nLab Grothendieck fibration

Idea

A Grothendieck fibration (named after Alexander Grothendieck, also called a fibered category or just a fibration) is a functor $p \colon E\to B$ such that the fibers $E_b = p^{-1}(b)$ depend (contravariantly) pseudofunctorially on $b\in B$.

One also says that $E$ is a fibered category over $B$. One calls $E$ the total category of the fibration (not to be confused with the independent notion of total category) and $B$ its base category.

Dually, in a (Grothendieck) opfibration the dependence is covariant.

There is an equivalence of 2-categories

$$Fib(B) \overset {\simeq} {\leftrightarrow} [B^{op}, Cat] \;\colon\; \textstyle{\int}$$

between the 2-category of fibrations, cartesian functors, and vertical natural transformations over $B$, and the 2-category $[B^{op},Cat]$ of contravariant pseudofunctors from $B$ to Cat, also called $B$-indexed categories.

The “Grothendieck construction” $\int \colon [B^{op}, Cat] \to Fib(B) \colon F \mapsto \int F$ of a fibration from a pseudofunctor.

Those fibrations corresponding to pseudofunctors that factor through Grpd are called categories fibered in groupoids.

Definition

Definition

(cartesian morphism)
Given a functor $P \,\colon\, \mathcal{E} \longrightarrow \mathcal{B}$, a morphism $f \colon x \to y$ in its domain category $\mathcal{E}$ is called cartesian if for any morphism $g \colon z\to y$ in $\mathcal{E}$ and $w \colon P(z)\to P(x)$ in $\mathcal{B}$ such that $P(f)\circ w = P(g)$, there exists a unique $\widehat{w} \,\colon\, z\to x$ such that $g = f \circ \widehat{w}$ and $P(\widehat{w}) = w$:

Definition

(Grothendieck fibration)
A functor $p \colon \mathcal{E} \to \mathcal{B}$ is a fibration if for all objects $y \in \mathcal{E}$ and morphisms $f_0 \colon x_0 \to P(y)$, there is a cartesian morphism $f \colon x \to y$ (Def. ) such that $P(f) = f_0$.

Such $f$ is also called a “cartesian lifting” of $Pf$ to $y$, and a choice of cartesian lifting for every $y$ and $Pf$ is called a cleavage. Thus, assuming the axiom of choice, a functor is a fibration iff it admits some cleavage.

Remark

(weakly cartesian morphisms)
We say that a morphism $f$ is weakly cartesian if it has the property of a Cartesian morphism (Def. ) only when $w$ is an identity morphism.

One can prove that $P$ is a fibration (Def. ) 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 $f$ of $Pf$ to $y$ is in fact cartesian (as one can show by combining the universal properties of $f$ and of a given cartesian lifting 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 not necessarily closed under composition, one sometimes speaks of a prefibered category.

Definition

(morphism of fibrations – fibered functor)
For $P$ and $P'$ fibrations (Def. ), a commuting square of functors of the form

(1)$$\array{ \mathcal{E}' & \longrightarrow & \mathcal{E} \\ \mathllap{{}^{P'}}\big\downarrow && \big\downarrow \mathrlap{{}^{P}} \\ \mathcal{B}' & \longrightarrow & \mathcal{B} }$$

is a morphism of fibrations (also called a cartesian square or fibred functor) if the top functor preserves Cartesian morphisms (Def. ).

Definition

(2-morphisms of fibrations – fibered natural transformation)
A 2-morphism (or fibered natural transformation) between morphisms of fibrations (Def. ) is a pair of natural transformations, one lying over the other in (1).

Remark

If the functor on base categories is an identity, then a 2-morphism (Def. ) is a natural transformations between total categories which is “vertical” in that its component morphisms lie in fibers.

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

• fibrations over $B$, morphisms of fibrations over $Id_B$, and 2-cells over $id_{Id_B}$, and

• pseudofunctors of the form $B^{op}\to Cat$, pseudonatural transformations, and modifications.

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 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 $B\to Prof$.

Fibrations versus presheaves of categories

The Grothendieck construction is an equivalence of bicategories from the bicategory of presheaves of categories to the bicategory of Grothendieck fibrations.

As explained there, this equivalence, interpreted as a functor between 1-categories, has both a left and a right adjoint (which are equivalent in the bicategorical context). Roughly speaking, the left adjoint strictifies a Grothendieck fibration by adding formal pullbacks of objects, which themselves pullback strictly, whereas the right adjoint strictifies a Grothendieck fibration by equipping object with a functorial choice of a pullback for each possible morphism. See Grothendieck construction for more details.

These two adjunctions can be turned into Quillen equivalences of model categories. This can be deduced, for example, from two Quillen equivalences between cartesian fibrations and presheaves of marked simplicial sets on the quasicategory given by the nerve of $C$. See straightening functor for more details.

Remarks

• 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.

• 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.

Examples

Example

Let $\VBun$ denote the category whose objects are pairs $(M,V)\in\Man\times\Man$ of manifolds where $q:V\longrightarrow M$ is a vector bundle over $M$. A morphism $(M_{1},V_{1})\longrightarrow (M_{2},V_{2})$ is a commutative square:

where $f$ is a smooth map, and $g$ is a smooth map such that $g\mid_{q_{1}^{-1}(m)}:q_{1}^{-1}(m)\longrightarrow q_{2}^{-1}(f(m))$ is linear for every $m\in M_{1}$. Composition is given as in the arrow category. There is a forgetful functor $\mathcal{P}:\VBun\longrightarrow\Man$ which projects onto the base data. To see that this functor is a Grothendieck fibration, first consider a morphism of the form:

given by the pullback bundle of a vector bundle $V_{2}\longrightarrow M_{2}$ along a smooth map $M_{1}\longrightarrow M_{2}$. Certainly, the universal property of the pullback will supply the conditions for this square to define a cartesian morphism in $\VBun$ with respect to the the functor $\mathcal{P}$. We show that these are in fact all the cartesian morphisms in $\VBun$. Let $e' = (V_{1},M_{1})$, $e = (V_{2},M_{2})$, and $e'' = (V,M)$, then by applying the definition from above directly, we see that a morphism $\phi:e'\longrightarrow e$ in $\VBun$, i.e., a square:

is cartesian, if given any morphism $\psi:e''\longrightarrow e$ in $\VBun$, i.e., a square (which we draw around the first):

and any morphism $g:\mathcal{P}(e'')\longrightarrow \mathcal{P}(e')$ in $\Man$ such that $\mathcal{P}(\phi)\circ g = \mathcal{P}(\psi)$, i.e. a smooth map $g:M\longrightarrow M_{1}$ that fills in the above diagram to a commuting diagram:

then there exists a unique morphism $\chi:e''\longrightarrow e'$ in $\VBun$ such that $\psi = \phi\circ\chi$ and $\mathcal{P}(\chi) = g$, i.e., there exists a unique morphism $V\longrightarrow V_{1}$ that further fills the diagram to a commuting diagram:

It immediately follows from the above unfolding of the definition that $\VBun(e'',e')\cong\VBun(e'',p)$ naturally in $e''$, where $p$ is the pullback bundle of $V_{2}\longrightarrow M_{2}$ along $M_{1}\longrightarrow M_{2}$, and thus by the Yoneda lemma $e'\cong p$, and thus $V_{1}\cong\mathcal{P}(e')^{*}V_{2}$. Note that the isomorphisms $e'\cong p$ preserve the commutativity of the diagrams we have drawn above by the universal property of the pullback, and by the definition of a cartesian morphism. Now that we have identified the cartesian morphisms, the statement that $\mathcal{P}:\VBun\longrightarrow\Man$ is a Grothendieck fibration is equivalent to the statement that vector bundles can be pulled back along smooth maps. This is of course true, and thus $\mathcal{P}:\VBun\longrightarrow\Man$ is a Grothendieck fibration. The associated functors $f^{*}$ are the pullback psuedofunctors $f^{*}:\VBun_{M_{2}}\longrightarrow \VBun_{M_{1}}$ that pullback vector bundles over a manifold $M_{2}$ to vector bundles over a manifold $M_{1}$ along a smooth map $f:M_{1}\longrightarrow M_{2}$.

Example

Let Ring be the category of 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.

Example

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. The term “cartesian” is motivated by this example, which is usually called the codomain fibration over $C$.

Example

Let $C$ be a category with binary cartesian products. Then for each object $\Gamma$, we can construct an indexed comonad $C \to Comonad$, the environment comonad $\Gamma \times -$ on $C$, and taking Kleisli categories we get a functor $C^{op} \to Cat$. The corresponding fibration is called the simple fibration? (e.g., in Jacobs 1998) over $C$ because it can be used to give semantics to simply-typed lambda calculus. The simple fibration can be seen as a full subfibration of the codomain fibration, with objects being the projections $\Gamma \times A \to \Gamma$.

Example

(families, free coproduct completion)
Let $C$ be any category and let $Fam(C)$ be the category of set-indexed families of objects of $C$ (the free coproduct completion of $C$, which is a Grothendieck construction by the discussion there). 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.

Properties

• 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.

• Every fibration (or opfibration) is a Conduché functor, and therefore an exponentiable morphism in Cat.

• Every fibration or opfibration is an isofibration. In particular, strict 2-pullbacks of fibrations are also 2-pullbacks.

• Suppose that $p\colon A\to B$ is a fibration, and $B$ has $I$-indexed limits. Then $A$ has $I$-indexed limits and $p$ strictly preserves them if and only if $p$ has $I$-indexed fibred limits. This is in Gray 66; for a general perspective, see Hermida 99, Corollary 4.9. Concretely, for instance, from right to left, 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.

• Suppose that $p\colon A\to B$ is a fibred cartesian closed category with simple products?. Then if $B$ is cartesian closed, so is $A$ and $p$ preserves the structure. (See Hermida 99, Corollary 4.12.) But the converse does not hold, see Section 4.3.2 of Hermida’s thesis.

• 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.

• Under suitable hypotheses, versions of the preceding fact can be shown weak factorization systems and model structures as well.

• 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 cosinks along an opfibration. To lift in the “opposite” way we require more of $p$; see topological concrete category.

Variations

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 categories fibered in groupoids. A fibration $E\to B$ is fibered in groupoids precisely when 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 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 unique lifting of $f$ to a morphism $e'\to e$.

Opfibrations and bifibrations

We say that $p\colon E\to B$ is an 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 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 cofibered category, and if the fibers are groupoids a category cofibered in groupoids (cf. 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 lifting property, as is characteristic of other notions of fibration, as opposed to the extension property exhibited by cofibrations 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 not the opposite category of $E_1$.

Version respecting the Principle of equivalence

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 equal, rather than merely isomorphic, to $a$. This is connected with the fact that we use strict fibers, rather than essential fibers, 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 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 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.

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

$$\array{ K(X,E) & \to & K(Y,E) \\ \downarrow && \downarrow \\ K(X,B) & \to & K(Y,B)}$$

is a morphism of fibrations. There is an alternate characterization in terms of comma objects 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.

Two-sided version

A two-sided fibration, or a 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 $E_2$.

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)-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.

Versions for other categorical structures

• There is an analog for multicategories. See fibration of multicategories.

• For higher multicategories, see Cartesian fibration of dendroidal sets.

• For double categories, see double fibration.

Alternate definitions

There are several alternate characterizations of when a functor is a fibration, some of which are more convenient for internalization. Here we mention a few.

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.

Lemma

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 $Cat / B$.

Proof

First, recall that the strict slice 2-category $Cat/X$ has objects the functors $C \to X$, as morphisms the commuting triangles

$$\array{C & \stackrel{h}{\to} & C' \\ & f \searrow \swarrow g & \\ & X, & }$$

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 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

$$\array{ x & \overset{1}{\to} & x \\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e }$$

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

$$\array{ p y & \stackrel{1}{\to} & p y \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{p g} \\ x & \underset{k}{\to} & p e }$$

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

$$\array{ p y & \stackrel{1}{\to} & p y \\ \mathllap{p h} \downarrow & & \downarrow \mathrlap{p h} \\ \mathllap{p r k =} x & \stackrel{1}{\to} & x \mathrlap{= p r k}\\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e. }$$

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.

In terms of pseudoalgebras

Fibrations are pseudoalgebras for a lax-idempotent pseudomonad (see there for more details).

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 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).

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.

Related concepts

• opfibration, bifibration

  two-sided fibration

• monoidal fibration

• fibration of 2-categories

• n-fibration

• Cartesian fibration

• foliated category

• Grothendieck construction, category of elements

References

The concept was introduced in the context of descent theory and in the guise now known as indexed categories, in

• Alexander Grothendieck, p. 300 in: Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [numdam:SB_1958-1960__5__299_0]

and then elaborated on in

• Alexander Grothendieck, exposé VI of: Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1) , LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

Another important early reference:

• John W. Gray, Fibred and Cofibred Categories, pp. 21-83 in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]

Early development of the theory:

• Jean Bénabou, Fibrations petites et localement petites, C. R. Acad. Sci. Paris 281 Série A (1975) 897-900 [gallica]

• Jean Bénabou, Fibered Categories and the Foundations of Naive Category Theory, The Journal of Symbolic Logic, Vol. 50 1 (1985) 10-37 [doi:10.2307/2273784]

reviewed in:

• Thomas Streicher, Fibred Categories à la Jean Bénabou, [arXiv:1801.02927]

Further review with emphasis on descent and Grothendieck fibrations as incarnations of stacks:

• Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory [math.AG/0412512, MR2223406] in: Fantechi et al. (eds.), Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, MR2007f:14001]

• Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Chapter 3 of: Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) [MR2007f:14001, ISBN:978-0-8218-4245-4, lecture notes]

• Garth Warner: Fibrations and Sheaves, EPrint Collection, University of Washington (2012) [hdl:1773/20977, pdf, pdf]

Textbook accounts:

• Michael Barr, Charles Wells, Chapter 12 of: Category Theory for Computing Science , Prentice Hall 1995³. (TAC reprints no.22 (2012))

• Francis Borceux, Chapter 8 of: Handbook of Categorical Algebra vol. 2, Cambridge UP 1994.

• Bart Jacobs, Chapters 1 and 9 in: Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]

  (in the context of categorical semantics of dependent types)

• Peter Johnstone, Sketches of an Elephant vol.1, Oxford UP 2002. (Part B)

  (beware that this uses the non-standard terms “prone” and “supine” for “cartesian” and “opcartesian” morphisms)

• Niles Johnson, Donald Yau, Chapter 9 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

See also:

• André Joyal, Grothendieck fibrations

• Wikipedia, Fibered category

On the 2-category of fibrations and application to computer science:

• Claudio Hermida, Some properties of $Fib$ as a fibred 2-category, Journal of Pure and Applied Algebra 134 (1999) 83-109 [doi:10.1016/S0022-4049(97)00129-1]

• Claudio Hermida, PhD thesis, University of Edinburgh

A 2-comonad characterizing Grothendieck fibrations:

• Jacopo Emmenegger, Luca Mesiti, Giuseppe Rosolini, Thomas Streicher, A comonad for Grothendieck fibrations [arXiv:2305.01474]

On the connection between Grothendieck fibrations and factorisation systems (see also stable factorisation system):

• J. Hughes and Bart Jacobs, Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem, Electr. Notes in Theor. Comp. Sci., 69 (2002)

• Jiří Rosický, Walter Tholen, Factorization, fibration and torsion, arxiv/0801.0063, Journal of homotopy and Related Structures

• Miloslav Štěpán, Factorization systems and double categories, arXiv:2305.06714.

See also:

• R. Brown, R. Sivera, Algebraic colimit calculations in homotopy theory using fibred and cofibred categories, TAC 22 (2009) pp.222-251. (pdf)