nLab
two-sided fibration

Two-sided fibrations

Idea

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

Similarly, a 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).

Definition

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

In terms of 2-monads

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 $p:E\to A$ in $K$, $Lp$ is given by composing the span $A\stackrel{p}{\leftarrow }E\to 1$ with the canonical span $\Phi A=A\stackrel{\mathrm{dom}}{\leftarrow }{A}^2\stackrel{\mathrm{cod}}{\to }A$, so that $Lp:E/p\to A$ is the canonical projection. This can equivalently be described as the comma object $(1_A/p)$. This 2-monad is lax-idempotent, so that $p:E\to B$ is a fibration if and only if the unit $p\to Lp$ has a left adjoint with invertible counit.

More generally, the same construction gives a 2-monad $L$ on $\mathrm{Span}K(B,A)$, whose algebras we call left fibrations. In Cat, a span $C\stackrel{p}{\leftarrow }H\stackrel{q}{\to }D$ 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 $\mathrm{Span}K(B,A)$ whose algebras are called right fibrations, the special case of $\mathrm{Span}\mathrm{Cat}(D,1)$ yielding cloven opfibrations.

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

Proposition

A two-sided Street fibration from $B$ to $A$ in $\mathrm{Cat}$ is given by a span $p:E\to A$, $q:E\to B$ such that

1. each $i:a\to px$ in $A$ has a $p$-cartesian lift ${\kappa }_i:i^{*}x\to x$ in $E$ that is $q$-vertical (that is, $E$ is a left fibration)

2. each $j:qx\to b$ in $B$ has a $q$-opcartesian lift ${\kappa }^j:x\to j_{!}x$ in $E$ that is $p$-vertical ($E$ is a right fibration)

3. 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.

Proof

By the usual theory of distributive laws, an $M$-algebra $m:ME\to E$ gives rise to $L$- and $R$-algebras $m\cdot (\Phi A\circ {\eta }_E^R)$ and $m\cdot ({\eta }_E^L\circ \Phi B)$, 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 $\overline{r}:r\cdot (\ell \circ \Phi B)\Rightarrow \ell \cdot (\Phi A\circ r)$ that makes $r$ a colax morphism of $L$-algebras. So we want to show that in the case of $\mathrm{Cat}$, the components of this natural transformation are the canonical morphisms of (3).

The 2-cell $\overline{r}$ is given by $\ell \cdot (\Phi A\circ r)\cdot (ϵ\circ \Phi B)$, where $ϵ$ is the counit of the adjunction ${\eta }_E^L⊣\ell$. Its components are thus given, for each $i:a\to px$ in $A$ and $j:px\to b$ 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 $j_{!}{i}^{*}x\to i^{*}{j}_{!}x$.

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

A representable definition

Another definition of internal fibration is that a (cloven) fibration in $K$ is a morphism $p:E\to B$ such that $K(X,p):K(X,E)\to K(X,B)$ is a (cloven) fibration in $\mathrm{Cat}$, for any $X\in K$, and for any $X\to Y$ the corresponding square is a morphism of fibrations in $\mathrm{Cat}$. To adapt this definition to two-sided fibrations, we therefore need only to say what is a two-sided fibration in $\mathrm{Cat}$. For this we can use the characterization of Proposition 1.

As iterated fibrations

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

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

Theorem

A span $A\stackrel{p}{\leftarrow }E\stackrel{q}{\to }B$ is a two-sided fibration from $B$ to $A$ if and only if

1. $p:E\to A$ is a fibration and
2. $(p,q):E\to A\times B$ is an opfibration in $\mathrm{Fib}(A)$.

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

Therefore, $(p,q)$ is an opfibration in $\mathrm{Fib}(A)$ iff $q$ is an opfibration in $K$ and the left adjoint of $E\to (q/1_B)$ is a morphism in $\mathrm{Fib}(A)$. 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 1 is satisfied.

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

This result appears in Bourn–Penon; it was noticed independently and recorded here by Mike Shulman.

Two-sided discrete fibrations

Definition

A two-sided fibration $A\leftarrow E\to B$ in $K$ is discrete if it is 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 $\mathrm{Fib}(A,B)$ of two-sided fibrations.

For Grothendieck fibrations in Cat, this means the following.

Definition

A two-sided discrete fibration is a span $q:E\to A$, $p:E\to B$ of categories and functors such that

1. each $b\to pe$ in $B$ has a unique lift in $E$ that has codomain $e$ and is in the fiber over $qe$
2. each $qe\to a$ in $A$ has a unique lift in $E$ that has domain $e$ and is in the fiber over $pe$
3. for each $f:e\to e\prime$ in $E$, the codomain of the lift of $qf$ equals the domain of the lift of $pf$ and their composite is $f$.

We write

$$\mathrm{DFib}(A,B)\subset \mathrm{Span}(A,B)$$DFib(A,B) \subset Span(A,B)

for the full subcategory on the 2-category $\mathrm{Span}K(A,B)$ of spans 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 $\mathrm{Fib}(A,B)$. And since they are discrete objects, this 2-category is actually (equivalent to) a 1-category.

Properties

Profunctors and collages

Definition

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

$$p:K_F\to \Delta [1]$$p : K_F \to \Delta[1]

With $p^{-1}(0)=B$, $p^{-1}(1)=A$, $K_F(b,a)=F(b,a)$ and $K_F(a,b)=\varnothing$ for all $b\in B$, $a\in A$, where

• the composite of $b\stackrel{e}{\to }a$ with $a\stackrel{f}{\to }a\prime$ is given by $F(b,f)(e)$;

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

Proposition

There is an equivalence of categories

$$[B^{\mathrm{op}}\times A,\mathrm{Set}]\stackrel{\simeq }{\to }\mathrm{DFib}(A,B)$$[B^{op} \times A, Set] \stackrel{\simeq}{\to} DFib(A,B)

$$F\mapsto E_F,$$F \mapsto E_F \,,

pseudo-natural in $A,B\in \mathrm{Cat}$, between profunctors in Set and discrete fibrations from $A$ to $B$, where $E_F$ is the category whose

• objects are sections $\sigma :\Delta [1]\to K_F$ of the collage $p:K_F\to \Delta [1]$

• morphisms are natural transformations between such sections;

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

Proof

First we write out $E_F$ in detail. In the following $b,b\prime ,\cdots \in B$ and $a,a\prime ,\dots \in A$.

The objects of $E_F$ are morphisms

$$\begin{array}{c}b\\ {}^e\downarrow \\ a\end{array}$$\array{ b \\ {}^{\mathllap{e}}\downarrow \\ a }

in $K_F$, hence triples $(b\in B,a\in A,e\in F(b,a))$.

Morphisms are commuting diagrams

$$\begin{array}{ccc}b& \stackrel{g}{\to }& b\prime \\ {}^e\downarrow & & {\downarrow }^{e\prime }\\ a& \stackrel{f}{\to }& a\prime \end{array}$$\array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e}}\downarrow && \downarrow^{\mathrlap{e'}} \\ a &\stackrel{f}{\to}& a' }

in $K_F$. We may identify these with pairs $((b\stackrel{g}{\to }b\prime )\in B,(a\stackrel{f}{\to }a\prime )\in A)$ such that

$$F(g,a\prime )(e\prime )=F(b,f)(e).$$F(g,a')(e') = F(b,f)(e) \,.

We check that this construction yields a two-sided fibration. The three conditions are

1. For

   $$\begin{array}{c}b\\ {}^e\downarrow \\ a\end{array}$$\array{ b \\ {}^{\mathllap{e}}\downarrow \\ a }

   an object of $E_F$ and $a\stackrel{f}{\to }a\prime$ a morphism in $A$, we have that

   $$\begin{array}{ccc}b& \stackrel{\mathrm{Id}}{\to }& b\\ {}^e\downarrow & & {\downarrow }^{fe}\\ a& \underset{f}{\to }& a\prime \end{array}$$\array{ b &\stackrel{Id}{\to}& b \\ {}^{\mathllap{e}}\downarrow & & \downarrow^{\mathrlap{f e}} \\ a &\underset{f}{\to}& a' }

   is the unique lift to a morphism in $E$ that maps to ${\mathrm{Id}}_b$.

2. Analogously, for

   $$\begin{array}{c}b\prime \\ {}^{e\prime }\downarrow \\ a\prime \end{array}$$\array{ b' \\ {}^{\mathllap{e'}}\downarrow \\ a' }

   an object of $E_F$ and $b\stackrel{g}{\to }b\prime$ a morphism in $B$, we have that

   $$\begin{array}{ccc}b& \stackrel{g}{\to }& b\prime \\ {}^{e\prime g}\downarrow & & {\downarrow }^{e\prime }\\ a\prime & \underset{\mathrm{id}}{\to }& a\prime \end{array}$$\array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e' g}}\downarrow & & \downarrow^{\mathrlap{e'}} \\ a' &\underset{id}{\to}& a' }

   is the unique lift to a morphism in $E$ that maps to ${\mathrm{Id}}_{a\prime }$.

3. For

   $$\begin{array}{ccc}b& \stackrel{g}{\to }& b\prime \\ {}^e\downarrow & & {\downarrow }^{e\prime }\\ a& \underset{f}{\to }& a\prime \end{array}$$\array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e}}\downarrow && \downarrow^{\mathrlap{e'}} \\ a &\underset{f}{\to}& a' }

   an arbitrary morphism in $E_F$, these two unique lifts of its $A$- and its $B$-projection, respectively, are

   $$\begin{array}{ccc}b& \stackrel{\mathrm{Id}}{\to }& b\\ {}^e\downarrow & & {\downarrow }^{fe}\\ a& \underset{f}{\to }& a\prime \end{array}$$\array{ b &\stackrel{Id}{\to}& b \\ {}^{\mathllap{e}}\downarrow & & \downarrow^{\mathrlap{f e}} \\ a &\underset{f}{\to}& a' }

   and

   $$\begin{array}{ccc}b& \stackrel{g}{\to }& b\prime \\ {}^{e\prime g}\downarrow & & {\downarrow }^{e\prime }\\ a\prime & \underset{\mathrm{Id}}{\to }& a\prime \end{array}.$$\array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e' g}}\downarrow & & \downarrow^{\mathrlap{e'}} \\ a' &\underset{Id}{\to}& a' } \,.

   The codomain and domain do match, since $fe=e\prime g$ by the existence of the original morphism, and their composite is the original morphism

   $$\begin{array}{ccccc}b& \stackrel{\mathrm{Id}}{\to }& b& \stackrel{g}{\to }& b\\ {}^e\downarrow & & {}^{fe}{\downarrow }^{e\prime g}& & {\downarrow }^{e\prime }\\ a& \underset{f}{\to }& a\prime & \stackrel{\mathrm{Id}}{\to }& a\prime \end{array}.$$\array{ 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' } \,.

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

• $F_E(b,a):=E_{b,a}$;

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

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

One checks that this yields an equivalence of categories.

Observation

The category $E_F$ is equivalently characterizd as being the comma category of the diagram $B\to K_F\leftarrow A$.

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?.

Related concepts

• Grothendieck fibration, Street fibration, discrete fibration, fibration in a 2-category, two-sided fibration,

• Cartesian fibration

References

The notion is originally discussed in

• Ross Street. 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.

• Ross Street, Fibrations in bicategories. Cahiers Topologie Géom. Différentielle, 21(2):111–160, 1980. (Corrections in 28(1):53–56, 1987)

Some further discussion of discrete fibrations can be found in

• Dominique Bourn and Jacques Penon. 2-catégories réductibles. Preprint, University of Amiens Department of Mathematics, 1978. Reprinted as TAC Reprints no. 19, 2010 (link).

A useful review of discrete fibrations is in

• Emily Riehl, Two-sided discrete fibrations in 2-categories and bicategories 2010 (pdf)