3 I have added the link to these papers

It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck.

Actually, Kontsevich and Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf $\text{Alg}^{op} \rightarrow \text{Set}$ as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian, group scheme, general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology".

I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold".

There are the following references:

All these papers are available in Max Plank preprint series (search Kontsevich or Rosenberg in "author" and leave other blank empty).

If you take a look at the first paper, just disregard the notion of "Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general.

2 formatting

It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck.

Actually,Kontsevich-Rosenberg

Actually, Kontsevich and Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf : Alg^op---->Set $\text{Alg}^{op} \rightarrow \text{Set}$ as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian,group scheme,general grassmannian, group scheme, general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology"topology".

I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold"hold".

There are the following references:

• Noncommutative spaces (from page 15)
• Noncommutative stacks
• Noncommutative Grassmannian and related constructions.(from constructions (from page 12 is interesting)

All these papers are available in Max Plank preprint series.(just series (search Kontsevich or Rosenberg in "author" and leave other blank empty) I do not know how to get a link from that paper list:(empty).

If you take a look at the first paper, just disregard the notion"Q-categorynotion of "Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general.

1 [made Community Wiki]

It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck.

Actually,Kontsevich-Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf: Alg^op---->Set as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian,group scheme,general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology"

I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold"

There are following references: Noncommutative spaces (from page 15) Noncommutative stacks Noncommutative Grassmannian and related constructions.(from page 12 is interesting)

All these papers are available in Max Plank preprint series.(just search Kontsevich or Rosenberg in "author" and leave other blank empty) I do not know how to get a link from that paper list:(

If you take a look at the first paper, just disregard the notion"Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general.