Higher category theory

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1-categorical presentations



An nn-fibration is the version of a Grothendieck fibration appropriate for nn-categories.

The idea is that a functor p:EBp:E\to B between nn-categories is an nn-fibration if the assignation xE x=p 1(x)x\mapsto E_x = p^{-1}(x) of an object xBx\in B to its fiber can be made into a (contravariant) functor from BB to the (n+1)(n+1)-category nCatn Cat.


(This definition is schematic, and needs to be adapted to be made precise for any particular definition of nn-category.)

Let p:EBp:E\to B be a functor between (weak) nn-categories.


A morphism ϕ:ba\phi:b\to a in EE is cartesian (relative to pp) if for any cEc\in E, the following square:

E(c,b) ϕ E(c,a) p p B(pc,pb) pϕ B(pc,pa)\array{E(c,b) & \overset{\phi\circ -}{\to} & E(c,a)\\ ^p \downarrow && \downarrow ^p\\ B(p c, p b) & \underset{p\phi\circ -}{\to} & B(p c, p a)}

(which commutes, up to equivalence, by functoriality of pp) is a (weak) pullback of (n1)(n-1)-categories.


We say that p:EBp:E\to B is an nn-fibration (or just a fibration) if

  1. For any object aEa\in E and morphism f:xpaf:x\to p a in BB, there exists a cartesian ϕ:ba\phi:b\to a and an equivalence pϕfp\phi \simeq f in the slice nn-category B/paB/p a,
  2. For any objects a,bEa,b\in E, the functor p:E(b,a)B(pb,pa)p:E(b,a) \to B(p b, p a) is an (n1)(n-1)-fibration, and
  3. For any a,b,cEa,b,c\in E, the square
    E(b,c)×E(a,b) E(a,c) p p B(pb,pc)×B(pa,pb) B(pa,pc)\array{E(b,c)\times E(a,b) &\overset{\circ}{\to} & E(a,c)\\ ^p\downarrow && \downarrow^p\\ B(p b, p c)\times B(p a, p b) & \underset{\circ}{\to} & B(p a, p c)}

    is a morphism of (n1)(n-1)-fibrations.


If p 1:E 1B 1p_1:E_1\to B_1 and p 2:E 2B 2p_2:E_2\to B_2 are nn-fibrations, a commutative square

E 1 h E 2 p 1 p 2 B 1 g B 2\array{E_1 & \overset{h}{\to} & E_2\\ p_1 \downarrow && \downarrow p_2\\ B_1 & \underset{g}{\to} & B_2}

is a morphism of nn-fibrations if

  1. Whenever ϕ\phi is cartesian for p 1p_1, h(ϕ)h(\phi) is cartesian for p 2p_2, and
  2. For any a,bE 1a,b\in E_1, the square
    E 1(b,a) h E 2(hb,ha) p 1 p 2 B 1(p 1b,p 1a) g B 2(gp 1b,gp 1a)\array{E_1(b,a) & \overset{h}{\to} & E_2(h b, h a)\\ p_1 \downarrow && \downarrow p_2\\ B_1(p_1 b, p_1 a)& \underset{g}{\to} & B_2(g p_1 b, g p_1 a)}

    is a morphism of (n1)(n-1)-fibrations.


  • The definition is recursive in nn, but if we unravel it, it makes perfect sense for n=ωn=\omega. That is, saying that ff is a fibration requires some things about cartesian 1-cells, and also that its action on hom-categories be a fibration–which in turn requires some things about cartesian 2-cells, and also that its action on hom-categories be a fibration—which in turn which requires some things about cartesian 3-cells, and so on. After ω\omega steps of unraveling, we are left with a list of conditions on cartesian nn-cells for every nn.

    An equivalent, conciser way to say this is that we interpret the definition in the case n=ωn=\omega as a coinductive definition.

  • When n=1n=1, this reduces to a Street fibration, a weakened version of a Grothendieck fibration. We recover Grothendieck’s original notion by requiring that for any aEa\in E and f:xpaf:x\to p a in BB, there exists a cartesian ϕ:ba\phi:b\to a such that pϕp\phi and ff are equal. This condition violates the principle of equivalence as stated, but not if it is rephrased to apply to displayed categories instead.

  • When n=2n=2, so that weak 2-categories are bicategories, this notion of fibration can be found in (Buckley). A strict version for strict 2-categories (though with one condition missing) was originally studied by (Hermida).

  • In general, given any notion of (semi)strict nn-category, we can expect to appropriately strictify the definition to make it correspond to stricter notions of pseudofunctor.

Fibrations versus functors

If p:EBp:E\to B is an nn-fibration, we define a functor (or ‘nn-pseudofunctor’) from BB to nCatn Cat as follows. (Like the above definition, this is only a schematic sketch.)

  • Send xBx\in B to the essential fiber E xE_x, whose objects are objects aEa\in E equipped with a equivalence paxp a \simeq x.

  • For a morphism f:yxf:y\to x in BB, define f *:E xE yf^*:E_x\to E_y by choosing, for each aE xa\in E_x, a cartesian ϕ:ba\phi:b\to a over ff and defining f *(a)=bf^*(a)=b. The universal property of cartesian arrows makes f *f^* a functor.

  • For a 2-cell α:fg:yx\alpha:f\to g:y\to x in BB, define a transformation α *:g *f *\alpha^*:g^*\to f^* as follows. Given aE xa\in E_x, we have a cartesian arrow ϕ:g *aa\phi:g^*a\to a over gg. Now choose a cartesian 2-cell μ:ψϕ\mu:\psi\to \phi over α\alpha in E(g *a,a)E(g^*a,a). Since pψ=fp \psi = f, ψ\psi factors essentially uniquely through the cartesian arrow χ:f *aa\chi:f^*a\to a, giving a morphism g *af *ag^*a \to f^*a; we define this to be the component of the transformation α *\alpha^* at aa.

  • and so on…

Note that the functor we obtain is “totally contravariant:” it is contravariant on kk-cells for all 1kn1\le k\le n.

Conversely, if we have a totally contravariant ‘nn-pseudofunctor’ from BB to nCatn Cat, we define p:EBp:E\to B by a generalization of the Grothendieck construction as follows:

  • The objects of EE over xBx\in B are those of FxnCatF x \in n Cat.

  • The morphisms of EE over f:yxf:y\to x in BB from bFyb\in F y to aFxa\in F x are the morphisms from bb to F f(a)F_f(a) in FyF y.

  • The 2-cells of EE over α:fg:yx\alpha:f\to g:y\to x in BB from ψ:bF f(a)\psi : b \to F_f(a) to ϕ:bF g(a)\phi : b \to F_g(a) are the 2-cells in FyF y from ψ\psi to the composite bϕF g(a)F α(a)F f(a)b \xrightarrow{\phi} F_g(a) \xrightarrow{F_\alpha(a)} F_f(a).

  • and so on…

One expects that in this way, the (n+1)(n+1)-category of fibered nn-categories over BB is equivalent to the (n+1)(n+1)-category of functors BnCatB\to n Cat. These constructions are known precisely only for n=2n=2.


  • The prototypical example is that if KK is an nn-category with finite limits (or at least pullbacks), then the nn-category Fib KFib_K of internal (n1)(n-1)-fibrations in KK should admit an nn-fibration cod:Fib KKcod : Fib_K \to K. Of course this requires defining the notion of internal (n1)(n-1)-fibration in an nn-category; this is usually done representably. For n=2n=2 this gives the notion of fibration in a 2-category, and the fact that cod:Fib KKcod : Fib_K \to K is a 2-fibration is in (Hermida). For n=1n=1 it is just the standard fact that the codomain fibration is a fibration, i.e. every morphism in a 1-category is an “internal 0-fibration”.


Claudio Hermida introduced 2-fibrations in:

  • Claudio Hermida, “Some properties of Fib as a fibred 2-category,” J. Pure Appl. Algebra 134 (1), 83–109, 1999; preprint version ps.gz

In fact they also appeared earlier, in some form, in Gray's book.

However, Hermida’s definition was missing the stability of cartesian 2-cells under postcomposition, which is necessary for the “Grothendieck construction” turning a pseudofunctor into a fibration to have an inverse. This was rectified, and the definition generalized to bicategories, in

  • Igor Bakovic, “Fibrations of bicategories”, preprint
  • Mitchell Buckley, “Fibred 2-categories and bicategories”, doi, arxiv

A definition for strict nn-categories due to Hermida is unpublished, but it is used and presented in another joint work with Marta Bunge, presented at CATS07 at Calais:

  • Marta Bunge, “Intrinsic nn-stack completions over a topos,” slides here

nn-pseudofunctors may be viewed (and defined) as anafunctors. For nn-groupoids such an approach to nn-pseudofunctors has been studied in

  • D. Bourn, Pseudo functors and non abelian weak equivalences, in “Categorical algebra and its applications”, Springer LNM 1348 (1988), 55–70.

Revised on June 12, 2017 12:30:24 by Mike Shulman (