Q-category
In mathematics, a Q-category or almost quotient category[1] is a category that is a "milder version of a Grothendieck site."[2] 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.[2] 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[1]
where 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.[1]
- For any category, one can define the Q-category of cones.[1]
- There is a Q-category of sieves.[1]
References
- Škoda, Zoran; Schreiber, Urs; Mrđen, Rafael; Fritz, Tobias (14 September 2017). "Q-category". nLab. Retrieved 25 March 2023.
- Kontsevich & Rosenberg 2004a, § 1.
- Kontsevich, Maxim; Rosenberg, Alexander (2004a). "Noncommutative spaces" (PDF). ncatlab.org. Retrieved 25 March 2023.
- Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988.
Further reading
- Kontsevich, Maxim; Rosenberg, Alexander (2004b). "Noncommutative stacks". ncatlab.org. Retrieved 25 March 2023.
- Brzezinski, Tomasz (29 October 2007). Brzeziński, Tomasz; Pardo, José Luis Gómez; Shestakov, Ivan; Smith, Patrick F. (eds.). Notes on formal smoothness. Modules and Comodules. arXiv:0710.5527. doi:10.1007/978-3-7643-8742-6.
- Lawvere, F. William (2007). "Axiomatic Cohesion" (PDF). Theory and Applications of Categories. 19 (3): 41–49.
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