Michael Shulman
fibrational slice

Definition

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

Basic properties

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

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

Any pullback of an (op)fibration is again an (op)fibration. Therefore, any morphism f:YXf:Y\to X induces a pullback functor f *:Fib(X)Fib(Y)f^*:Fib(X)\to Fib(Y), which restricts to a functor f *:DFib(X)DFib(Y)f^*:DFib(X)\to 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 XX is groupoidal, Fib(X)K/XOpf(X)Fib(X)\simeq K/X\simeq Opf(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)Opf(X) by K/XK/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)Opf(X) is ff in Opf(XOpf(X) iff its underlying morphism in KK is ff.

Proof

Suppose that a:AXa:A\to X and b:BXb:B\to X are opfibrations, and that fmapsABf\maps A\to B is a morphism in Opf(X)Opf(X) which is ff in KK. Then since Opf(X)KOpf(X)\to K is homwise faithful, ff is clearly faithful in Opf(X)Opf(X). And given a 2-cell α:hk:TB\alpha: h\to k:T\rightrightarrows B in Opf(X)Opf(X), since ff is ff in KK, we must have α=fβ\alpha = f\beta for some 2-cell β:jl:TA\beta: j\to l:T\rightrightarrows A. But then aβbfβ=bαa\beta \cong b f\beta = b\alpha, so β\beta must also be a 2-cell in Opf(X)Opf(X).

Conversely, suppose that ff is ff in Opf(X)Opf(X), and that we have 2-cells α,β:xy:TA\alpha,\beta: x\rightrightarrows y: T\rightrightarrows A in KK such that fα=fβf\alpha=f\beta. Then since bfab f\cong a we have aalph=aβa\alph= a\beta; call this 2-cell ξ:axay\xi: a x\to a y. Since aa and bb are opfibrations and ff is a map of opfibrations, we have an opcartesian 2-cell xξ !(x)x\to \xi_!(x) lying over ξ\xi such that fxfξ !(x)f x \to f \xi_!(x) is also opcartesian. Then α\alpha and β\beta both factor through xξ !(x)x\to \xi_!(x) to give 2-cells γ,δ:ξ !(x)y\gamma,\delta: \xi_!(x)\rightrightarrows y in Opf(X)Opf(X) whose images under ff are equal. Since ff is faithful in Opf(X)Opf(X), we have γ=δ\gamma=\delta, and hence α=β\alpha=\beta. Thus, ff is faithful in KK. The proof of fullness is analogous.

Therefore, if AXA\to X is an (op)fibration in KK, we have Sub Opf(K)(A)Sub K(A)Sub_{Opf(K)}(A) \subset Sub_K(A), although in general the inclusion is proper. Of course, the dual result about Fib(X)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)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 analogue of the standard equivalence K/A(K/X)/(AX)K/A \simeq (K/X)/(A\to X) for ordinary slice categories. It also implies that any morphism between discrete fibrations over XX is itself a (discrete) fibration in KK, since in Fib(X)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 ApEqBA \overset{p}{\leftarrow} E \overset{q}{\to} B is called a (two-sided) fibration from BB to AA if

  1. qq is an opfibration and pp takes qq-opcartesian 2-cells to isomorphisms,
  2. pp is a fibration and qq takes pp-cartesian 2-cells to isomorphisms, and
  3. for any e:XEe:X\to E, 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 !)(\alpha^* e)\beta_! \to \alpha^* (e \beta_!) is an isomorphism.

Such two-sided fibrations in CatCat correspond to functors B×A opCatB\times A^{op} \to Cat. The third condition corresponds precisely to the “interchange” equality (β,1)(1,α)=(1,α)(β,1)(\beta,1)(1,\alpha) = (1,\alpha)(\beta,1) in B×A opB\times A^{op}. We write Fib(B,A)Fib(B,A) for the 2-category of two-sided fibrations from BB to AA.

Theorem

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

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, by the remarks above, 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, 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 Definition 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.

It follows that the 2-categories Fib(B,A)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)Fib(X) and Opf(X)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 August 14, 2014 05:06:53 by Mike Shulman (192.195.154.58)