Skip to main content
MathOverflow

Return to Answer

added 240 characters in body
Source Link
Harry Gindi
Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

I would split this problem up into two parts (here, 'sheaf (of groupoids)' is used instead of stack in order to disambiguate between Algebraic stacks (geometric objects) and mere (pseudo-)functors satisfying descent). :

1.) Show that the inclusion of algebraic stacks into the category of fppf stackssheaves of groupoids on Sch preserves coproducts. This follows immediately from the fact that algebraic stacks are a full subcategory(2,1)-subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in 2(2,1)-Stackstack topoi. This follows from the left-exactness of the stackification functor(2,1)-functor together with the altogether more obvious version of this fact for 2(2,1 topoi)-topoi of groupoid fibrations (also called (2,1)-presheaf topoi), where one can immediately reduce to proving the statement for groupoids pointwise.

I would split this problem up into two parts:

1.) Show that the inclusion of algebraic stacks into the category of fppf stacks on Sch preserves coproducts. This follows immediately from the fact that algebraic stacks are a full subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in 2,1-Stack topoi. This follows from the left-exactness of the stackification functor together with the altogether more obvious version of this fact for 2,1 topoi of groupoid fibrations, where one can immediately reduce to proving the statement for groupoids pointwise.

I would split this problem up into two parts (here, 'sheaf (of groupoids)' is used instead of stack in order to disambiguate between Algebraic stacks (geometric objects) and mere (pseudo-)functors satisfying descent). :

1.) Show that the inclusion of algebraic stacks into the category of fppf sheaves of groupoids on Sch preserves coproducts. This follows immediately from the fact that algebraic stacks are a full (2,1)-subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in (2,1)-stack topoi. This follows from the left-exactness of the stackification (2,1)-functor together with the altogether more obvious version of this fact for (2,1)-topoi of groupoid fibrations (also called (2,1)-presheaf topoi), where one can immediately reduce to proving the statement for groupoids pointwise.

Source Link
Harry Gindi
Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

I would split this problem up into two parts:

1.) Show that the inclusion of algebraic stacks into the category of fppf stacks on Sch preserves coproducts. This follows immediately from the fact that algebraic stacks are a full subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in 2,1-Stack topoi. This follows from the left-exactness of the stackification functor together with the altogether more obvious version of this fact for 2,1 topoi of groupoid fibrations, where one can immediately reduce to proving the statement for groupoids pointwise.