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IPM > School of Mathematics > IPM-Isfahan > Workshop

There is a short course on: 

“2-Categoriecal Covering Theory and Derived Equivalences”

The lecturer is Professor Hideto Asashiba from Shizuoka University, Japan.


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:


with the subject: Categoriecal Covering Theory

For information on accomodation please click here.


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.



IPM-Isfahan Branch,
University of Isfahan,
Isfahan, Iran
P. O. Box: 81745-157

Phone: 0098 (311) 793 2319
Email: ipm-isfahan@ipm.ir


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