Michael Shulman
fibrational slice

Definition

The fibrational slice of a 2-category K over an object X is the homwise-full sub-2-category Fib(X) of the slice 2-category K/X whose objects are fibrations over X and whose morphisms are morphisms of fibrations. Likewise, we have the opfibrational slice Opf(X) consisting of opfibrations.

The fibrational and opfibrational slices in a 2-category often play the role of the ordinary slice categories of a 1-category, replacing the ordinary slice 2-category. On this whole page we assume that K has finite limits.

Basic properties

The 2-category Opf(X) is monadic over K/X. The relevant monad on K/X takes p:AX to the comma object (p/1 X), or equivalently the pullback A× XX 2. It is lax-idempotent, so a morphism p:AX is an opfibration if and only if A(p/1 X) has a left adjoint with invertible counit in K/X. Likewise, p is a fibration iff A(1 X/p) has a right adjoint with invertible unit in K/X.

Since the fibrational slices are monadic over K/X, they inherit all limits from it. It follows that a fibration is a discrete object in Fib(X) iff it is discrete in K/X. These are unsurprisingly called discrete fibrations; we write DFib(X) for the category of such. Every morphism in K/X between discrete fibrations is a morphism of fibrations; thus DFib(X) is a full subcategory of both Fib(X) and K/X.

Any pullback of an (op)fibration is again an (op)fibration. Therefore, any morphism f:YX induces a pullback functor f *:Fib(X)Fib(Y), which restricts to a functor f *:DFib(X)DFib(Y), and dually. Regarding the existence of adjoints to these functors, see comprehensive factorization and exponentials in a 2-category.

Any morphism with groupoidal codomain is a fibration and opfibration. Therefore, if X is groupoidal, Fib(X)K/XOpf(X). In particular, for (2,1)-categories and thus also for 1-categories, the fibrational slices are no different from the ordinary slices.

Subobjects

Central for us is the following fact, which would be false if we replaced Opf(X) by K/X. It underlies the inheritance of all sorts of structure by fibrational slices, such as regularity, coherency, extensivity, and exactness.

Theorem

A morphism in Opf(X) is ff in Opf(X) iff its underlying morphism in K is ff.

Proof

Suppose that a:AX and b:BX are opfibrations, and that fmapsAB is a morphism in Opf(X) which is ff in K. Then since Opf(X)K is homwise faithful, f is clearly faithful in Opf(X). And given a 2-cell α:hk:TB in Opf(X), since f is ff in K, we must have α=fβ for some 2-cell β:jl:TA. But then aβbfβ=bα, so β must also be a 2-cell in Opf(X).

Conversely, suppose that f is ff in Opf(X), and that we have 2-cells α,β:xy:TA in K such that fα=fβ. Then since bfa we have aalph=aβ; call this 2-cell ξ:axay. Since a and b are opfibrations and f is a map of opfibrations, we have an opcartesian 2-cell xξ !(x) lying over ξ such that fxfξ !(x) is also opcartesian. Then α and β both factor through xξ !(x) to give 2-cells γ,δ:ξ !(x)y in Opf(X) whose images under f are equal. Since f is faithful in Opf(X), we have γ=δ, and hence α=β. Thus, f is faithful in K. The proof of fullness is analogous.

Therefore, if AX is an (op)fibration in K, we have Sub Opf(K)(A)Sub K(A), although in general the inclusion is proper. Of course, the dual result about Fib(X) is also true.

Iterated fibrations

It is well-known that a composite of fibrations is a fibration. Moreover:

Theorem

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

This is a standard result, at least in the case K=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 AX in K we have Fib K(A)Fib Fib K(X)(AX), and similarly for opfibrations. This is a fibrational analogue of the standard equivalence K/A(K/X)/(AX) for ordinary slice categories. It also implies that any morphism between discrete fibrations over X is itself a (discrete) fibration in K, since in Fib(X) it has a discrete (hence groupoidal) target and thus is a fibration there.

We will also need the corresponding result for mixed-variance iterated fibrations, which seems to be less well-known. First we recall:

Definition

A span ApEqB is called a (two-sided) fibration from B to A if

  1. q is an opfibration and p takes q-opcartesian 2-cells to isomorphisms,

  2. p is a fibration and q takes p-cartesian 2-cells to isomorphisms, and

  3. for any e:XE, and any square

    α *e pcartesian e qopcartesian qopcartesian (α *e)β ! α *(eβ !) pcartesian eβ !\array{&& \alpha^*e & \overset{p-cartesian}{\to} & e\\ ^{q-opcartesian} & \swarrow && & \downarrow & ^{q-opcartesian}\\ (\alpha^* e)\beta_! & \to & \alpha^* (e \beta_!) & \overset{p-cartesian}{\to} & e \beta_!}

    of 2-cells, (α *e)β !α *(eβ !) is an isomorphism.

Such two-sided fibrations in Cat correspond to functors B×A opCat. The third condition corresponds precisely to the “interchange” equality (β,1)(1,α)=(1,α)(β,1) in B×A op. We write Fib(B,A) for the 2-category of two-sided fibrations from B to A.

Theorem

A span ApEqB is a two-sided fibration from B to A if and only if

  1. p:EA is a fibration and
  2. (p,q):EA×B is an opfibration in Fib(A).
Proof

Recall that the projection A×BA 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 Fib(A), i.e. that it preserves cartesian 2-cells, says precisely that q takes p-cartesian 2-cells to isomorphisms.

Now, by the remarks above, q is an opfibration in K iff E(q/1 B) has a left adjoint with invertible counit in K/B, and (p,q) is an opfibration in Fib(A) iff E((p,q)/1 A×B) has a left adjoint with invertible counit in Fib(A)/(A×B). Of crucial importance is that here ((p,q)/1 A×B) denotes the comma object calculated in the 2-category Fib(A), or equivalently in K/A, 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 Fib(A) iff q is an opfibration in K and the left adjoint of E(q/1 B) is a morphism in 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 Definition 1 is satisfied.

In particular, we have Fib(B,A)Opf Fib(A)(A×B). By duality, Fib(B,A)Fib Opf(B)(A×B), and therefore Fib Opf(B)(A×B)Opf Fib(A)(A×B), a commutation result that is not immediately obvious.

It follows that the 2-categories Fib(B,A) of two-sided fibrations also inherit any properties that can be shown to be inherited by the “one-sided” fibrational slices Fib(X) and Opf(X). Thus, we will usually concentrate on the latter, although two-sided fibrations will make an appearance in our treatment of duality involutions.

Revised on June 12, 2012 11:10:00 by Andrew Stacey? (129.241.15.200)