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There is a
short course on:
“2-Categoriecal Covering Theory and Derived
Equivalences”
The
lecturer is Professor Hideto Asashiba from
Shizuoka University, Japan.
Organizer:
Javad
Asadollahi
(University of Isfahan and IPM)
Rasool Hafezi
(IPM)
The program is as
follows:
Sunday, May 18, 10:30-12:00
and 14:30-15:30
Monday, May 19, 10:30-12:00 and 14:30-15:30.
To register for the short course, send an
e-mail to:
registration@ipm.ir
with the
subject: Categoriecal Covering Theory
For
information on accomodation please click
here.
Abstract:
We fix a
base field k and all algebras are assumed to
be basic k-algebras and regarded as finite
locally bounded k-categories. We start from
an original Galois covering of locally
bounded k-categories with a group G defined
by Gabriel, and generalize it to a
G-covering C --> C' for an arbitrary
k-categories with a G-action, which we
characterize by using a canonical G-covering
(P, f) from C to the orbit category C/G.
The
orbit category construction will be extended
to a 2-functor (-/G) from the 2-category
G-Cat of small k-categories with G-action to
the 2-category k-Cat of small k-categories,
which will be characterized as a left
adjoint to the diagonal 2-functor k-Cat -->
G-Cat sending each small k-category to
itself with the trivial G-action, and will
be shown that the canonical G-covering (P,
f) is a component of the unit of the
adjoint. Noting that a G-action of a
category C is nothing but a functor X from G
as the category with a single object * to
k-Cat with X(*) = C, the consideration above
will be generalized to functors (or more
generally colax functors) X from a small
category I to the 2-category k-Cat, which is
regarded as an I-diagram in k-Cat or an
I-action on X(i)'s (i objects in I), and the
collection of them can be extended to a
2-category Colax(I, k-Cat).
Then the
orbit category construction is generalized
to the Grothendieck construction Colax(I,
k-Cat) --> k-Cat, X |--> Gr(X) (Note that
Gr(X) is a single k-category for each X).
The class of objects of Gr(X) is the union
of the classes of objects of X(i)'s (i
objects in I) and Gr(X) is regarded as a
"gluing" of X(i)'s. We define a "module
category" Mod X and a "derived module
category" D(Mod X) of X in a natural way
that leads us a definition of derived
equivalences between X's. The main theorem
states that if two colax functors X and X'
from I to k-Cat are derived equivalent, then
so are Gr(X) and Gr(X'), which enables us to
glue derived equivalences between X(i) and
X'(i) (i objects in I) together to have a
derived equivalence between Gr(X) and
Gr(X').
Please
look at the following three papers
(preprints can be downloaded from arXiv):
[1]
Asashiba, Hideto: A generalization of
Gabriel's Galois covering functors and
derived equivalences, J. Algebra 334 (2011),
no. 1, 109--149. (preprint arXiv: 0807.4706)
[2]
Asashiba, Hideto: Derived equivalences of
actions of a category, Applied Categorical
Structures, DOI: 10.1007/s10485-012-9284-5.
(preprint arXiv:1111.2239)
[3]
Asashiba, Hideto: Gluing derived
equivalences together, Adv. Math. 235 (2013)
134--160. (preprint arXiv:1204.0196)
You can
download the Photos
here.
Address:
IPM-Isfahan Branch,
University of Isfahan,
Isfahan, Iran
P. O. Box: 81745-157
Phone: 0098 (311) 793 2319
Email:
ipm-isfahan@ipm.ir
http://math.ipm.ac.ir/Isfahan/
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