The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche conjectured that, for any small category $\cal B$, the category ${\bf DCF}/{\cal B}$ of discrete Conduché fibrations over $\cal B$ should be a topos. In this note we show that, although for suitable categories $\cal B$ the discrete Conduché fibrations over $\cal B$ may be presented as the `sheaves' for a family of coverings on a category ${\cal B}_{tw}$ constructed from $\cal B$, they are in general very far from forming a topos.
Theory and Applications of Categories, Vol. 5, 1999, No. 1, pp 1-11 http://www.tac.mta.ca/tac/volumes/1999/n1/n1.dvi http://www.tac.mta.ca/tac/volumes/1999/n1/n1.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n1/n1.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n1/n1.psTAC Home