Generating and cogenerating structures

Abstract:

A functor $T:\mathcal {A} \to \mathcal {B}$ acts faithfully on the right of a class of objects $\mathcal {A}’$ of $\mathcal {A}$ if it distinguishes morphisms out of objects of $\mathcal {A}’$ (that is, $A’ \in \mathcal {A}’,X \in \mathcal {A},f,g \in \mathcal {A}(A’,X)$ and $f \ne g$ implies $T(f) \ne T(g))$. We define a full subcategory $\mathcal {R}\mathcal {F}(T)$ such that $T$ acts faithfully on the right of the objects of $\mathcal {R}\mathcal {F}(T)$. An object $U \in \mathcal {A}$ is a generator if ${H^U}:\mathcal {A} \to \mathcal {E}ns$ is faithful, and if ${H^U}$ is not faithful, we may still consider $\mathcal {R}\mathcal {F}({H^U})$. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of $R$-modules. Relationships between these can be used in the homological classification of rings.