fibration in a 2-category

Fibrations in a 2-category


Recall the notion of a Grothendieck fibration: a functor p:EBp \colon E \to B whose fibres E bE_b are (contravariantly) functorial in bBb \in B.

This idea may be generalized to work in any suitable 2-category, although if the 2-category is not strict, then one has to generalize instead the non-strict notion of Street fibration. The generalized definition can be given in any of several equivalent ways, in such a way that

When KK is Cat, strict fibrations in this sense are precisely Grothendieck fibrations, while non-strict ones are precisely Street fibrations.


Fix a finitely complete (non-strict) 2-category KK. Recall that for any two morphisms f:ACf: A \to C and g:BCg: B \to C, f/gf/g denotes their comma object. Let f/ gf/_{\cong} g be their 2-pullback.

A morphism p:EBp \colon E \to B in KK is a (non-strict) fibration when the following equivalent conditions hold:

  • p *=K(X,p):K(X,E)K(X,B)p_* = K(X,p) \colon K(X,E) \to K(X,B) is a Street fibration in CatCat for each XKX \in K, and for all f:YXf \colon Y \to X in KK

    K(X,E) f * K(Y,E) p * p * K(X,B) f * K(Y,B) \array{ K(X,E) & \overset{f^*}{\to} & K(Y,E) \\ \mathllap{p_*} \downarrow & & \downarrow \mathrlap{p_*} \\ K(X,B) & \overset{f^*}{\to} & K(Y,B) }

    is a morphism of fibrations.

  • For every morphism f:XBf: X \to B, the canonical map i:f/ pf/pi: f/_{\cong} p \to f/p has a right adjoint in the slice 2-category K/XK / X.

  • The canonical map i:pB/pi \colon p \to B/p has a right adjoint in K/BK / B.

  • p:EBp \colon E \to B is an algebra for the 2-monad LL on K/BK/B given by Lp=B/pL p = B/p.

If KK is a strict 2-category with finite strict 2-limits, then we say that p:EBp \colon E \to B is a strict fibration when the corresponding conditions hold where “Street fibration” is replaced by “Grothendieck fibration”, slice 2-categories are replaced by strict-slice 2-categories, comma objects are replaced by strict comma objects, and 2-pullbacks are replaced by strict 2-pullbacks. In this case the 2-monad LL is in fact a strict 2-monad, but we do not require pp to be a strict algebra, only a pseudoalgebra; strict algebras for LL would instead be split fibrations.

Note also that the first definition makes perfect sense regardless of whether KK has any limits, although this is not true of the others.


By way of spelling out the first definition, we may define a 2-cell η:ee:XE\eta \colon e \to e' \colon X \to E to be pp-cartesian if f *η=ηff^*\eta = \eta f is p *p_*-cartesian for every f:YXf \colon Y \to X. Since this definition already incorporates stability under pullback, we can then say that pp is a fibration in KK if for every 2-cell β:bpe\beta \colon b \to p e there is a cartesian β^:ee\hat\beta \colon e' \to e such that pe=bp e' = b and pβ^=βp \hat\beta = \beta.

The third definition is perhaps the simplest. Of course, it is implied by the second (take f=1f = 1), but the converse is also true by the pasting lemma for comma and pullback squares.

We can show that the first and third definitions are equivalent by using the representability of fibrations and adjunctions, plus the following lemma, whose proof can be found at Street fibration.


A functor p:EBp \colon E \to B is a cloven Street fibration if and only if the canonical functor i:BE/pi \colon B \to E/p has a right adjoint rr in Cat/BCat / B.

It follows that a morphism p:EBp \colon E \to B in any 2-category KK is representably a fibration (i.e. satisfies the first definition) if and only if the adjunction iri \dashv r exists in K/BK/B (i.e. it satisfies the third condition).

Now we connect the first three conditions with the fourth. Because KK is finitely complete, we may form the tricategory Span(K)Span(K) of spans in KK. In particular, K/BK/B is equivalent to the hom-2-category Span(K)(B,1)Span(K)(B,1). Now recall that B/pB/p can be expressed as a pullback or span composite

B/p B 2 E B B 1 \array{ & & & & B/p & & & & \\ & & & \swarrow & & \searrow & & & \\ & & B^{\mathbf{2}} & & & & E & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & \\ B & & & & B & & & & 1 }

Write ΦB=B 2B\Phi B = B^{\mathbf{2}}\rightrightarrows B. The functor L:pB/pL \colon p \mapsto B/p is then given by composition: Lp=ΦBpL p =\Phi B \circ p. To show that pp is a fibration iff it is an LL-algebra, it suffices to show

Lemma. ΦB\Phi B is a colax-idempotent monad in Span(K)Span(K) with unit i:BB 2=B/Bi \colon B \to B^{\mathbf{2}} = B/B.

Proof. Write Δ\mathbf{\Delta} for the monoidal 2-category whose underlying 1-category is the augmented simplex category (q.v.) Δ a\Delta_a. Recall that for n1n \geq 1 each [n]Δ[n] \in \mathbf{\Delta} is given by the (n1)(n-1)-fold composite of the cospan [1][2][1][1] \to [2] \leftarrow [1] with itself, where we take a 0-fold composite to denote the identity [1][1][1][1] \to [1] \leftarrow [1]. Recall also that the monoidal structure [n][m][n] \oplus [m] on Δ\mathbf{\Delta} is generated by composing the cospans [n],[m]:[1][1][n],[m] \colon [1] ⇸ [1] (together with the fact that [0],[1][0],[1] are the initial and terminal objects).

Thus for each n1n \geq 1, there is a span B [n]:BBB^{[n]} \colon B ⇸ B, which is canonically equivalent to the (n1)(n-1)-fold composite B [2]B [2]B [2]:BBB^{[2]} \circ B^{[2]} \circ \cdots \circ B^{[2]} \colon B ⇸ B, because cotensor preserves finite limits. For the same reason, B :Cat fp opKB^- \colon Cat^{op}_{fp} \to K almost restricts to a monoidal 2-functor Δ opSpan(K)(B,B)\mathbf{\Delta}^{op} \to Span(K)(B,B), except that [0][0] does not yield a span from BB to BB. Precomposing B B^- with the functor Δ coΔ op\mathbf{\Delta}^{co} \to \mathbf{\Delta}^{op} that sends [n][n] to [n+1][n+1], while δ iσ i\delta_i \mapsto \sigma_i and σ iδ i+1\sigma_i \mapsto \delta_{i+1} does yield a monoidal 2-functor Δ coSpan(K)(B,B)\mathbf{\Delta}^{co} \to Span(K)(B,B), which by the universal property of Δ\mathbf{\Delta} corresponds to a unique colax-idempotent monad in Span(K)(B,B)Span(K)(B,B).

In detail, the monoid object [0][1][2][0] \to [1] \leftarrow [2] in Δ\mathbf{\Delta} is sent to the monad ΦB=B [2]\Phi B = B^{[2]} in Span(K)Span(K), with structure maps i=(σ 0 1) *:BΦBi = (\sigma^1_0)^* \colon B \to \Phi B and c=(δ 1 2) *:ΦBΦBΦBc = (\delta^2_1)^* \colon \Phi B \circ \Phi B \to \Phi B. Moreover, because of the adjunction δ 1 2σ 0 2=σ 0 1[1]\delta^2_1 \dashv \sigma^2_0 = \sigma^1_0 \oplus [1] in Δ\mathbf{\Delta}, we have iΦBci \circ \Phi B \dashv c in Span(K)(B,B)Span(K)(B,B) with identity unit.

It follows that L=ΦBL = \Phi B \circ - is a monad, and is colax-idempotent because, for a span H:BAH \colon B ⇸ A, we have η H=iH\eta_H = i \circ H, μ H=cH\mu_H = c \circ H, and

η LH=iΦBHcH \eta_{L H} = i \circ \Phi B \circ H \dashv c \circ H

with identity unit. It is clear, too, that the morphism i:pB/pi \colon p \to B/p as above is given by η p =ip \eta_{p^\circ} = i \circ p^\circ, where p p^\circ denotes the obvious span B1B ⇸ 1. Thus, by the definition of a colax-idempotent monad, a morphism p:EBp \colon E \to B carries the structure of an LL-algebra if and only if the unit i=η p :pB/pi = \eta_{p^\circ} \colon p \to B/p has a right adjoint.


The 2-fibration of fibrations

For a strict 2-category KK, let Fib KFib_K be the 2-category of strict fibrations in KK, morphisms of fibrations, and 2-cells between them. Then the codomain functor cod:Fib KKcod : Fib_K \to K is a strict 2-fibration. (This is arguably an instance of the microcosm principle.)

Similarly, for any bicategory KK, the bicategory Fib KFib_K of weak fibrations in KK, morphisms of fibrations, and 2-cells admits a weak 2-fibration cod:Fib KKcod : Fib_K \to K.

Iterated fibrations

It is easy to show that a composite of fibrations is a fibration. Moreover, if Fib(X)=Fib K(X)Fib(X)= Fib_K(X) denotes the 2-category of fibrations over XKX\in K, then we have:


A morphism in Fib(X)Fib(X) is a fibration in the 2-category Fib(X)Fib(X) iff its underlying morphism in KK is a fibration.

This is a standard result, at least in the case K=CatK=Cat, and is apparently due to Benabou. References include:

  • J. Benabou, “Fibered categories and the foundations of naive category theory”
  • B. Jacobs, Categorical Logic and Type Theory, Chapter 9
  • C. Hermida, “Some properties of Fib as a fibred 2-category”

Therefore, for any fibration AXA\to X in KK we have Fib K(A)Fib Fib K(X)(AX)Fib_K(A) \simeq Fib_{Fib_K(X)}(A\to X), and similarly for opfibrations. This is a fibrational 2-categorical analogue of the standard equivalence K/A(K/X)/(AX)K/A \simeq (K/X)/(A\to X) for ordinary slice categories.


  • Ross Street, Fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21 no. 2 (1980), p. 111–160 (numdam).

  • Mark Weber, Yoneda structure from 2-toposes. Applied Categorical Structures 15:259–323 (2007).

Revised on June 12, 2017 12:18:44 by Mike Shulman (