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(Terminology of pseudofunctors and fibrations is mixed throughout.)

The Beck-Chevalley condition is defined on the nlab as a property of a quadruple of functors: starting from an invertible 2-cell of functors (a commutative-up-to-isomorphism square), if opposite sides have adjoints then their universal properties induces another 2-cell which we want to be invertible. For the case of bifibrations one considers commutative squares on the base and asks whether a cartesian lifting of one pair of opposite sides and a cocartesian lifting of the other opposite pair might give an invertible 2-cell.

On the other hand, another (?) thing called the Beck-Chevalley condition appears when discussing fiberwise vs fibered adjunctions. In this setting we have a morphism $F$ of fibrations over $\mathsf C$, and for the restriction $F_c$ of this morphism to each fiber we have an adjoint. The universal properties of Cartesian arrows and the adjoints again induce a 2-cell $F_cf^\ast\Rightarrow f^\star F_{c^\prime}$ which we want to be invertible.

The second Beck-Chevalley condition codifies "preservation of structure" of the adjunctions by the base-change functors, as exemplified by the fibered adjunction defining limits of a certain shape in a fibration. However, I don't really understand the first morally (though the example of bundles for monadic descent is instructive).

Are these two Beck-Chevalley conditions related somehow? Are they both special cases of a more general one? I ask because the second 2-cell does not seem to "come" from a commutative square in the base category.

Added. The prototype situation seems involve a pair of fibrations $\varphi,\psi$ over the same base, and a pair of adjunctions $$\varphi ^{-1}(a)\begin{smallmatrix} \overset{F}\rightarrow \\ \perp \\ \underset{G}\leftarrow \end{smallmatrix}\psi ^{-1}(b),\; \; \varphi^{-1}(c)\begin{smallmatrix} \overset{F^\prime}\rightarrow \\ \perp \\ \underset{G^\prime}\leftarrow \end{smallmatrix}\psi ^{-1}(d)$$ related by base-change functors $$f^\ast :\varphi^{-1}(c)\to \varphi^{-1}(a),\; \; g^\star:\psi^{-1}(d)\to \psi^{-1}(b).$$

My problem is that I can't find a general mechanism furnishing some canonical 2-cell $$F \circ f^\ast\Rightarrow g^\star \circ F^\prime$$ which encompasses both of the examples below.

  1. There's a bifibration $\varphi$ and a commutative square $k\circ f=g\circ h$ in the base; we consider the adjunctions induced by $k,h$ related by the inverse-image functors $f^\ast ,g^\ast$. Here the 2-cell $h_!\circ f^\ast \Rightarrow g^\ast \circ k_!$ comes from the invertible 2-cell of inverse-image functors furnished by pseudofunctoriality.

  2. There are fibrations $\varphi,\psi$ over a common base and adjunctions over fibers of the same object such that both $G,G^\prime$ come from a fibered functor $\varphi \leftarrow \psi$ $$\varphi ^{-1}(a)\begin{smallmatrix} \overset{F}\rightarrow \\ \perp \\ \underset{G}\leftarrow \end{smallmatrix}\psi ^{-1}(a),\; \; \varphi^{-1}(b)\begin{smallmatrix} \overset{F^\prime}\rightarrow \\ \perp \\ \underset{G^\prime}\leftarrow \end{smallmatrix}\psi ^{-1}(b)$$ and these are connected by the base-change functors induced by the same arrow $f$ in the base. In this case, the fibered functor preserves cartesian arrows and this induces an invertible 2-cell of the $G,G^\prime$ which, by pasting with the unit and counit, gives a 2-cell involving the fiberwise left adjoints.

Particularly, the first example is "local" and the 2-cell comes from a commutative diagram in the base, while in the other one a very global creature - a fibered functor between the fibrations - is needed to induce a 2-cell.

Arrow
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  • I haven't worked out the details but I believe that the first view is the second except that we think of the (cloven) fibration instead as an indexed category (i.e. a pseudofunctor into $\mathbf{Cat}$). – Derek Elkins left SE Oct 26 '17 at 02:00
  • @DerekElkins I don't I understand how. The 2-cell $F_cf^\ast\Rightarrow f^\star F_{c^\prime}$ does not seem to generally come from a commutative square in the domain (thinking of a $F$ as a pseudofunctor).. – Arrow Oct 26 '17 at 08:58
  • The definition you linked to on the nLab is the general mechanism. You start from a commutative square of functors of which two have adjoints, and produce a 2-cell relating the two adjoints to the other two functors in the commutative square. In the first case the original commutative square comes from pseudofunctoriality; in the second case it comes from the fact that a fibered functor preserves the restriction functors (since it preserves cartesian arrows). – Mike Shulman Nov 01 '17 at 15:29
  • @MikeShulman thank you for the comment. I understand the nlab explains the 'mate' mechanism given an invertible 2-cell of functors (two of which have adjoint). I was hoping perhaps for a general criterion as to when such a 2-cell might exist, encompassing both examples mentioned in my question. – Arrow Nov 02 '17 at 17:42
  • I don't think there could be a general criterion for when a square commutes (whether it be strictly or up to isomorphism). Commutative squares arise all over category theory for all sorts of different reasons. It's like asking for a general criterion for why you might have an equation $g_1 g_2 = h_1 h_2$ in a group. – Mike Shulman Nov 03 '17 at 17:40
  • Although like most anything in category theory, you can come up with tautological ways to make it true. For instance, your first example is obtained by applying a (pseudo) functor to a square that's assumed to commute, whereas your second example is a (pseudo) naturality square for a (pseudo) natural transformation. If we view a natural transformation between functors $F,G: C\to D$ as a single functor $C\times 2 \to D$, then both examples would be obtained by applying some functor to a commutative square. (cont.) – Mike Shulman Nov 03 '17 at 17:42
  • But in fact any commutative square is obtained by applying a functor to some other commutative square, either tautologically by taking the functor to be the identity functor, or only slightly less tautologically by taking the domain of the functor to be the "free-living commutative square" $2\times 2$. (Here by $2$ I mean the interval category, with two objects and one nonidentity morphism between them.) – Mike Shulman Nov 03 '17 at 17:43
  • I am not asking for such a general criterion - only wondering whether there is a joint generalizations of both examples in the questions. – Arrow Nov 03 '17 at 18:02

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