Q-category

In mathematics, a Q-category or almost quotient category  is a category that is a "milder version of a Grothendieck site." A Q-category is a coreflective subcategory. The Q stands for a quotient.

The concept of Q-categories was introduced by Alexander Rosenberg in 1988. The motivation for the notion was its use in noncommutative algebraic geometry; in this formalism, noncommutative spaces are defined as sheaves on Q-categories.

Definition
A Q-category is defined by the formula $$\mathbb{A} : (u^* \dashv u_*) :  \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$$where $$u^*$$ is the left adjoint in a pair of adjoint functors and is a full and faithful functor.

Examples

 * The category of presheaves over any Q-category is itself a Q-category.
 * For any category, one can define the Q-category of cones.
 * There is a Q-category of sieves.