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Arshak Aivazian
Arshak Aivazian
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Is the category of continuous presheaves Necessary and sufficient conditions for all sheaves on a Grothendieck topossite to be continuous functors?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)
  1. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous?
  2. Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)

Is the category of continuous presheaves a Grothendieck topos?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)

Necessary and sufficient conditions for all sheaves on a site to be continuous functors?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous?
  2. Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)
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Source Link
Arshak Aivazian
Arshak Aivazian
  • 7k
  • 1
  • 13
  • 44

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site I$I$ for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site I for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site $I$ for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)
Source Link
Arshak Aivazian
Arshak Aivazian
  • 7k
  • 1
  • 13
  • 44

Is the category of continuous presheaves a Grothendieck topos?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.

  1. What are the sufficient and what are the necessary conditions on the category $I$ for the category of continuous presheaves on $I$ to be a Grothendieck topos?
  2. What are the sufficient and what are the necessary conditions on the subcanonical site I for all sheaves to be continuous? Is there a canonical way to introduce the structure of such a site on an arbitrary small category? (by canonical I mean functorial with respect to category equivalences)