1204.0196Apr 1, 201228 pages. 2nd version: many changes with oplax --> colax. 3rd version: minor changes including "The k-flatness assumption was added to apply Keller's theorem on derived equivalences of categories."
The Grothendieck construction of a diagram X of categories can be seen as a process to construct a single category (X) by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category I to the 2-category of small -categories for a fixed commutative ring . In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors X, X' I → are derived equivalent if there is a derived equivalence from X(i) to X′(i)
Nearby in the stack
for all objects
i
in
I
satisfying some "
I
-equivariance" conditions. In this paper we glue the derived equivalences between
X(i)
and
X′(i)
together to obtain a derived equivalence between Grothendieck constructions
(X)
and
(X')
, which shows that if colax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two
-categories with a
G
-action for a group
G
are "
G
-equivariantly" derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two
k
-algebras
A
and
A′
are derived equivalent, then so are the path categories
AQ
and
A′Q
for any quiver
Q
; so are the incidence categories
AS
and
A′S
for any poset
S
; and so are the monoid algebras
AG
and
A′G
for any monoid
G
. Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence.