two-sided fibration

Two-sided fibrations

Two-sided fibrations


Recall that a functor EBE \to B is called a fibration if its fibres E bE_b vary (pseudo-)functorially in bb. 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 AEBA \leftarrow E \to B is a pair of functors whose fibres E(a,b)E(a,b) vary functorially in both aa and bb (contravariantly in one and covariantly in the other).


Let KK be a bicategory with finite 2-limits, and recall that fibrations in KK 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 EBE \to B in KK are the (pseudo)algebras for a (pseudo) 2-monad LL on K/BK/B. For a morphism p:EAp \colon E \to A in KK, LpL p is given by composing the span ApE1A \overset{p}{\leftarrow} E \to 1 with the canonical span ΦA=AdomA 2codA\Phi A = A \overset{dom}{\leftarrow} A^{\mathbf{2}} \overset{cod}{\to} A, so that Lp:E/pAL p \colon E/p \to A is the canonical projection. This can equivalently be described as the comma object (1 A/p)(1_A/p). This 2-monad is lax-idempotent, so that p:EBp\colon E\to B is a fibration if and only if the unit pLpp\to L p has a left adjoint with invertible counit.

More generally, the same construction gives a 2-monad LL on SpanK(B,A)Span K(B,A), whose algebras we call left fibrations. In Cat, a span CpHqDC \overset{p}{\leftarrow} H \overset{q}{\to} D is a left fibration if pp is a cloven fibration whose chosen cartesian lifts are qq-vertical. (Since we are working bicategorically, “qq-vertical” means that they map to isomorphisms under qq.)

Dually, there is a colax-idempotent 2-monad RR on each SpanK(B,A)Span K(B,A) whose algebras are called right fibrations, the special case of SpanCat(D,1)Span Cat(D,1) yielding cloven opfibrations.

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


A two-sided Street fibration from BB to AA in CatCat is given by a span p:EAp \colon E \to A, q:EBq \colon E \to B such that

  1. each i:apxi \colon a \to p x in AA has a pp-cartesian lift κ i:i *xx\kappa_i \colon i^* x \to x in EE that is qq-vertical (that is, EE is a left fibration)

  2. each j:qxbj \colon q x \to b in BB has a qq-opcartesian lift κ j:xj !x\kappa^j \colon x \to j_! x in EE that is pp-vertical (EE is a right fibration)

  3. for every cartesian–opcartesian composite i *xxj !xi^* x \to x \to j_! x in EE, the canonical morphism j !i *xi *j !xj_! i^* x \to i^* j_! x is an isomorphism.


By the usual theory of distributive laws, an MM-algebra m:MEEm \colon M E \to E gives rise to LL- and RR-algebras m(ΦAη E R)m \cdot (\Phi A \circ \eta^R_E) and m(η E LΦB)m \cdot (\eta^L_E \circ \Phi B), and conversely an LL-algebra \ell and an RR-algebra rr underlie an MM-algebra if and only if there is an isomorphism r(ΦB)(ΦAr)r \cdot (\ell \circ \Phi B) \cong \ell \cdot (\Phi A \circ r) that makes rr a morphism of LL-algebras.

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

The 2-cell r¯\bar r is given by (ΦAr)(ϵΦB)\ell \cdot (\Phi A \circ r) \cdot (\epsilon \circ \Phi B), where ϵ\epsilon is the counit of the adjunction η E L\eta^L_E \dashv \ell. Its components are thus given, for each i:apxi \colon a \to p x in AA and j:pxbj \colon p x \to b in BB, by first factoring κ jκ i\kappa^j \kappa_i through the opcartesian i *xj !i *xi^* x \to j_! i^* x and then factoring the result through the cartesian i *j !xj !xi^* j_! x \to j_! x, to obtain exactly the canonical morphism j !i *xi *j !xj_! i^* x \to i^* j_! x.

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

A representable definition

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

As iterated fibrations

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

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


A span ApEqBA \overset{p}{\leftarrow} E \overset{q}{\to} B is a two-sided fibration from BB to AA if and only if 1. p:EAp\colon E\to A is a fibration and 1. (p,q):EA×B(p,q)\colon E\to A\times B is an opfibration in Fib(A)Fib(A).


Recall that the projection A×BAA\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 BB is an isomorphism. Therefore, saying that (p,q)(p,q) is a morphism in Fib(A)Fib(A), i.e. that it preserves cartesian 2-cells, says precisely that qq takes pp-cartesian 2-cells to isomorphisms.

Now qq is an opfibration in KK iff E(q/1 B)E\to (q/1_B) has a left adjoint with invertible counit in K/BK/B, and (p,q)(p,q) is an opfibration in Fib(A)Fib(A) iff E((p,q)/1 A×B)E\to ((p,q)/1_{A\times B}) has a left adjoint with invertible counit in Fib(A)/(A×B)Fib(A)/(A\times B). Of crucial importance is that here ((p,q)/1 A×B)((p,q)/1_{A\times B}) denotes the comma object calculated in the 2-category Fib(A)Fib(A), or equivalently in K/AK/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)(q/1_B) calculated in KK.

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

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

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

Two-sided discrete fibrations


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

For Grothendieck fibrations in Cat, this means the following.


A two-sided discrete fibration is a span q:EAq \colon E \to A, p:EBp \colon E \to B of categories and functors such that 1. each bpeb \to p e in BB has a unique lift in EE that has codomain ee and is in the fiber over qeq e 1. each qeaq e \to a in AA has a unique lift in EE that has domain ee and is in the fiber over pep e 1. for each f:eef\colon e \to e' in EE, the codomain of the lift of qfq f equals the domain of the lift of pfp f and their composite is ff.

We write

DFib(A,B)Span(A,B) DFib(A,B) \subset Span(A,B)

for the full subcategory on the 2-category SpanK(A,B)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 Fib(A,B)Fib(A,B). And since they are discrete objects, this 2-category is actually (equivalent to) a 1-category.


Profunctors and collages


Given a profunctor F:B op×ASetF : B^{op} \times A \to Set, its collage is the category K FK_F over the interval category

p:K FΔ[1] p : K_F \to \Delta[1]

With p 1(0)=Bp^{-1}(0) = B, p 1(1)=Ap^{-1}(1) = A, K F(b,a)=F(b,a)K_F(b,a) = F(b,a) and K F(a,b)=K_F(a,b) = \emptyset for all bBb \in B, aAa \in A, where

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

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


There is an equivalence of categories

[B op×A,Set]DFib(A,B) [B^{op} \times A, Set] \stackrel{\simeq}{\to} DFib(A,B)
FE F, F \mapsto E_F \,,

pseudo-natural in A,BCatA, B \in Cat, between profunctors in Set and discrete fibrations from AA to BB, where E FE_F is the category whose

  • objects are sections σ:Δ[1]K F\sigma : \Delta[1] \to K_F of the collage p:K FΔ[1]p : K_F \to \Delta[1]

  • morphisms are natural transformations between such sections;

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


First we write out E FE_F in detail. In the following b,b,Bb, b', \cdots \in B and a,a,Aa,a', \dots \in A.

The objects of E FE_F are morphisms

b e a \array{ b \\ {}^{\mathllap{e}}\downarrow \\ a }

in K FK_F, hence triples (bB,aA,eF(b,a))(b \in B, a \in A, e \in F(b,a)).

Morphisms are commuting diagrams

b g b e e a f a \array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e}}\downarrow && \downarrow^{\mathrlap{e'}} \\ a &\stackrel{f}{\to}& a' }

in K FK_F. We may identify these with pairs ((bgb)B,(afa)A)((b \stackrel{g}{\to}b') \in B,(a \stackrel{f}{\to} a') \in A) such that

F(g,a)(e)=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

    b e a \array{ b \\ {}^{\mathllap{e}}\downarrow \\ a }

    an object of E FE_F and afaa \stackrel{f}{\to} a' a morphism in AA, we have that

    b Id b e fe a f a \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 EE that maps to Id bId_b.

  2. Analogously, for

    b e a \array{ b' \\ {}^{\mathllap{e'}}\downarrow \\ a' }

    an object of E FE_F and bgbb \stackrel{g}{\to} b' a morphism in BB, we have that

    b g b eg e a id a \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 EE that maps to Id aId_{a'}.

  3. For

    b g b e e a f a \array{ b &\stackrel{g}{\to}& b' \\ {}^{\mathllap{e}}\downarrow && \downarrow^{\mathrlap{e'}} \\ a &\underset{f}{\to}& a' }

    an arbitrary morphism in E FE_F, these two unique lifts of its AA- and its BB-projection, respectively, are

    b Id b e fe a f a \array{ b &\stackrel{Id}{\to}& b \\ {}^{\mathllap{e}}\downarrow & & \downarrow^{\mathrlap{f e}} \\ a &\underset{f}{\to}& a' }


    b g b eg e a Id a. \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=egf e = e' g by the existence of the original morphism, and their composite is the original morphism

    b Id b g b e fe eg e a f a Id a. \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 (AEB)(F E:B op×ASet)(A\leftarrow E \to B) \mapsto (F_E : B^{op} \times A \to Set) by setting

  • F E(b,a):=E b,aF_E(b,a) := E_{b,a};

  • for a morphism bgbb \stackrel{g}{\to} b' let F E(g,a):F E(b,a)F E(b,a)F_E(g,a') : F_E(b',a') \to F_E(b,a') be the function that sends beab' \stackrel{e'}{\to} a' to the domain of the unique lift of bgbb \stackrel{g}{\to} b' with this codomain and mapping to Id aId_{a'};

  • for a morphism afaa \stackrel{f}{\to} a' let F E(b,f):F E(b,a)F E(b,a)F_E(b,f) : F_E(b,a) \to F_E(b,a') be the function that sends beab \stackrel{e}{\to} a to the codomain of the unique lift of afaa \stackrel{f}{\to} a' with this domain and mapping to Id bId_{b};.

One checks that this yields an equivalence of categories.


The category E FE_F is equivalently characterized as being the comma category of the diagram BK FAB \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?.


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)

Revised on March 2, 2015 07:38:50 by Tim Porter (