View source for Q-category

{{Short description|Concept in mathematical category theory}}

In mathematics, a '''Q-category''' or '''almost quotient category<ref name=":0" />''' is a [[category (mathematics)|category]] that is a "milder version of a Grothendieck site."<ref name=":1">{{harvnb|Kontsevich|Rosenberg|2004a|loc=§ 1.}}</ref> A Q-category is a [[coreflective subcategory]].<ref name=":0">{{Cite web |last1=Škoda |first1=Zoran |last2=Schreiber |first2=Urs |last3=Mrđen |first3=Rafael |last4=Fritz |first4=Tobias |date=14 September 2017 |title=Q-category |url=https://ncatlab.org/nlab/show/Q-category |access-date=25 March 2023 |website=[[nLab]]}}</ref>{{Clarification needed|reason=subcategory of what?|date=March 2023}} The Q stands for a quotient. 

The concept of Q-categories was introduced by Alexander Rosenberg in 1988.<ref name=":1" /> The motivation for the notion was its use in [[noncommutative algebraic geometry]]; in this formalism, [[noncommutative space]]s are defined as [[Sheaf (mathematics)|sheaves]] on Q-categories.

== Definition ==
A Q-category is defined by the formula<ref name=":0" />{{Further explanation needed|reason=notation is unclear|date=March 2023}} <math display="block">\mathbb{A} : (u^* \dashv u_*) :  \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A</math>where <math>u^*</math> is the left adjoint in a pair of [[adjoint functors]] and is a [[Full and faithful functors|full and faithful functor]].

== Examples ==

*The category of [[presheaves]] over any Q-category is itself a Q-category.<ref name=":0" />
*For any category, one can define the ''Q-category of cones''.<ref name=":0" />{{Further explanation needed|reason=what is a "cone"?|date=March 2023}}
*There is a Q-category of sieves.<ref name=":0" />{{Clarification needed|reason=what is a sieve?|date=March 2023}}

== References ==
{{reflist}}
* {{Cite web |last1=Kontsevich |first1=Maxim |last2=Rosenberg |first2=Alexander |date=2004a |title=Noncommutative spaces |url=https://ncatlab.org/nlab/files/KontsevichRosenbergNCSpaces.pdf |access-date=25 March 2023 |website=ncatlab.org}}
* Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988.

== Further reading ==
* {{Cite web |last1=Kontsevich |first1=Maxim |last2=Rosenberg |first2=Alexander |date=2004b |title=Noncommutative stacks |url=http://www.mpim-bonn.mpg.de/preblob/2333 |access-date=25 March 2023 |website=ncatlab.org}}
* {{Cite conference |title=Notes on formal smoothness |language=en |last=Brzezinski|first=Tomasz|date=29 October 2007|doi=10.1007/978-3-7643-8742-6 |arxiv=0710.5527 |conference=Modules and Comodules|conference-url=https://link.springer.com/book/10.1007/978-3-7643-8742-6|editor-last=Brzeziński|editor-first=Tomasz|editor2-last=Pardo|editor2-first=José Luis Gómez|editor3-last=Shestakov|editor3-first=Ivan|editor4-last=Smith|editor4-first=Patrick F.}}
* {{Cite journal |last=Lawvere |first=F. William |date=2007 |title=Axiomatic Cohesion |url=http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf |journal=Theory and Applications of Categories |volume=19 |issue=3 |pages=41–49}}

[[Category:Category theory]]
[[Category:Noncommutative geometry]]


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