nLab two-sided fibration

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 here 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 span of functors whose joint fibres E(a,b)E(a,b) vary functorially in both aa and bb (contravariantly in one and covariantly in the other).

This notion should not be confused with a bifibration, which is a functor that is both a fibration and a cofibration.


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 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 .

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

  2. (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 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 bp(e)b \to p(e) in BB has a unique lift in EE that has codomain ee and is in the fiber over q(e)q(e)
  2. each q(e)aq(e) \to a in AA has a unique lift in EE that has domain ee and is in the fiber over p(e)p(e)
  3. for each f:eef\colon e \to e' in EE, the codomain of the lift of q(f)q(f) equals the domain of the lift of p(f)p(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.

Note, though, that the two legs of a two-sided discrete fibration are not necessarily individually discrete as a fibration and an opfibration.


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 two-sided 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?.

Relation to distributive laws

Fibrations and opfibrations on a category CC (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If CC has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the two-sided fibrations satisfying the Beck-Chevalley condition; see von Glehn (2015).


An early reference is the notion of “regular span” on page 535 of:

Original discussion:

  • 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)

Further discussion of discrete fibrations

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

Useful reviews are in

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

In relation to categorical semantics of dependent types:

Last revised on December 29, 2023 at 18:21:03. See the history of this page for a list of all contributions to it.