Beck-Chevalley for presheaves on groupoids?

Beck-Chevalley for presheaves on groupoids?

[John Baez]

Hi -

One of you must know the answer to this!

Suppose we have a weak pullback (= pseudo-pullback) square of
groupoids:

G -> H
|    |
v    v
K -> L

Suppose we take presheaves on all four.  We can get a square

hom(G^{op},Set) -> hom(H^{op},Set)
      ^                 ^
      |                 |
hom(K^{op},Set) -> hom(L^{op},Set)

where the arrows pointing forward - in the same direction as
the original arrows - are defined using pushforward, and the arrows
pointing backward are defined using pullback.

Does this square commute up to natural isomorphism?  Do you
know a reference somewhere?

Some side remarks:

1) This seems related to the "Beck-Chevalley condition".

2) It may work for categories as well as groupoids, but I happen to
need it only for groupoids.

3) I really need it with the category Vect replacing Set, so
if you know a general result for any sufficiently nice category
playing the role of Set here, that would be wonderful.

Best,
jb

Re: Beck-Chevalley for presheaves on groupoids?

[RJ Wood]

If

G -> H
| <= |
v    v
K -> L

is a square (with a 2-cell as shown) in cat then

applying ^ =((-)^{op},Set) gives

G^<- H^
^ => ^
|    |
K^<- L^

and taking the mate with respect to the horizontal adjunctions
given by left Kan extension gives

G^-> H^
^ => ^
|    |
K^-> L^

If the original square is a comma square then the 2-cell in
the third square is invertible. There are squares other than
comma squares, cocomma squares for example, for which the
2-cell in the third square is invertible. See Rene Guitart's
early work on exact squares.

Best, Rj Wood


> One of you must know the answer to this!
>
> Suppose we have a weak pullback (= pseudo-pullback) square of
> groupoids:
>
> G -> H
> |    |
> v    v
> K -> L
>
> Suppose we take presheaves on all four.  We can get a square
>
> hom(G^{op},Set) -> hom(H^{op},Set)
>       ^                 ^
>       |                 |
> hom(K^{op},Set) -> hom(L^{op},Set)
>
> where the arrows pointing forward - in the same direction as
> the original arrows - are defined using pushforward, and the arrows
> pointing backward are defined using pullback.
>
> Does this square commute up to natural isomorphism?  Do you
> know a reference somewhere?
>
> Some side remarks:
>
> 1) This seems related to the "Beck-Chevalley condition".
>
> 2) It may work for categories as well as groupoids, but I happen to
> need it only for groupoids.
>
> 3) I really need it with the category Vect replacing Set, so
> if you know a general result for any sufficiently nice category
> playing the role of Set here, that would be wonderful.
>
> Best,
> jb