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.
Definition
A Q-category is defined by the formula[1][further explanation needed] [math]\displaystyle{ \mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A }[/math]where [math]\displaystyle{ u^* }[/math] 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][further explanation needed]
- There is a Q-category of sieves.[1][clarification needed]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Škoda, Zoran; Schreiber, Urs; Mrđen, Rafael; Fritz, Tobias (14 September 2017). "Q-category". https://ncatlab.org/nlab/show/Q-category.
- ↑ 2.0 2.1 Kontsevich & Rosenberg 2004a, § 1.
- Kontsevich, Maxim; Rosenberg, Alexander (2004a). "Noncommutative spaces". https://ncatlab.org/nlab/files/KontsevichRosenbergNCSpaces.pdf.
- 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". http://www.mpim-bonn.mpg.de/preblob/2333.
- Brzezinski, Tomasz (29 October 2007). "Notes on formal smoothness". in Brzeziński, Tomasz; Pardo, José Luis Gómez; Shestakov, Ivan et al. (in en). Modules and Comodules. doi:10.1007/978-3-7643-8742-6.
- Lawvere, F. William (2007). "Axiomatic Cohesion". Theory and Applications of Categories 19 (3): 41–49. http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf.
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