Q-category explained

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.^[1] 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 $$\backslash mathbb\; :\; (u^*\; \backslash dashv\; u\_*)\; :\; \backslash bar\; A\; \backslash stackrel\; A$$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.

References

• Web site: Kontsevich . Maxim . Rosenberg . Alexander . 2004a . Noncommutative spaces . 25 March 2023 . ncatlab.org.
• Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988.

Further reading

• Web site: Kontsevich . Maxim . Rosenberg . Alexander . 2004b . Noncommutative stacks . 25 March 2023 . ncatlab.org.
• Notes on formal smoothness . en . Brzezinski. Tomasz. 29 October 2007. 10.1007/978-3-7643-8742-6 . 0710.5527 . Modules and Comodules. https://link.springer.com/book/10.1007/978-3-7643-8742-6. Brzeziński. Tomasz. Pardo. José Luis Gómez. Shestakov. Ivan. Smith. Patrick F..
• Lawvere . F. William . 2007 . Axiomatic Cohesion . Theory and Applications of Categories . 19 . 3 . 41–49.

Notes and References

1. Web site: Škoda . Zoran . Schreiber . Urs . Mrđen . Rafael . Fritz . Tobias . 14 September 2017 . Q-category . 25 March 2023 . nLab.