The fibrational slice of a 2-category over an object is the homwise-full sub-2-category of the slice 2-category whose objects are fibrations over and whose morphisms are morphisms of fibrations. Likewise, we have the opfibrational slice 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 has finite limits.
The 2-category is monadic over . The relevant monad on takes to the comma object , or equivalently the pullback . It is lax-idempotent, so a morphism is an opfibration if and only if has a left adjoint with invertible counit in . Likewise, is a fibration iff has a right adjoint with invertible unit in .
Since the fibrational slices are monadic over , they inherit all limits from it. It follows that a fibration is a discrete object in iff it is discrete in . These are unsurprisingly called discrete fibrations; we write for the category of such. Every morphism in between discrete fibrations is a morphism of fibrations; thus is a full subcategory of both and .
Any pullback of an (op)fibration is again an (op)fibration. Therefore, any morphism induces a pullback functor , which restricts to a functor , 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 is groupoidal, . In particular, for (2,1)-categories and thus also for 1-categories, the fibrational slices are no different from the ordinary slices.
Central for us is the following fact, which would be false if we replaced by . It underlies the inheritance of all sorts of structure by fibrational slices, such as regularity, coherency, extensivity, and exactness.
A morphism in is ff in ) iff its underlying morphism in is ff.
Suppose that and are opfibrations, and that is a morphism in which is ff in . Then since is homwise faithful, is clearly faithful in . And given a 2-cell in , since is ff in , we must have for some 2-cell . But then , so must also be a 2-cell in .
Conversely, suppose that is ff in , and that we have 2-cells in such that . Then since we have ; call this 2-cell . Since and are opfibrations and is a map of opfibrations, we have an opcartesian 2-cell lying over such that is also opcartesian. Then and both factor through to give 2-cells in whose images under are equal. Since is faithful in , we have , and hence . Thus, is faithful in . The proof of fullness is analogous.
Therefore, if is an (op)fibration in , we have , although in general the inclusion is proper. Of course, the dual result about is also true.
It is well-known that a composite of fibrations is a fibration. Moreover:
A morphism in is a fibration in the 2-category iff its underlying morphism in is a fibration.
This is a standard result, at least in the case , and is apparently due to Benabou. References include:
Therefore, for any fibration in we have , and similarly for opfibrations. This is a fibrational analogue of the standard equivalence for ordinary slice categories. It also implies that any morphism between discrete fibrations over is itself a (discrete) fibration in , since in 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:
A span is called a (two-sided) fibration from to if
of 2-cells, is an isomorphism.
Such two-sided fibrations in correspond to functors . The third condition corresponds precisely to the “interchange” equality in . We write for the 2-category of two-sided fibrations from to .
A span is a two-sided fibration from to if and only if
Recall that the projection is a fibration (and also an opfibration, although that is irrelevant here), and the cartesian 2-cells are precisely those whose component in is an isomorphism. Therefore, saying that is a morphism in , i.e. that it preserves cartesian 2-cells, says precisely that takes -cartesian 2-cells to isomorphisms.
Now, by the remarks above, is an opfibration in iff has a left adjoint with invertible counit in , and is an opfibration in iff has a left adjoint with invertible counit in . Of crucial importance is that here denotes the comma object calculated in the 2-category , or equivalently in , and it is easy to check that this is in fact equivalent to the comma object calculated in .
Therefore, is an opfibration in iff is an opfibration in and the left adjoint of is a morphism in . It is then easy to check that this left adjoint is a morphism in iff inverts -opcartesian arrows, and that it is a morphism of fibrations iff the final condition in Definition 1 is satisfied.
In particular, we have . By duality, , and therefore , a commutation result that is not immediately obvious.
It follows that the 2-categories of two-sided fibrations also inherit any properties that can be shown to be inherited by the “one-sided” fibrational slices and . Thus, we will usually concentrate on the latter, although two-sided fibrations will make an appearance in our treatment of duality involutions.