Colinear map

In coalgebra theory, the notion of colinear map is dual to the notion for linear map of vector space, or more generally, for morphism between R-module. Specifically, let R be a ring, M,N,C be R-modules, and

${\displaystyle \rho _{M}:M\rightarrow M\otimes C,\rho _{N}:N\rightarrow N\otimes C}$

be right C-comodules. Then an R-linear map ${\displaystyle f:M\rightarrow N}$ is called a (right) comodule morphism, or (right) C-colinear, if

${\displaystyle \rho _{N}\circ f=(f\otimes 1)\circ \rho _{M}}$

References

• Khaled AL-Takhman, Equivalences of Comodule Categories for Coalgebras over Rings, J. Pure Appl. Algebra,.V. 173, Issue: 3, September 7, 2002, pp. 245–271