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 $p:A\to X$ to the comma object $(p/1_X)$, or equivalently the pullback $A\times_X X ^{\mathbf{2}}$. It is lax-idempotent, so a morphism $p:A\to X$ is an opfibration if and only if $A\to (p/1_X)$ has a left adjoint with invertible counit in $K/X$. Likewise, $p$ is a fibration iff $A\to (1_X/p)$ 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 $f:Y\to X$ induces a pullback functor $f^*:Fib(X)\to Fib(Y)$, which restricts to a functor $f^*:DFib(X)\to DFib(Y)$, 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 $X$ is groupoidal, $Fib(X)\simeq K/X\simeq Opf(X)$. 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 $Opf(X)$ 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 $a:A\to X$ and $b:B\to X$ are opfibrations, and that $f: A\to B$ is a morphism in $Opf(X)$ which is ff in $K$. Then since $Opf(X)\to K$ is homwise faithful, $f$ is clearly faithful in $Opf(X)$. And given a 2-cell $\alpha: h\to k:T\rightrightarrows B$ in $Opf(X)$, since $f$ is ff in $K$, we must have $\alpha = f\beta$ for some 2-cell $\beta: j\to l:T\rightrightarrows A$. But then $a\beta \cong b f\beta = b\alpha$, so $\beta$ must also be a 2-cell in $Opf(X)$.

Conversely, suppose that $f$ is ff in $Opf(X)$, and that we have 2-cells $\alpha,\beta: x\rightrightarrows y: T\rightrightarrows A$ in $K$ such that $f\alpha=f\beta$. Then since $b f\cong a$ we have $a\alpha= a\beta$; call this 2-cell $\xi: a x\to a y$. Since $a$ and $b$ are opfibrations and $f$ is a map of opfibrations, we have an opcartesian 2-cell $x\to \xi_!(x)$ lying over $\xi$ such that $f x \to f \xi_!(x)$ is also opcartesian. Then $\alpha$ and $\beta$ both factor through $x\to \xi_!(x)$ to give 2-cells $\gamma,\delta: \xi_!(x)\rightrightarrows y$ in $Opf(X)$ whose images under $f$ are equal. Since $f$ is faithful in $Opf(X)$, we have $\gamma=\delta$, and hence $\alpha=\beta$. Thus, $f$ is faithful in $K$. The proof of fullness is analogous.

Therefore, if $A\to X$ is an (op)fibration in $K$, we have $Sub_{Opf(K)}(A) \subset Sub_K(A)$, 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 $A\to X$ in $K$ we have $Fib_K(A) \simeq Fib_{Fib_K(X)}(A\to X)$, and similarly for opfibrations. This is a fibrational analogue of the standard equivalence $K/A \simeq (K/X)/(A\to X)$ 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 $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ 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 $e:X\to E$, and any square
   $$\array{&& \alpha^*e & \overset{p-cartesian}{\to} & e\\ ^{q-opcartesian} & \swarrow && & \downarrow & ^{q-opcartesian}\\ (\alpha^* e)\beta_! & \to & \alpha^* (e \beta_!) & \overset{p-cartesian}{\to} & e \beta_!}$$

   of 2-cells, $(\alpha^* e)\beta_! \to \alpha^* (e \beta_!)$ is an isomorphism.

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

Theorem

A span $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ is a two-sided fibration from $B$ to $A$ if and only if

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

Proof

Recall that the projection $A\times B \to A$ is a fibration (and also an opfibration, although that is irrelevant here), and the cartesian 2-cells are precisely those whose component in $B$ 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 $E\to (q/1_B)$ has a left adjoint with invertible counit in $K/B$, and $(p,q)$ is an opfibration in $Fib(A)$ iff $E\to ((p,q)/1_{A\times B})$ has a left adjoint with invertible counit in $Fib(A)/(A\times B)$. Of crucial importance is that here $((p,q)/1_{A\times B})$ 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 $E\to (q/1_B)$ 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 is satisfied.

In particular, we have $Fib(B,A) \simeq Opf_{Fib(A)}(A\times B)$. By duality, $Fib(B,A) \simeq Fib_{Opf(B)}(A\times B)$, and therefore $Fib_{Opf(B)}(A\times B) \simeq Opf_{Fib(A)}(A\times B)$, a commutation result that is not immediately obvious.

It follows that the 2-categories $Fib(B,A)$ 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.