Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Recall that a functor is called a fibration if its fibres vary (pseudo-)functorially in . Taking here fibre to mean strict fibre results in the notion of Grothendieck fibration, while taking it to mean essential fibre gives the notion of Street fibration.
Similarly, a two-sided fibration is a span of functors whose joint fibres vary functorially in both and (contravariantly in one and covariantly in the other).
This notion should not be confused with a bifibration, which is a functor that is both a fibration and a cofibration.
Let be a bicategory with finite 2-limits, and recall that fibrations in may be defined in any of several ways. Each of these has an analogous version for two-sided fibrations.
Recall that (cloven) fibrations in are the (pseudo)algebras for a (pseudo) 2-monad on . For a morphism in , is given by composing the span with the canonical span , so that is the canonical projection. This can equivalently be described as the comma object . This 2-monad is lax-idempotent, so that is a fibration if and only if the unit has a left adjoint with invertible counit.
More generally, the same construction gives a 2-monad on , whose algebras we call left fibrations. In Cat, a span is a left fibration if is a cloven fibration whose chosen cartesian lifts are -vertical. (Since we are working bicategorically, “-vertical” means that they map to isomorphisms under .)
Dually, there is a colax-idempotent 2-monad on each whose algebras are called right fibrations, the special case of yielding cloven opfibrations.
There is then a composite 2-monad that takes a span from to to , and -algebras are called two-sided fibrations. Although is neither lax- nor colax-idempotent, it is still property-like?.
A two-sided Street fibration from to in is given by a span , such that
each in has a -cartesian lift in that is -vertical (that is, is a left fibration)
each in has a -opcartesian lift in that is -vertical ( is a right fibration)
for every cartesian–opcartesian composite in , the canonical morphism is an isomorphism.
By the usual theory of distributive laws, an -algebra gives rise to - and -algebras and , and conversely an -algebra and an -algebra underlie an -algebra if and only if there is an isomorphism that makes a morphism of -algebras.
Now given and , because is colax-idempotent, there is a unique 2-cell that makes a colax morphism of -algebras. So we want to show that in the case of , the components of this natural transformation are the canonical morphisms of (3).
The 2-cell is given by , where is the counit of the adjunction . Its components are thus given, for each in and in , by first factoring through the opcartesian and then factoring the result through the cartesian , to obtain exactly the canonical morphism .
If is a two-sided fibration, then the operation sending to the corresponding (essential) fiber of defines a pseudofunctor . The third condition in Proposition corresponds to the “interchange” equality in . We write for the 2-category of two-sided fibrations from to .
Another definition of internal fibration is that a (cloven) fibration in is a morphism such that is a (cloven) fibration in , for any , and for any the corresponding square is a morphism of fibrations in . To adapt this definition to two-sided fibrations, we therefore need only to say what is a two-sided fibration in . For this we can use the characterization of Proposition .
Let denote the 2-category of fibrations over . It is a well-known fact (apparently due to Benabou) that a morphism in is a fibration in if and only if its underlying morphism in is a fibration. See fibration in a 2-category. Thus, for any fibration , we have .
Of course there is a dual result for opfibrations: for any opfibration we have . When we combine variance of iteration, however, we obtain two-sided fibrations.
A span is a two-sided fibration from to if and only if
is a fibration and
is an opfibration in .
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 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 (since monadic forgetful functors create limits), 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 Proposition is satisfied.
In particular, we have . By duality, , and therefore , a commutation result that is not immediately obvious.
This result appears in Bourn–Penon; it was noticed independently and recorded here by Mike Shulman.
A two-sided fibration in is discrete if it is discrete as an object of . Since discreteness is a limit construction, it is created by monadic forgetful functors; hence this is equivalent to being discrete as an object of the 2-category of two-sided fibrations.
For Grothendieck fibrations in Cat, this means the following.
A two-sided discrete fibration is a span , of categories and functors such that
We write
for the full subcategory on the 2-category of spans on the 2-sided discrete fibrations. Since a morphism of spans between discrete fibrations is automatically a morphism of fibrations, this is also the full sub-2-category of the 2-category of two-sided fibrations . And since they are discrete objects, this 2-category is actually (equivalent to) a 1-category.
Note, though, that the two legs of a two-sided discrete fibration are not necessarily individually discrete as a fibration and an opfibration.
Given a profunctor , its collage is the category over the interval category
With , , and for all , , where
the composite of with is given by ;
the composite of with is given by .
There is an equivalence of categories
pseudo-natural in , between profunctors in Set and two-sided discrete fibrations from to , where is the category whose
morphisms are natural transformations between such sections;
the two projections are the two functors induced by restriction along .
First we write out in detail. In the following and .
The objects of are morphisms
in , hence triples .
Morphisms are commuting diagrams
in . We may identify these with pairs such that
We check that this construction yields a two-sided fibration. The three conditions are
For
an object of and a morphism in , we have that
is the unique lift to a morphism in that maps to .
Analogously, for
an object of and a morphism in , we have that
is the unique lift to a morphism in that maps to .
For
an arbitrary morphism in , these two unique lifts of its - and its -projection, respectively, are
and
The codomain and domain do match, since by the existence of the original morphism, and their composite is the original morphism
To see that this construction indeed yields an equivalence of categories, define a functor by setting
;
for a morphism let be the function that sends to the domain of the unique lift of with this codomain and mapping to ;
for a morphism let be the function that sends to the codomain of the unique lift of with this domain and mapping to ;.
One checks that this yields an equivalence of categories.
The category is equivalently characterized as being the comma category of the diagram .
Note that profunctors can also be characterized by their collages, these being the two-sided codiscrete cofibrations; and the collage corresponding to a two-sided fibration is its cocomma object?.
Fibrations and opfibrations on a category (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the two-sided fibrations satisfying the Beck-Chevalley condition; see von Glehn (2015).
two-sided fibration,
An early reference is the notion of “regular span” on page 535 of:
Original discussion:
Ross Street. Fibrations and Yoneda’s lemma in a 2-category. In Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 104 133. Lecture Notes in Math., Vol. 420. Springer, Berlin, 1974.
Ross Street, Fibrations in bicategories. Cahiers Topologie Géom. Différentielle,
21(2):111–160, 1980. (Corrections in 28(1):53–56, 1987)
Further discussion of discrete fibrations
Useful reviews are in
In relation to categorical semantics of dependent types:
Last revised on December 29, 2023 at 18:21:03. See the history of this page for a list of all contributions to it.