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][clarification needed]} 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.

A Q-category is defined by the formula^{[1][further explanation needed]} $${\displaystyle \mathbb {A} :(u^{*}\dashv u_{*}):{\bar {A}}{\stackrel {\overset {u^{*}}{\leftarrow }}{\underset {u_{*}}{\to }}}A}$$where ${\displaystyle u^{*}}$ is the left adjoint in a pair of adjoint functors and is a full and faithful functor.

• 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][further explanation needed]}
• There is a Q-category of sieves.^{[1][clarification needed]}

• 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.

• 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.