Dear category theorists, Thanks a lot for your answers. I decided to prepare a short summary of the discussion about (co)completeness of fibrations. Fibrations (arise, e.g., by a flattening construction from an indexed category) provide an appropriate mathematical tool for describing morphisms between objects of different types. In the case of my research, these objects are typed graphs and typed graph transformation rules or productions comprised in graph transformation systems (GTS). (Co)completeness property becomes very important under the construction of new GTS from the existing ones, e.g., different kinds of compositions of two GTSs over the third one. 1. Completeness of flattened categories: If C:Ind_Op -> Cat is an indexed category such that 1. Ind is complete, 2. C_i is complete for all i in (Ind)_obj, and 3. C_m : C_j -> C_i is continuous for all index morphisms m: i -> j, then Flat(C) is complete. 2. Cocompleteness of flattened categories: If C:Ind_Op -> Cat is an indexed category such that 1. Ind is cocomplete, 2. C_i is cocomplete for all i in (Ind)_obj, and 3. C is locally reversible (i.e., if for each index morphism m: i -> j in Ind, the translation functor C_m : C_j -> C_i has a left adjoint), then Flat(C) is cocomplete. Finally, I have collected all the references containing in the answers to my question. 1. @InCollection(Gray66, Author={Gray, J. W.}, Title={Fibred and Cofibred categories}, Booktitle={Proceedings of the Conference on Categorical Algebra}, Editor={Eilenberg, S.}, Publisher= sv, Year=1966) 2 @article(fun3, title = "Some Fundamental Algebraic Tools for the Semantics of Computation, Part 3: Indexed Categories", author = "Andrzej Tarlecki and Rod Burstall and Joseph Goguen", journal = "Theoretical Computer Science", year = 1991, volume = 91, pages = "239--264") 3. Agusti Roig. Model category structures in bifibred categories JPAA 95, (1994), 203 - 223 4. @Article{Hermida96a, author = {Hermida, C.}, title = {Some properties of {\textbf{{F}ib}} as a fibred 2-category}, journal = jpaa, year = {1999}, volume = {134}, number = {1}, pages = {83-109}, note = {Presented at ECCT'94, Tours, France.} } 5. @BOOK{Jacobs99, author = "Jacobs, B.", title = "Categorical logic and type theory", publisher = nh, volume = "141", series = "Studies in Logic and the Foundations of Mathematics", year = 1999 } Best regards, Alexey Cherchago 13-Dec-2004 13:27:38 -0400,3032;000000000000-00000000