  We provide a construction of minimal injective resolutions of simple
comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau
condition of algebras, we introduce the Calabi-Yau condition to coalgebras.
Then we give some descriptions of Calabi-Yau coalgebras with lower global
dimensions. An appendix is included for listing some properties of cohom
functors.
