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.Template:Clarification needed 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 formulaTemplate:Further explanation needed $$𝔸:(u^{*}⊣u_{*}):\overline{A}\stackrel{\stackrel{u^{*}}{\leftarrow }}{\underset{u_{*}}{\to }}A$$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.Template:Further explanation needed
• There is a Q-category of sieves.Template:Clarification needed

References

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• Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988.

Further reading

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