in $A$ can be constructed out of those in $B$ and in the fiber categories. \item If $p\colon A\to B$ is a fibration and $B$ admits an [[orthogonal factorization system]] $(E,M)$, then there is a factorization system $(E',M')$ on $A$, where $M'$ is the class of cartesian arrows whose image in $B$ lies in $M$, and $E'$ is the class of all arrows whose image in $B$ lies in $E$. A dual construction is possible if $p$ is an opfibration. If it is a bifibration (or more generally, an [[ambifibration]] over $(E,M)$), then these together form a [[ternary factorization system]]. \item Under suitable hypotheses, versions of the preceding fact can be shown [[weak factorization systems]] and [[model structures]] as well. \item Generalizing in a different direction, if $p\colon A\to B$ is a fibration and $(E,M)$ is a [[factorization structure for sinks]] on $B$, then $A$ admits a factorization structure for sinks $(E',M')$, where $M'$ is the class of cartesian arrows whose image in $B$ lies in $M$, and $E'$ is the class of all arrows whose image in $B$ lies in $E$. Similarly, we can lift factorization structures for \emph{cosinks} along an opfibration. To lift in the ``opposite'' way we require more of $p$; see [[topological concrete category]]. \end{itemize} \hypertarget{variations}{}\subsection*{{Variations}}\label{variations} \hypertarget{discrete_and_groupoidal_fibrations}{}\subsubsection*{{Discrete and groupoidal fibrations}}\label{discrete_and_groupoidal_fibrations} One important special case of a fibration is when each fiber is a [[groupoid]]; these correspond to pseudofunctors $B^{op}\to Grpd$. These are also called \emph{categories fibered in groupoids}. A fibration $E\to B$ is fibered in groupoids precisely when \emph{every} morphism in $E$ is cartesian. Another important special case is when each fiber is a [[discrete category]]; these correspond to functors $B^{op}\to Set$. These are also called \emph{[[discrete fibrations]]}. A functor $p\colon E\to B$ is a discrete fibration precisely when for every $e\in E$ and $f\colon b\to p(e)$ there is a \emph{unique} lifting of $f$ to a morphism $e'\to e$. \hypertarget{opfibrations_and_bifibrations}{}\subsubsection*{{Opfibrations and bifibrations}}\label{opfibrations_and_bifibrations} We say that $p\colon E\to B$ is an \textbf{opfibration} if $p^{op}:E^{op}\to B^{op}$ is a fibration. These correspond to covariant pseudofunctors $B\to Cat$. A functor that is both a fibration and an opfibration is called a \textbf{[[bifibration]]}. It is not hard to see that a fibration is a bifibration iff each functor $f^*$ has a left adjoint, written $f_!$ or $\Sigma_f$. In many cases $f^*$ also has a right adjoint, written $f_*$ or $\Pi_f$, but this is not as easily expressible in fibrational language. Grothendieck originally called an opfibration a \emph{cofibered category}, and if the fibers are groupoids a [[category cofibered in groupoids]] (SGA I, [[Higher Topos Theory]]). However, that term has fallen out of favor in the homotopy-theory and category-theory communities (though still used sometimes in algebraic geometry), because an opfibration still has a \emph{lifting} property, as is characteristic of other notions of [[fibration]], as opposed to the \emph{extension} property exhibited by [[cofibration]]s in [[homotopy theory]]. Note that an opfibration is the same as an internal fibration in the 2-category $Cat^{co}$, while it is the fibrations in the 2-category $Cat^{op}$ which are more deserving of the name ``cofibration.'' Note also that a given pseudofunctor $B^{op}\to Cat$ can be represented both by a fibration $E_1\to B$ and by an opfibration $E_2\to B^{op}$. However $E_2$ is \emph{not} the opposite category of $E_1$. \hypertarget{StreetFibration}{}\subsubsection*{{Version respecting the Principle of equivalence}}\label{StreetFibration} There is something apparently in violation of the [[principle of equivalence]] about the notion of fibration, namely the requirement that for every $f:a\to b$ and $e\in E_b$ there exists a $\phi:e'\to e$ such that $p(e')$ is \emph{[[equality|equal]]}, rather than merely [[isomorphism|isomorphic]], to $a$. This is connected with the fact that we use strict fibers, rather than [[essential fiber]]s, and that fibrations and pseudofunctors can be recovered from each other up to isomorphism rather than merely equivalence. Note that almost any fibration between ``concrete'' categories that arises in practice does satisfy this strict property. However, even stating the strict property requires our categories to be [[strict categories]] (i.e. to have a notion of equality of objects), so when working in a context where not all categories are strict (such as [[internal categories in homotopy type theory]], or with objects in a [[bicategory]]) it is problematic. Sometimes (such as in type theory) this can be avoided by working with [[displayed categories]] instead, but in some cases (such as internally in a bicategory) one does not have classifying objects either, so it is sometimes useful to have a version which manifestly accords to the [[principle of equivalence]]. The correct modification, first given by [[Ross Street]], is simply to require that for any $f:a\to b$ and $e\in E_b$ there exists a cartesian $\phi:e'\to e$ and an \emph{isomorphism} $h:p(e') \cong a$ such that $f\circ h = p(\phi)$; the definition of ``cartesian'' is unchanged; this gives the notion of [[Street fibration]]. Every [[equivalence of categories]] is a Street fibration, which is not true for the concept of Grothendieck fibrations according to the [[principle of equivalence]], but every Street fibration can be factored as an equivalence of categories followed by a Grothendieck fibration. We might also think that it violates the [[principle of equivalence]] to say that the target of the cartesian arrow $\phi$ is equal to the given object $e$, akin to the topological distinction between [[Serre fibrations]] and [[Dold fibrations]], where the initial point of a lifted path can only be specified up to homotopy. However, this part of the definition is really better regarded as a typing assertion, akin to saying, in the definition of a [[product]] of two objects $A$ and $B$, that the target of the two projections $A\times B\to A$ and $A\times B \to B$ are \emph{equal} to $A$ and $B$. Moreover, any weakening along these lines would actually be equivalent to the version above: if we demand only that for any $f\colon a\to b$ in $B$ and $e\in E_b$, there exists a cartesian $\phi\colon e' \to \hat{e}$ with $p(\phi)=f$ and an isomorphism $\hat{e}\cong e$ in the fiber $E_b$, then you can just compose $\phi$ with the isomorphism $\hat{e}\cong e$ to get a cartesian arrow $\hat{\phi}\colon e'\to e$ with $p(\phi)=f$ still. The reason this doesn't work in topology is that paths come with parametrizations, and requiring the lower triangle (in the square drawn at [[Dold fibration]]) to commute strictly prevents the reparametrization necessary to compose with a vertical homotopy. The idea of [[proto-fibration]] is closely related to this. \hypertarget{internal_version}{}\subsubsection*{{Internal version}}\label{internal_version} In a [[strict 2-category]] $K$, a morphism $p:E\to B$ is called a fibration if for every object $X$, $K(X,E)\to K(X,B)$ is a fibration of categories, and for every morphism $f:Y\to X$, the square \begin{displaymath} \itexarray{ K(X,E) & \to & K(Y,E) \\ \downarrow && \downarrow \\ K(X,B) & \to & K(Y,B)} \end{displaymath} is a morphism of fibrations. There is an alternate characterization in terms of [[comma object]]s and adjoints, see [[fibration in a 2-category]]. The same definition works in a [[bicategory]], as long as we use the version in accord with the [[principle of equivalence]] above. Interpreted in [[Cat]] we obtain the explicit notion we started with. \hypertarget{twosided_version}{}\subsubsection*{{Two-sided version}}\label{twosided_version} A \textbf{two-sided fibration}, or a \textbf{fibration from $B$ to $A$}, is a span $A \leftarrow E \rightarrow B$ such that $E\to A$ is a fibration, $E\to B$ is an opfibration, and the structure ``commutes'' in a natural way. Such two-sided fibrations correspond to pseudofunctors $A^{op}\times B \to Cat$. If they are discrete, they correspond to functors $A^{op}\times B\to Set$, i.e. to [[profunctors]] from $B$ to $A$. See [[two-sided fibration]] for more details. Note that a pseudofunctor $A^{op}\times B \to Cat$ can also be represented by an opfibration $E_1\to A^{op}\times B$ and by a fibration $E_2\to A\times B^{op}$, but there is no simple relationship between the three categories $E$, $E_1$, and $E_2$. \hypertarget{multicategory_version}{}\subsubsection*{{Multicategory version}}\label{multicategory_version} There is an analog for [[multicategories]]. See \begin{itemize}% \item [[fibration of multicategories]] \end{itemize} \hypertarget{higher_categorical_versions}{}\subsubsection*{{Higher categorical versions}}\label{higher_categorical_versions} There is also a notion of fibration for 2-categories that has been studied by Hermida. See [[n-fibration]] for a general version. For [[(∞,1)-category|(∞,1)-categories]] the notion of fibered category is modeled by the notion of [[Cartesian fibration]] of simplicial sets. The corresponding analog of the Grothendieck construction is discussed at [[(∞,1)-Grothendieck construction]]. \hypertarget{higher_multicategory_version}{}\subsubsection*{{Higher multicategory version}}\label{higher_multicategory_version} \begin{itemize}% \item [[Cartesian fibration of dendroidal sets]] \end{itemize} \hypertarget{alternate_definitions}{}\subsection*{{Alternate definitions}}\label{alternate_definitions} There are several alternate characterizations of when a functor is a fibration, some of which are more convenient for [[fibration in a 2-category|internalization]]. Here we mention a few. \hypertarget{in_terms_of_adjoints}{}\subsubsection*{{In terms of adjoints}}\label{in_terms_of_adjoints} Since Grothendieck fibrations are a strict notion, in what follows we denote by $B\downarrow p$ the strict [[comma category]] (i.e. determined up to isomorphism, not merely up to equivalence) and by $Cat/B$ the [[strict slice 2-category]] (elsewhere denoted $Cat/_s B$). \begin{ulemma} A functor $p \colon E \to B$ is a cloven fibration if and only if the canonical functor $i \colon E \to B\downarrow p$ has a right adjoint $r$ in $Cat / B$. \end{ulemma} \begin{proof} First, recall that the strict slice 2-category $Cat/X$ has objects the functors $C \to X$, as morphisms the commuting triangles \begin{displaymath} \itexarray{C & \stackrel{h}{\to} & C' \\ & f \searrow \swarrow g & \\ & X, & } \end{displaymath} and as 2-cells the natural transformations $\alpha : h_1 \to h_2$ such that $g\alpha = id_f$. Next, recall that the [[comma category]] $B\downarrow p$ has objects the triples $(x, e, k)$, with $k \colon x \to p e$. Let $\pi \colon B\downarrow p \to B$ denote the projection $(x, e, k) \mapsto x$. The canonical morphism $i:E \to B\downarrow p$ is simply the inclusion functor of identity maps $i e = 1_{p e} \colon p e \to p e$. Somewhat imprecisely, seeing both categories $E$ and $B\downarrow p$ as sitting over $B$ means that functors between those should be the identity on the $b$ component, and natural transformations should have the identity as their $b$ component. To give an adjunction $i \dashv r$ it suffices to give, for each $k \colon x \to p e$ in $B\downarrow p$, an object $r k$ in $E$ such that $p r k = x$ and an arrow $i r k = 1_x \to k$ in $B\downarrow p$ that is [[universal arrow|universal]] from $i$ to $k$. For the adjunction to live in $Cat / B$ we must have that $\pi \circ i r k = 1_{p r k} = 1_x$, so the universal arrow must be of the form \begin{displaymath} \itexarray{ x & \overset{1}{\to} & x \\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e } \end{displaymath} and thus amounts to a choice of $\epsilon_k \colon r k \to e$ in $E$ such that $p \epsilon_k = k$. The universal property of $\epsilon_k$ tells us that for any other morphism in $B\downarrow p$ from some $i y$ to $k$, i.e., for any $y$ and any pair $(f,g)$ making the square \begin{displaymath} \itexarray{ p y & \stackrel{1}{\to} & p y \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{p g} \\ x & \underset{k}{\to} & p e } \end{displaymath} commute, there is a unique map $h \colon y \to r k$ in $B$ such that the above square factors in $B\downarrow p$ as \begin{displaymath} \itexarray{ p y & \stackrel{1}{\to} & p y \\ \mathllap{p h} \downarrow & & \downarrow \mathrlap{p h} \\ \mathllap{p r k =} x & \stackrel{1}{\to} & x \mathrlap{= p r k}\\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e. } \end{displaymath} In other words, the universal property provides a unique $h$ such that $\epsilon_k h = g$ and $p h = f$, which exactly asserts that $\epsilon_k$ is a cartesian lift of $k$. So the existence of a right adjoint to $i$ means precisely that for each morphism $k \colon x \to p e$ a choice is given of a cartesian lift of $k$, which means in turn that $p$ is a cloven fibration. \end{proof} \hypertarget{discussions}{}\subsection*{{Discussions}}\label{discussions} The following discussion brings out some interesting points about the equivalence between fibrations and pseudofunctors. [[Sridhar Ramesh]]: I have a (possibly stupid) question about the nature of this equivalence. I assume the idea here is that moving from a cloven fibration to the corresponding pseudofunctor is in some sense ``inverse'' to carrying out the Grothendieck construction in the other direction. But, in trying to get a good intuition for the nuances of non-splittable fibrations, I seem to be stumbling upon just in what sense this is so. For example, consider the nontrivial group homomorphism from Z (integer addition) to Z\_2 (integer addition modulo 2); this gives us a non-splittable fibration (and, for that matter, an opfibration), for which a cleavage can be readily selected. No matter what cleavage is selected, the corresponding (contravariant) pseudofunctor from Z\_2 to Cat, it would appear to me, is the one which sends the unique object in Z\_2 to the subcategory of Z containing only even integers (let us call this 2Z), and which sends both of Z\_2's morphisms to identity; thus, it is actually a genuine functor, and indeed, a ``constant'' functor. Applying the Grothendieck construction now, I would seem to get back the projection from Z\_2 X 2Z onto Z\_2. But can this really be equivalent to the fibration I started with? After all, Z and Z\_2 X 2Z are very different groups. So either ``equivalence'' means something trickier here than I realize, or I keep making a mistake somewhere along the line. Either way, it'd be great if someone could help me see the light. [[Mike Shulman]]: Good question! I think the missing subtlety is that a pseudofunctor is not uniquely determined by its action on objects and morphisms, even if its domain is a mere category (or a mere group); there are also natural coherence isomorphisms $g^* f^* \cong (g f)^*$ to take into account. For instance, if $g$ is the nonidentity element of $\mathbb{Z}/2$, then $g g = 1$, so even if $g$ acts by the identity on $2\mathbb{Z}$, a pseudofunctor also contains the additional data of a natural automorphism of the identity of $2\mathbb{Z}$, i.e. a (central) element of $2\mathbb{Z}$. If you start from $\mathbb{Z}$, then depending on your cleavage your element can be anything that's 2 mod 4, while if you start from $\mathbb{Z}/2\times 2\mathbb{Z}$, your element can be anything that's 0 mod 4. Finally, there is a pseudonatural equivalence between two such pseudofunctors just when their corresponding elements differ by a multiple of 4, so you get exactly two equivalence classes of pseudofunctors, corresponding to the two groups $\mathbb{Z}$ and $\mathbb{Z}/2\times 2\mathbb{Z}$. Of course we are reproducing the classification of group extensions via group cohomology. By the way, this sort of thing (by which I mean, the cohomology class that classifies some categorical structure arising as the trace of a coherence isomorphism) happens in lots of other places too. For instance, a [[2-group]] is classified by a group $G$, an abelian group $H$, an action of $G$ on $H$, and an element in $H^3(G;H)$. If you replace a 2-group by a [[skeleton|skeletal]] one, then $G$ is the group of objects (which is strictly associative and unital, by skeletality), $H$ is the group of endomorphisms of the unit, and the action is defined by ``whiskering''. The cohomology class comes from the [[associator]] isomorphism, which can (and often must) still be nontrivial even though the multiplication is ``strictly associative'' at the level of objects (by skeletality). \emph{Toby}: So the multiplication is strictly associative, but the $2$-group itself is not a strict $2$-group, since it uses a different associator than the identity. As in the example of the pseudofunctor from $Z_2$ to $Cat$, there is some additional structure here which is not trivial, even though it seems like it could be. [[Sridhar Ramesh]]: Ah, of course, that's what I was missing. Thanks, both of you; that clears it all up. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item \textbf{Grothendieck fibration}, [[two-sided fibration]] \item [[monoidal fibration]] \item [[n-fibration]] \item [[Cartesian fibration]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The concept was introduced in the context of [[descent theory]] by [[Alexander Grothendieck]] in a [[Bourbaki]] seminar in 1959-60 (\href{http://www.numdam.org/numdam-bin/item?id=SB_1958-1960__5__299_0}{numdam}) and then elaborated in expos\'e{} VI of \begin{itemize}% \item [[Alexander Grothendieck]] , \emph{Rev\^e{}tements Etales et Groupe Fondamental - S\'e{}minaire de G\'e{}ometrie Alg\'e{}brique du Bois Marie 1960/61} ([[SGA 1]]) , LNM \textbf{224} Springer Heidelberg 1971. (updated version with comments by M. Raynaud: \href{http://arxiv.org/abs/math/0206203}{arxiv.0206203}) \end{itemize} Another important early reference is \begin{itemize}% \item [[John W. Gray]], \emph{Fibred and Cofibred Categories} , pp.21-83 in Eilenberg, Harrison, MacLane, R\"o{}hrl (eds.), \emph{Proceedings of the Conference on Categorical Algebra - La Jolla 1965} , Springer Heidelberg 1966. \end{itemize} A concise and didactic introduction to fibrations can be found in ch.12 of \begin{itemize}% \item [[Michael Barr]], [[Charles Wells]], \emph{Category Theory for Computing Science} , Prentice Hall 1995$^3$. (\href{http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html}{TAC reprints no.22 (2012)}) \end{itemize} More thorough is ch.8 of \begin{itemize}% \item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]] vol. 2} , Cambridge UP 1994. \end{itemize} The elephant contains a lot of basic information and some good intuition\footnote{But beware that Johnstone uses the non-standard words ``prone'' and ``supine'' where most people say ``cartesian'' and ``opcartesian'' morphism!} : \begin{itemize}% \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol.1, Oxford UP 2002. (Part B)} \end{itemize} A standard reference that focusses on the interplay with [[type theory]] is \begin{itemize}% \item [[Bart Jacobs]], \emph{Categorical Logic and Type Theory} , Elsevier Amsterdam 1999. \end{itemize} B\'e{}nabou's perspective on fibrations and some results related to Moens' thesis are documented in the highly recommendable lecture notes: \begin{itemize}% \item [[Thomas Streicher]], \emph{Fibred Categories \`a{} la Jean B\'e{}nabou} , arXiv:1801.02927 (2018). (\href{https://arxiv.org/abs/1801.02927}{abstract}) \end{itemize} B\'e{}nabou \emph{en personne} on the role of fibrations in the foundations of category theory and on differences to [[indexed categories]]: \begin{itemize}% \item [[Jean Bénabou]], \emph{Fibered Categories and the Foundations of Naive Category Theory} , JSL \textbf{50} (1985) pp.10-37. \end{itemize} The following lecture notes stress the original perspective of [[algebraic geometry]]: \begin{itemize}% \item [[Angelo Vistoli]], \emph{Notes on Grothendieck topologies, fibered categories and descent theory} . (\href{http://homepage.sns.it/vistoli/descent.pdf}{pdf}) \end{itemize} For the use of fibrations in homotopy see e.g. \begin{itemize}% \item [[Ronnie Brown|R. Brown]], R. Sivera, \emph{Algebraic colimit calculations in homotopy theory using fibred and cofibred categories}, TAC \textbf{22} (2009) pp.222-251. (\href{http://www.tac.mta.ca/tac/volumes/22/8/22-08.pdf}{pdf}) \end{itemize} [[André Joyal]]`s take on fibrations can be found here: \begin{itemize}% \item \emph{[[joyalscatlab:Grothendieck fibrations]]} \end{itemize} The Wikipedia entry on fibered / fibred categories is okay, and contains a number of other references: \begin{itemize}% \item \href{http://en.wikipedia.org/wiki/Fibred_category}{Wikipedia entry} \end{itemize} [[!redirects Grothendieck fibrations]] [[!redirects Grothendieck opfibration]] [[!redirects Grothendieck cofibration]] [[!redirects Grothendieck opfibrations]] [[!redirects Grothendieck cofibrations]] [[!redirects opfibration]] [[!redirects opfibrations]] [[!redirects bifibration]] [[!redirects bifibrations]] [[!redirects fibered category]] [[!redirects fibred category]] [[!redirects fibered categories]] [[!redirects fibred categories]] [[!redirects (Grothendieck) fibration]] [[!redirects (Grothendieck) fibrations]] [[!redirects morphism of fibrations]] [[!redirects morphisms of fibrations]] \end{document}