Q-category

In mathematics, a Q-category or almost quotient category is a category that is a "milder version of a Grothendieck site."{{harvnb|Kontsevich|Rosenberg|2004a|loc=§ 1.}} A Q-category is a coreflective subcategory.{{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}}{{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. 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{{Further explanation needed|reason=notation is unclear|date=March 2023}} $\mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ 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.{{Further explanation needed|reason=what is a "cone"?|date=March 2023}}
There is a Q-category of sieves.{{Clarification needed|reason=what is a sieve?|date=March 2023}}

References

{{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.

Q-category

Definition

Examples

References

Further reading