Does the Grothendieck construction satisfy Fubini's thorem
$begingroup$
Suppose we are given a functor
$F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$int_{Atimes B}F = (Atimes B)/F$.
We could also apply this construction pointwise to obtain a functor
$int_A F:B^{op}to operatorname{Cat}$
sending $bmapsto A/F(b)$
and similarly
$int_B F:A^{op}to operatorname{Cat}$
We can apply the Grothendieck construction again to each of these functors to obtain categories
$int_Aint_B F$
and
$int_Bint_A F$
Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?
at.algebraic-topology ct.category-theory grothendieck-construction
edited Feb 9 at 15:46
David White
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
$endgroup$
add a comment |
$begingroup$
Suppose we are given a functor
$F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$int_{Atimes B}F = (Atimes B)/F$.
We could also apply this construction pointwise to obtain a functor
$int_A F:B^{op}to operatorname{Cat}$
sending $bmapsto A/F(b)$
and similarly
$int_B F:A^{op}to operatorname{Cat}$
We can apply the Grothendieck construction again to each of these functors to obtain categories
$int_Aint_B F$
and
$int_Bint_A F$
Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?
at.algebraic-topology ct.category-theory grothendieck-construction
edited Feb 9 at 15:46
David White
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
$endgroup$
add a comment |
$begingroup$
Suppose we are given a functor
$F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$int_{Atimes B}F = (Atimes B)/F$.
We could also apply this construction pointwise to obtain a functor
$int_A F:B^{op}to operatorname{Cat}$
sending $bmapsto A/F(b)$
and similarly
$int_B F:A^{op}to operatorname{Cat}$
We can apply the Grothendieck construction again to each of these functors to obtain categories
$int_Aint_B F$
and
$int_Bint_A F$
Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?
at.algebraic-topology ct.category-theory grothendieck-construction
edited Feb 9 at 15:46
David White
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
$endgroup$
Suppose we are given a functor
$F:(Atimes B)^{operatorname{op}}to operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$int_{Atimes B}F = (Atimes B)/F$.
We could also apply this construction pointwise to obtain a functor
$int_A F:B^{op}to operatorname{Cat}$
sending $bmapsto A/F(b)$
and similarly
$int_B F:A^{op}to operatorname{Cat}$
We can apply the Grothendieck construction again to each of these functors to obtain categories
$int_Aint_B F$
and
$int_Bint_A F$
Is it the case that $int_A int_B Fcong int_B int_A Fcong int_{Atimes B} F$?
at.algebraic-topology ct.category-theory grothendieck-construction
at.algebraic-topology ct.category-theory grothendieck-construction
edited Feb 9 at 15:46
David White
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
edited Feb 9 at 15:46
David White
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
edited Feb 9 at 15:46
David White
13.1k462105
edited Feb 9 at 15:46
David White
13.1k462105
edited Feb 9 at 15:46
David White
13.1k462105
13.1k462105
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
asked Feb 9 at 13:50
Harry GindiHarry Gindi
9,720778174
9,720778174
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.
Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
$endgroup$
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.
Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
$endgroup$
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
add a comment |
$begingroup$
Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.
Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
$endgroup$
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
add a comment |
$begingroup$
Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.
Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
$endgroup$
Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.
Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
edited Feb 9 at 15:45
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
answered Feb 9 at 15:19
David WhiteDavid White
13.1k462105
13.1k462105
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
add a comment |
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
$begingroup$
Great, thanks a bunch!
$endgroup$
– Harry Gindi
Feb 9 at 15:27
add a comment |
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